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Yangians of Lie Superalgebras
Lucy Gow
A thesis submitted in fulfillmentof the requirements for
the degree of Doctor of Philosophy
School of Mathematics and StatisticsThe University of Sydney
November 25, 2007
ii
Abstract
This thesis is concerned with extending some well-known results about the Yan-gians Y (glN) and Y (slN) to the case of super-Yangians.
First we produce a new presentation of the Yangian Y (glm|n), using the Gaussdecomposition of a matrix with non-commuting entries. Then, by writing thequantum Berezinian in terms of generators from the new presentation we provethat its coefficients generate the centre Zm|n of Y (glm|n). We show that the YangianY (slm|n) is isomorphic to a subalgebra of the Yangian Y (glm|n), and in particular ifm 6= n, then
Y (glm|n) ∼= Zm|n ⊗ Y (slm|n).
Finally, we show that a Yangian Y (psln|n) associated with the projective speciallinear Lie superalgebra may be obtained from Y (sln|n) by quotienting out the idealgenerated by the coefficients of the quantum Berezinian.
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Acknowledgements
I gratefully acknowledge the help of my supervisor Alex Molev, who provided theoriginal plan for this thesis project and has been helpful and supportive through-out its completion. I also acknowledge the help of my associate supervisor RuibinZhang, who made himself available to explain some mathematics to me on a num-ber of occasions.
The School of Mathematics and Statistics at the University of Sydney pro-vided a friendly community in which to carry out this research. I would partic-ularly like to thank my office-mates James Parkinson and Stephen Ward, as wellas fellow student Ben Wilson, for many interesting mathematical discussions andadvice on the use of LaTeX. David Easdown and Andrew Mathas, as postgraduatecoordinators, also gave very useful advice that helped me to complete this thesis.
Thanks also to Mark Fisher for providing me with a space in his office whileI added the finishing touches in Melbourne, and to my brother Ian who read mydraft and corrected various typographical errors. Finally, I’d like to thank twomathematicians from faraway places, Jon Brundan and Vladimir Stukopin, whokindly explained details of their work to me via email.
This thesis was supported financially by an Australian Postgraduate Award,a supplementary top-up scholarship from the School of Mathematics and Statis-tics, funds from the Postgraduate Student Support Scheme, and additional fundsfor conference expenses from the University of Sydney Algebra Group and AMSI.
I declare this thesis to be wholly my own work, unless stated otherwise. No partof this thesis has been used in the fulfilment of any other degree.
30/06/2007Lucy GowUniversity of Sydney
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vi
Contents
Abstract iii
Acknowledgements v
1 Introduction 1
1.1 Yangians of Classical Lie Algebras . . . . . . . . . . . . . . . . . . . 1
1.2 Yangians of Lie superalgebras . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Summary of Thesis Results . . . . . . . . . . . . . . . . . . . . . . . 4
2 The Yangian of glm|n 7
2.1 Definition of Y (glm|n) . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Hopf Superalgebra Structure . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Relationship with U(glm|n) . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 The Poincare-Birkhoff-Witt Theorem . . . . . . . . . . . . . . . . . . 15
3 The Gauss Decomposition 19
3.1 Quasideterminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 Quasideterminants in the Yangian . . . . . . . . . . . . . . . 20
3.2 Gauss Decomposition in Y (glm|n) . . . . . . . . . . . . . . . . . . . . 20
3.3 Relations between quasideterminants . . . . . . . . . . . . . . . . . 22
3.3.1 Two maps between Yangians . . . . . . . . . . . . . . . . . . 22
vii
3.3.2 Relations in the Yangian Y (glN) . . . . . . . . . . . . . . . . . 26
3.3.3 Relations in the Yangian Y (gl1|1) . . . . . . . . . . . . . . . . 27
3.3.4 Relations in the Yangian Y (gl2|1) . . . . . . . . . . . . . . . . 28
3.3.5 Relations in the Yangian Y (glm|n) . . . . . . . . . . . . . . . . 31
3.4 New Presentation of Y (glm|n) . . . . . . . . . . . . . . . . . . . . . . 32
4 The Centre of Y (glm|n) 39
4.1 The Quantum Berezinian . . . . . . . . . . . . . . . . . . . . . . . . . 39
5 Stukopin’s Presentation of Y (slm|n) 45
5.1 Quantization of Super Lie Bialgebras . . . . . . . . . . . . . . . . . . 45
5.1.1 Super Lie Bialgebras . . . . . . . . . . . . . . . . . . . . . . . 45
5.1.2 Co-Poisson Hopf Superalgebras . . . . . . . . . . . . . . . . 46
5.1.3 The h-adic topology . . . . . . . . . . . . . . . . . . . . . . . 47
5.1.4 Definition of Quantization . . . . . . . . . . . . . . . . . . . . 48
5.2 Stukopin’s Presentation . . . . . . . . . . . . . . . . . . . . . . . . . 49
6 New Presentation of Y (slm|n) 53
6.1 New presentation of Y (slm|n) . . . . . . . . . . . . . . . . . . . . . . 53
6.2 Isomorphism Between the Two Presentations . . . . . . . . . . . . . 55
6.3 The Yangian Y (psln|n) . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6.3.1 The Hopf structure on Y (psln|n) . . . . . . . . . . . . . . . . . 60
7 Conclusion 65
A Superalgebras 67
A.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A.2 The Rule of Signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
A.3 The Lie Superalgebra glm|n . . . . . . . . . . . . . . . . . . . . . . . . 69
viii
A.4 The Lie Superalgebras slm|n and psln|n . . . . . . . . . . . . . . . . . 70
A.5 Root Space Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 71
A.6 Cartan Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.7 The Killing Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A.8 Universal Enveloping Algebras . . . . . . . . . . . . . . . . . . . . . 74
A.9 Casimir Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
A.10 The Symmetric Group Acts on Cm|n ⊗ . . .⊗ Cm|n . . . . . . . . . . . 76
B Proof that φ is a homomorphism 77
ix
x
Chapter 1
Introduction
In this thesis we explore the structure of the Yangians Y (glm|n) and Y (slm|n)associated with the general linear and special linear Lie superalgebras, respec-tively. Our exploration is guided by the well-known theory of Yangians Y (glN)and Y (slN). So we begin with a brief review of this theory.
1.1 Yangians of Classical Lie Algebras
The Yangian of a Lie algebra g is a certain Hopf algebra that contains the uni-versal enveloping algebra U(g). Yangians were defined by Drinfeld [15] in 1985.They have a strong relationship with solutions of the Yang-Baxter equation, and soDrinfeld named them after the physicist C.N. Yang, who found the first solutionof the Yang-Baxter equation of a particular form.
The Yangian of the general linear Lie algebra is defined by an RTT presen-tation, which is also called the FRT formalism after Faddeev, Reshetikhin andTakhtajan [18].
Definition 1.1.1. The Yangian Y (glN) is the associative algebra with generators t(r)ij
where r ≥ 0 and 1 ≤ i, j ≤ N , with t(0)ij = δij , and
[t(r)ij , t
(s)kl ] =
min(r,s)∑t=0
(t(t)kj t
(r+s−t−1)il − t
(r+s−t−1)kj t
(t)il
)for all r, s ≥ 1. (1.1)
If we define the formal power series
tij(u) := δij + t(1)ij u
−1 + t(2)ij u
−2 + . . . ,
1
and the matrix T (u) = (tij(u))Ni,j=1 then we can write the defining relations as an
operator equation
R(u− v)T1(u)T2(v) = T2(v)T1(u)R(u− v),
where R(u − v) is a particular matrix satisfying the Yang-Baxter equation (see[9, 17, 18, 39]). Since the early eighties, the Yangian Y (glN) has been the subject ofintensive study and it is now well understood (see [39]).
There is a distinguished set of elements of Y (glN) that generates its centre ZN .This is the set of coefficients of the quantum determinant qdetT (u) , which is a formalpower series given by the following formula:
qdetT (u) :=∑
σ∈SN
sgn(σ) tσ(1),1(u) tσ(2),2(u− 1) . . . tσ(N),N(u−N + 1)
=∑
σ∈SN
sgn(σ) t1,σ(1)(u−N + 1) t2,σ(2)(u−N + 2) . . . tN,σ(N)(u).
There are also Yangians associated with simple Lie algebras. If we let g beany finite-dimensional simple Lie algebra, then its Yangian Y (g) is the uniquehomogeneous quantization of the Lie bialgebra U(g[u]). Drinfeld [15] gave a pre-sentation for the Yangian Y (g) and showed how its representations give solutionsto the Yang-Baxter equation. This presentation has finitely many generators, butthe defining relations are quite complicated. A new presentation for the Yangianwas given in [16], with a sequence of generators corresponding to each simpleroot of the Lie algebra g. This second realization facilitates a proof of the Poincare-Birkhoff-Witt theorem for Yangians [34], but the defining relations are still com-plicated, and there’s no neat formula for the coproduct in terms of its generators.
Fortunately there is a well-known relationship between the algebras Y (glN)and Y (slN) that allows us to use the results developed for the RTT presentationin the study of the Yangian Y (slN). Namely, Y (slN) is isomorphic to a Hopf subal-gebra of Y (glN). To show how this subalgebra is defined we need to define a classof automorphisms of Y (glN). Suppose that
f = 1 + f1u−1 + f2u
−2 + . . . (1.2)
is a formal power series with complex coefficients. Then the map
µf : Y (glN) → Y (glN)
defined byµf (tij(u)) = f(u)tij(u) for all i, j
2
is an automorphism of Y (glN). We take the set of elements of Y (glN) that are fixedby µf for every f of the form (1.2). Then this set forms a subalgebra isomorphic toY (slN) and
Y (glN) ∼= ZN ⊗ Y (slN),
where ZN is the centre of Y (glN) (see [39]). Then a presentation of Y (slN) maybe obtained from that of Y (glN) by setting the quantum determinant equal to 1.In particular this means that the representation theory of Y (slN) can be deducedfrom the representation theory of Y (glN). Thus our understanding of the Yangiansassociated with simple Lie algebras is founded on our understanding of Y (glN).
1.2 Yangians of Lie superalgebras
Lie superalgebras are a generalization of Lie algebras associated to the categoryof Z2-graded vector spaces. They have attracted a lot of attention from physicistssince the 1970s because of their ability to describe symmetries of systems involv-ing both bosons and fermions, as well as other things. A description of the basicproperties of Lie superalgebras used in this thesis is given in Appendix A.
In this thesis we consider analogues of the general and special linear Lie al-gebras. These are the general linear Lie superalgebra glm|n, special linear Lie su-peralgebra slm|n, and projective special linear Lie superalgebra psln|n. The generallinear Lie superalgebra glm|n is defined similarly to the general linear Lie algebra(see A.3) except that it relates to Z2-graded vector spaces or superspaces.
For linear operators on superspaces, the role of the trace is fulfilled by a lin-ear functional called the supertrace. The Lie superalgebra slm|n is defined as thesubalgebra of glm|n consisting of those elements with zero supertrace. The casewhere m = n is peculiar because the identity matrix in gln|n has zero supertraceand so the Lie superalgebra sln|n is not simple. This leads to the definition of theprojective special linear Lie superalgebra psln|n as the quotient of sln|n by its centre(see A.4).
Furthermore, instead of the determinant we have a superdeterminant calledthe Berezinian. For a matrix element of glm|n written in block form as
X =
(A BC D
),
whereA ism×m,B ism×n, C is n×m andD is n×n and invertible, its Berezinianis defined as:
Ber(X) = det(A−BD−1C)det(D−1) (1.3)
3
This is named after F.A. Berezin, who defined the Berezinian and studied its prop-erties [4].
In 1991, Max Nazarov [40] defined the Yangian Y (glm|n) associated with thegeneral linear Lie superalgebra using a super-analogue of the RTT -relation. Inthis work, he gave two formal power series with central coefficients - the quan-tum Berezinian (which is a generalization of the quantum determinant) and an-other power series called the quantum contraction. By demonstrating a relation-ship between these two series, Nazarov defined a quantum superanalogue of theLiouville theorem [3] and a super-analogue of the Capelli identity [27]. He thusestablished immediately that the study of super-Yangians could give insights intothe algebra U(glm|n). Nazarov also conjectured that the centre of Y (glm|n) wasgenerated by the coefficients of the quantum Berezinian.
Shortly after this, Vladimir Stukopin [45] defined a Yangian for the classicalLie superalgebra Y (slm|n) with m 6= n, by deforming the canonical super Lie bial-gebra structure of the algebra of polynomials with coefficients in slm|n. He gavepresentations analogous to both of those given by Drinfeld for the Yangians ofclassical simple Lie algebras. This work was followed in 2004 by a description ofYangians of all the classical Lie superalgebras [46]. It also received the attentionof V.N. Tolstoy [48] who gave another presentation of Y (slm|n) and related it to acertain two-parameter quantization of U(slm|n[x]) called the super-Drinfeldian.
Ruibin Zhang classified all the finite-dimensional representations of Y (glm|n)in [51, 52]. This work was taken as the main point of reference for the study of sys-tems with super-Yangian symmetry in physics. Many papers have appeared onapplications of super-Yangians to the study of quantum integrable systems [35].They have been applied to Calogero-Sutherland models [1, 26, 30, 47], formula-tions of the super-Schrodinger equation [8], and super Yang-Mills field theory[5, 13, 14, 36, 37, 50, 53]. Finding a super-Yangian symmetry for these modelsdemonstrates that they are integrable. In addition, the Yangian Y (gl1|1) and itsapplications have received detailed study in [7, 29].
Given the many applications of the super-Yangian Y (glm|n) in mathematicalphysics, it seems that a careful study of this Yangian from an algebraic perspectivemay be valuable. The goal of this thesis is to provide such a study.
1.3 Summary of Thesis Results
The main result of this thesis is a new presentation of the Yangian Y (glm|n) interms of quasideterminants (see Chapter 3). Quasideterminants [21, 20, 22] arealgebraic objects that provide convenient methods for dealing with matrices over
4
non-commutative rings. In particular, given such a matrix satisfying a few furtherproperties, one has its Gauss decomposition - a decomposition of the matrix into theproduct of an upper unitriangular, a diagonal and a lower unitriangular matrix.
Given an algebra like the Yangian, for which the defining relations are ex-pressed in terms of a matrix T (u), one may create new presentations by using theGauss decomposition of T (u). A clear exposition of this method as applied tothe Yangian Y (glN) is given by Brundan and Kleshchev in [6] where they demon-strate the relationship between this Yangian and Y (slN) mentioned in Section 1.1.We mimick their approach in the case of the the Yangian Y (glm|n).
In Chapter 4, we use the new presentation to prove Nazarov’s conjecture de-scribing the centre of Y (glm|n). The new presentation also allows us to prove thatthe Yangian Y (slm|n) defined by Stukopin is isomorphic to a Hopf subalgebra ofthe Yangian Y (glm|n) defined by Nazarov (in Chapter 6). In particular we find form 6= n that:
Y (glm|n) ∼= Zm|n ⊗ Y (slm|n),
where Zm|n is the centre of Y (glm|n). These results have appeared in [23] and [24].
We define the Yangian Y (psln|n) as the quotient of Y (sln|n) by the ideal Bgenerated by the coefficients of the quantum Berezinian. Then Y (psln|n) has triv-ial centre. Also since the ideal B is a Hopf ideal, the Hopf algebra structure onY (gln|n) induces a Hopf algebra structure on Y (psln|n). We show that the YangianY (psln|n) is then an homogeneous quantization of U(psln|n[w]).
5
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Chapter 2
The Yangian of glm|n
The Yangian of the general linear Lie superalgebra was defined by Nazarov [40].There are various ways of expressing the defining relations - in terms of the gen-erators explicitly, using formal power series, or using a matrix relation. In the firstsection of this chapter we explain how these are equivalent. In the second sectionwe show that the Yangian Y (glm|n) is a Hopf algebra. In Section 2.3 we definetwo different filtrations on the Yangian and in Section 2.5 we show that the as-sociated graded algebra is isomorphic to U(glm|n[x]). This will prepare us for thedevelopment of new results in subsequent chapters.
2.1 Definition of Y (glm|n)
The Yangian Y (glm|n) is defined to be the Z2-graded associative algebra over Cwith generators t(k)
ij and relations given by (2.2) below. We define the formal powerseries
tij(u) = δij + t(1)ij u
−1 + t(2)ij u
−2 + . . . ,
and a matrix
T (u) =m+n∑i,j=1
tij(u)⊗ Eij (−1)j(i+1), (2.1)
where Eij is the standard elementary matrix. (The sign is necessary in order toensure that the product of two matrices can still be calculated in the usual way).In analogy with the usual Yangian Y (gln), the defining relations are then expressedby the matrix product
R(u− v)T1(u)T2(v) = T2(v)T1(u)R(u− v) (2.2)
7
where
R(u− v) = 1− 1
(u− v)P12
with
P12 =m+n∑i,j=1
Eij ⊗ Eji(−1)j.
P12 is called the permutation matrix. It is the operator P(1,2) referred to in Ap-pendix A.10.
Proposition 2.1.1. We have three further equivalent forms of the defining relations:
[tij(u), tkl(v)] =(−1)ij+ik+jk
(u− v)(tkj(u)til(v)− tkj(v)til(u)), (2.3)
[t(r+1)ij , t
(s)kl ]− [t
(r)ij , t
(s+1)kl ] = (−1)ij+ik+jk (t
(r)kj t
(s)il − t
(s)kj t
(r)il ), (2.4)
and
[t(r)ij , t
(s)kl ] = (−1)ij+ik+jk
min(r,s)−1∑p=0
(t(p)kj t
(r+s−1−p)il − t
(r+s−1−p)kj t
(p)il ). (2.5)
Proof. To get (2.3) from (2.2) we simply apply separately the left and right handsides of (2.2) to ej ⊗ el and equate coefficients of ei ⊗ ek. On the left hand side wehave:
R(u− v)T1(u)T2(v) ej ⊗ el
= R(u− v)T1(u)∑
c
tcl(v)⊗ ej ⊗ ec (−1)l(c+1)+(c+l)j
= R(u− v)∑a,c
taj(u)tcl(v)⊗ ea ⊗ ec (−1)j(a+1)+l(c+1)+a(c+l)
=∑a,c
taj(u)tcl(v)⊗ ea ⊗ ec (−1)j(a+1)+l(c+1)+a(c+l)
− 1
(u− v)
∑a,c
taj(u)tcl(v)⊗ ec ⊗ ea (−1)j(a+1)+l(c+1)+a(c+l)+ac,
which gives as coefficient of ei ⊗ ek:
tij(u)tkl(v) (−1)j(i+1)+l(k+1)+i(k+l)
− 1(u−v)
tkj(u)til(v) (−1)j(k+1)+l(i+1)+k(i+l)+ik.
8
On the right hand side we have:
T2(v)T1(u)R(u− v) ej ⊗ el
= T2(v)T1(u)(ej ⊗ el −1
(u− v)el ⊗ ej (−1)jl
= T2(v)∑
a
taj(u)⊗ ea ⊗ el (−1)j(a+1)
− 1
(u− v)T2(v)
∑a
tal(u)⊗ ea ⊗ ej (−1)l(a+1)+jl
=∑a,c
tcl(v)taj(u)⊗ ea ⊗ ec (−1)j(a+1)+l(c+1)+j(c+l)
− 1
(u− v)
∑a,c
tcj(v)tal(u)⊗ ea ⊗ ec (−1)l(a+1)+j(c+1)+j l+l(j+c)
which gives as coefficient of ei ⊗ ek:
tkl(v)tij(u) (−1)j(i+1)+l(k+1)+j(k+l)
− 1(u−v)
tkj(v)til(u) (−1)l(i+1)+j(k+1)+j l+l(j+k).
Equating these gives the result (2.3). Now, using this result:
(u− v)[tij(u), tkl(v)] = (−1)ij+ik+jk(tkj(u)til(v)− tkj(v)til(u))
= (−1)ij+ik+jk(∞∑r,s
(t(r)kj t
(s)il − t
(s)kj t
(r)il )u−rv−s).
But this can be calculated directly as:
(u− v)[tij(u), tkl(v)] =∞∑
r,s=1
u[t(r)ij u−r, t(s)kl v−s]−
∞∑r,s=1
v[t(r)ij u−r, t(s)kl v−s)]
=∞∑
r,s=1
u−r+1v−s[t(r)ij , t(s)kl ]−
∞∑r,s=1
u−rv−s+1[t(r)ij , t(s)kl ]
=∞∑
r,s=1
u−rv−s([t(r+1)ij , t
(s)kl ]− [t(r)ij , t
(s+1)kl ])
and thus we arrive at (2.4). Finally, the set of equations (2.5) is obtained by addingtogether various applications of (2.4), as in §1.1.2 of [38].
Lemma 2.1.2. We also have the following relationship between entries of the matrix T (u)
and the entries of the inverse matrix T (v)−1 =(t′ij)m+n
i,j=1.
[tij(u), t′kl(v)] =
(−1)ij+ik+jk
(u− v)· ( δkj
∑s
tis(u)t′sl(v)− δil
∑s
t′ks(v)tsj(u) ). (2.6)
9
Proof. We rearrange the defining relations (2.2) into
T−12 (v)R(u− v)T1(u) = T1(u)R(u− v)T−1
2 (v). (2.7)
Applying the left hand side to ej ⊗ el gives:
T−12 (v)R(u− v)T1(u) ej ⊗ el
= T−12 (v)R(u− v)
∑a
taj(u)⊗ ea ⊗ el (−1)j(a+1)
= T−12 (v)
∑a
taj(u)⊗ ea ⊗ el (−1)j(a+1)
− 1(u− v)
T−12 (v)
∑a
taj(u)⊗ el ⊗ ea (−1)j(a+1)+al
=∑a,c
t′cl(v)taj(u)⊗ ea ⊗ ec (−1)j(a+1)+l(c+1)+j(c+l)
− 1(u− v)
∑a,c
t′ca(v)taj(u)⊗ el ⊗ ec (−1)j(a+1)+al+a(c+1)+(a+c)(a+j+l)
So that the coefficient of ei ⊗ ek is
t′kl(v)tij(u) (−1)(j+l)(k+l)+j(i+1)
− δi,l(u− v)
∑s
t′ks(v)tsj(u)(−1)j+j k+k l. (2.8)
Applying the right hand side of (2.7) to ej ⊗ el gives:
T1(u)R(u− v)T−12 (v) ej ⊗ el
= T1(u)R(u− v)∑
c
t′cl(v)⊗ ej ⊗ ec (−1)l(c+1)+j(c+l)
= T1(u)∑
c
t′cl(v)⊗ ej ⊗ ec (−1)l(c+1)+j(c+l)
− 1(u− v)
T1(u)∑
c
t′cl(v)⊗ ec ⊗ ej (−1)l(c+1)+j(c+l)+jc
=∑a,c
taj(u)t′cl(v)⊗ ea ⊗ ec (−1)l(c+1)+j(c+l)+j(a+1)+(a+j)(c+l)
− 1(u− v)
∑a,c
tac(u)t′cl(v)⊗ ea ⊗ ej (−1)l(c+1)+j(c+l)+jc+c(a+1)+(a+c)(c+l),
so that the coefficient of ei ⊗ ek is
tij(u)t′kl(v) (−1)(i+l)(k+l)+j(i+1)
− δkj
(u− v)
∑s
tis(u)t′sl(v) (−1)l+jl+il. (2.9)
Now equating (2.8) and (2.9) gives the result.
10
2.2 Hopf Superalgebra Structure
The definition of a Hopf superalgebra over a commutative ring k is that same asthat of a Hopf algebra [9, 33, 41] except that for a Hopf superalgebra we requirethat the underlying k-module is Z2-graded, and that the relevant maps are even.(We consider k to be Z2-graded in the trivial way where every element is even.)
Definition 2.2.1. A Hopf superalgebra over k is a Z2-graded k-module A = A0⊕A1,with:
a multiplication, m : A⊗ A→ A,
a comultiplication, ∆ : A→ A⊗ A,
a unit, ι : k → A,
a counit, ε : A→ k,
an antipode, S : A→ A,
such that all of these maps are even linear maps, and they satisfy the followingconditions (expressed in commuting diagrams):
1. m is associative:
A⊗ A⊗ A
A⊗ A
A⊗ A
A
m⊗id<<xxxxxxx
id⊗m ##FFFF
FFF
m
!!CCCC
CCC
m
<<yyyyyyy
.
2. ∆ is coassociative:
A⊗ A⊗ A
A⊗ A
A⊗ A
A||
∆⊗id xxxxxxx
cc
id⊗∆ FFFF
FFF
aa∆
CCCC
CCC
|| ∆
yyyyyyy
.
3. we have the unit condition:
A
A⊗ C
A
A⊗ A
A
C⊗ A
A
A⊗ AOO∼=
OO
m
id //
id⊗ι //OO
∼=
OO
m
ι⊗id //
id //
.
4. we have the counit condition:
A
A⊗ C
A
A⊗ A
A
C⊗ A
A
A⊗ A
∼=
∆
oo id
oo id⊗ ε
∼=
∆
oo ε⊗id
oo id
.
11
5. ∆ is an algebra homomorphism,
A⊗ A A A⊗ A
A⊗ A⊗ A⊗ A A⊗ A⊗ A⊗ A
∆⊗∆
OOid⊗τ⊗id //
m⊗m
∆ //m // .
6. ε is an algebra homomorphism,
k ⊗ k
A⊗ A
k
A
ε⊗ ε
ε
∼= //
m //
.
7. we have the antipode condition:
A⊗ A A A⊗ A
k
A⊗ A A A⊗ A
id⊗S
ε
ι
S⊗id
oo ∆ ∆ //
m// oo
m .
The following proposition is well known.
Proposition 2.2.1. The Yangian Y (glm|n) is a Hopf superalgebra with comultiplication
∆ : tij(u) 7→m+n∑k=1
tik(u)⊗ tkj(u), (2.10)
antipodeS : T (u) 7→ T−1(u)
and counit ε : T (u) 7→ 1.
Proof. Let p, q be positive integers and consider the algebra
W :=(Y (glm|n)[[u−1]]
)⊗p ⊗(EndCm|n)⊗q
.
Set:
T[b]a(u) :=m+n∑i,j=1
(1⊗(b−1) ⊗ tij(u)⊗ 1⊗(p−b))⊗ (1⊗(a−1) ⊗ Eij ⊗ 1⊗(q−a))(−1)j(i+1) ∈ W .
12
Then the coproduct ∆ is given by:
∆ : T (u) 7→ T[1]1(u)T[2]1(u). (2.11)
Indeed, if we consider the action of T[1]1(u)T[2]1(u) on an element 1 ⊗ 1 ⊗ ej ∈Y (glm|n)⊗2 ⊗ Cm|n, we find
T[1]1(u)T[2]1(u) · 1⊗ 1⊗ ej
= T[1]1(u)
(1⊗
∑k
tkj(u)⊗ Ekj(−1)j(k+1)
)· 1⊗ 1⊗ ej
=
(∑i
tik(u)⊗ 1⊗ Eik(−1)k(i+1)
)· 1⊗
∑k
tkj(u)⊗ ek(−1)j(k+1)
=∑i,k
tik(u)⊗ tkj(u)⊗ ei (−1)j(k+1)+k(i+1)+(i+k)(k+j)
=∑i,k
tik(u)⊗ tkj(u)⊗ ei (−1)j(i+1).
We need to show that the operator T[1]1(u)T[2]1(u) satisfies the relation (2.2). Bya simple calculation we show that T[2]1(u) and T[1]2(v) commute, and also T[1]1(u)and T[2]2(v) commute. Then:
R(u− v)T[1]1(u)T[2]1(u)T[1]2(v)T[2]2(v)
= R(u− v)T[1]1(u)T[1]2(v)T[2]1(u)T[2]2(v)
= T[1]2(v)T[1]1(u)R(u− v)T[2]1(u)T[2]2(v) by (2.2)= T[1]2(v)T[1]1(u)T[2]2(v)T[2]1(u)R(u− v) by (2.2)= T[1]2(v)T[2]2(v)T[1]1(u)T[2]1(u)R(u− v).
2.3 Filtrations
We may define two different filtrations on the Yangian Y (glm|n). These are definedby setting the degree of a generator as follows:
deg1(t(r)ij ) = r; deg2(t
(r)ij ) = r − 1.
Let gr1Y (glm|n) and gr2Y (glm|n), respectively, denote the corresponding gradedalgebras.
13
2.4 Relationship with U(glm|n)
The following observation suggests the possibility of applications for our study ofthe Yangian Y (glm|n).
Proposition 2.4.1. The mapping
π : tij(u) 7→ δij + Eij(−1)iu−1 (2.12)
defines an algebra onto homomorphism Y (glm|n) → U(glm|n). In addition,
e : Eij 7→ t(1)ij (−1)i (2.13)
is a one-to-one algebra homomorphism U(glm|n) → Y (glm|n).
Proof. To show π is an algebra homomorphism, we need to verify that it respectsthe relation (2.3). In other words, we need to show that
(u− v)[δij + (−1)iEij u−1, δkl + (−1)kEkl v
−1]
= (−1)i j+i k+j k
(δkj + Ekj(−1)ku−1)(δil + Eil(−1)iv−1)
− (δkj + Ekj(−1)kv−1)(δil + Eil(−1)iu−1).
This is clear, by the commutation relations in glm|n. The map π is clearly onto,because it maps the elements t(1)
ij of the Yangian to the generators Eij(−1)i of theuniversal enveloping algebra.
In order to prove the second part of the proposition, multiply (2.3) by
(u− v)−1 =∞∑
p=0
u−p−1vp
and equate coefficients of u−1v−1 to give
[t(1)ij , t
(1)kl ] = (−1)i j+j k+i k(δkjt
(1)il − δilt
(1)kj ),
which in turn gives
[t(1)ij (−1)i, t
(1)kl (−1)k] = δkj t
(1)il (−1)i − (−1)(i+j)(k+l)δil t
(1)kj (−1)k,
and thus shows that e is a homomorphism. That it is a one-to-one map followsfrom the fact that π e is the identity map on U(glm|n).
14
2.5 The Poincare-Birkhoff-Witt Theorem
We now prove the Poincare-Birkhoff-Witt theorem for the Yangian Y (glm|n). Theproof is based very closely on that of the corresponding theorem for Y (glN) givenin [6].
For each positive integer l ≥ 1, we define a homomorphism
κl := (π ⊗ · · · ⊗ π) ∆(l) : Y (glm|n) → U(glm|n)⊗l,
where ∆(l) : Y (glm|n) → Y (glm|n)⊗l is the coproduct iterated (l − 1) times and π isthe map given in (2.12). Then
κl(t(r)ij ) =
∑1≤s1<...<sr≤l
∑1≤i1,...,ir−1≤m+n
E[s1]ii1E
[s2]i1i2
· · ·E[sr]ir−1j(−1)i + i1 + i2 + ...+ir−1
where E[s]ij = 1⊗(s−1) ⊗ Eij ⊗ 1⊗(l−s). For any r > l ≥ 1, we have κl(t
(r)ij ) = 0.
Theorem 2.5.1. Suppose we have fixed some ordering on the generators t(r)ij (1 ≤ i, j ≤m + n; r ≥ 1) for the Yangian Y (glm|n). Then the ordered products of these, containingno second or higher order powers of the odd generators, form a basis for Y (glm|n).
Proof. 1 By relation (2.5), the graded algebra gr1Y (glm|n) is supercommutative, andthus the set of all ordered monomials in the generators t(r)ij (with no second orhigher order powers of the odd generators) span the Yangian Y (glm|n). It remainsto show that they are linearly independent. We show that, for every l ≥ 1, thecorresponding monomials in κl(t
(r)ij ) | 1 ≤ r ≤ l are linearly independent in
κl(Y (glm|n)). Consider the filtration
F0U(glm|n)⊗l ⊆ F1U(glm|n)⊗l ⊆ F2U(glm|n)⊗l ⊆ . . .
on U(glm|n)⊗l defined by setting each generator E[r]ij to be of degree 1. Then the
associated graded algebra grU(glm|n)⊗l is the polynomial algebra on supersym-metric generators
x[r]ij := gr1E
[r]ij ,
where x[r]ij is even if i+j = 0 and odd if i+j = 1. The map κl preserves the filtration
on the Yangian given by setting deg1(t(r)ij ) = r, and thus defines a homomorphism
between the corresponding graded algebras. It is enough to show that the samemonomials in the elements y(r)
ij := grrκl(t(r)ij ) in the graded algebra are linearly
1This theorem was stated in [52] but the proof there is incomplete.
15
independent. But for this, it is enough to show that the superderivatives dy(r)ij are
linearly independent at a point. We have:
y(r)ij =
∑1≤s1<...<sr<l
∑1≤i1,...,ir−1≤n
x[s1]ii1x
[s2]i1i2
· · ·x[sr]ir−1j(−1)i +i1+...+ir−1 .
We will show that the matrix dφ corresponding to the map(dx
[s]ij
)7→(dy
(r)ij
)has
non-zero determinant at a point. It suffices to show that the determinant of thismatrix is nonzero even when the variables are specialized to x(s)
kl = δklcs(−1)k forsome distinct cs (s ≥ 1). When the variables are specialized as described, we find:
dy(r)ij =
l∑s=1
∑1≤s1<...<sr−1≤l
si 6=s
cs1cs2 · · · csr−1(−1)idx[s]ij .
Let J be the (m+ n)× (m× n) matrix J =(δij(−1)i
). Then dφ = J ⊗Xl, where
Xl =
1 1 . . . 1
(c2 + c3 + . . .+ cl) (c1 + c3 + . . .+ cl) . . . (c1 + c2 + . . .+ cl−1)(∑
i,j 6=1 cicj) (∑
i,j 6=2 cicj) . . . (∑
i,j 6=l cicj)...
...c2c3 · · · cl c1c3c4 · · · cl . . . c2c3 · · · cl−1
.
We show by induction that det(Xl) = Π1≤i<j≤l(ci − cj) 6= 0, and hence det dφ 6= 0.Indeed, row-reducing Xl gives the following matrix:
1 . . . 1 1(cl − c1) . . . (cl − cl−1) 0
(cl − c1)∑
i,j 6=1i,j<l
cicj . . . (cl − cl−1)∑
i,j 6=l−1i,j<l
cicj 0
......
(cl − c1)c2c3 · · · cl−1 . . . (cl − cl−1)c1 · · · cl−2 0
,
which clearly has determinant (c1 − cl)(c2 − cl) · · · (cl−1 − cl)det(Xl−1).
Now suppose we have some non-trivial linear combination P of the orderedmonomials in t
(r)ij (with no second or higher order powers of the odd generators)
and take l to be any number greater than all the r that occur in P . Since the mono-mials in κl(t
(r)ij ) are linearly independent in κl(Y (glm|n)), we must have κl(P ) 6= 0.
Therefore, P 6= 0 in the Yangian.
Let glm|n[w] denote the algebra glm|n ⊗ C[w] with basis Eijwr1≤i,j≤m+n; r≥0.
16
Corollary 2.5.1. The graded algebra gr2Y (glm|n) is isomorphic to the universal envelop-ing algebra U(glm|n[w]), via the map
gr2Y (glm|n) → U(glm|n[w])
grr−12 t
(r)ij 7→ Eijw
r−1(−1)i (1 ≤ i, j ≤ m+ n, r ≥ 1).
Proof. To check that this map defines a homomorphism, we verify that it preservesthe defining relations (2.5). The graded version of this is:
[grr−12 t
(r)ij , grs−1
2 t(s)kl ] = (−1)i j+j k+i k(δkjgrr+s−2
2 t(r+s−1)il )− grr+s−2
2 t(r+s−1)kj δil.
The image of the left hand side of this equation under our map is:
[Eijwr−1, Eklw
s−1](−1)i+k = δkjEil wr+s−2(−1)i j+i k+j k+i−δilEkj wr+s−2(−1)i j+i k+j k+k,
which is clearly equal to the image of the right hand side. It is clear that this mapis surjective. Injectivity follows from the theorem.
17
18
Chapter 3
The Gauss Decomposition
We analyse the Yangian Y (glm|n) by considering a decomposition of the matrixT (u). We make use of the Gauss decomposition of matrices with non-commutingentries, as given in [22]. This is not a new technique. It has been applied to sev-eral different quantum groups including the Yangian Y (glN) in [6, 10], the super-Yangian Y (gl1|1) in [7] and other quantum groups [11].
3.1 Quasideterminants
Israel Gelfand and Vladimir Retakh developed the theory of quasideterminants tostudy matrices over non-commutative rings. Quasideterminants share some prop-erties with the usual determinants of matrices over commutative rings and allowus to manipulate the inverse matrix of arbitrary matrices. There is a rich theoryfor these objects, which is developed in [22]. We review only the definitions andresults that are needed for this thesis.
Definition 3.1.1. Let X be a square matrix over a ring with identity such that itsinverse matrix X−1 exists, and such that its (j, i)th entry is an invertible elementof the ring. Then the (i, j)th quasideterminant of X is defined by the formula
|X|ij =((X−1)ji
)−1.
We also use the following notation:
|X|ij =:
∣∣∣∣∣∣∣∣∣∣
x11 · · · x1j · · · x1n
· · · · · ·xi1 · · · xij · · · xin
· · · · · ·xn1 · · · xnj · · · xnn
∣∣∣∣∣∣∣∣∣∣.
19
Thus a matrix X has not just one quasideterminant, but rather anywhere up to n2
quasideterminants. These may be calculated using the following formula, so longas all the relevant expressions exist:
|X|ij = xij −∑
k 6=j, l 6=i
xik(|X ij|lk)−1xlj, (3.1)
where we denote by |X ij| the matrix obtained by deleting the ith row and jthcolumn fromX . It very often happens that this expression does not make sense, orthat it is not invertible. In such cases we will find it necessary to use the followingspecial cases of the homological relations [22] for quasideterminants.
Proposition 3.1.1 ([22]). The quasideterminants satisfy the following relationships:
|A|−1ij = − |Aii|−1
jj |Aij|ji|A|−1ii , (3.2)
|A|−1ij = − |A|−1
jj |Aij|ji|Ajj|−1ii (3.3)
(whenever the relevant expressions exist).
3.1.1 Quasideterminants in the Yangian
The matrix T (u) and all of its principal submatrices (those obtained by deletinga certain number of rows and columns from the bottom and from the right) areinvertible. To see this, note that they may be viewed as power series with coeffi-cients being the matrices with entries in the Yangian. For example,
T (u) = I + u−1(t(1)ij
)m+n
i,j=1+ u−2
(t(2)ij
)m+n
i,j=1+ . . .
This power series is clearly invertible since its constant term is the identity. Thuswe see that the inverse of any quasideterminant |T (u)|−1
ij exists. In the case wherei = j, we may, without any trouble, use the formula (3.1) to find expressions forboth the quasideterminant |T (u)|ii and its inverse t′ii(u). However, for i 6= j, thisexpression is not invertible in the algebra of formal power series Y (glm|n)[[u−1]]and so will not enable us to find the entries of the inverse matrix T (u)−1. Then wemust resort to the use of Proposition 3.1.1.
3.2 Gauss Decomposition in Y (glm|n)
We shall now apply a result about quasideterminants [22] that is the main ideabehind the results of this thesis. The result states that if we are given any square
20
matrix, such that the quasideterminants corresponding to the bottom right handcorner of any of its principal submatrices are defined and invertible, then we maydecompose the matrix as a product of a lower unitriangular matrix, a diagonalmatrix, and an upper unitriangular matrix. For the matrix T (u) in the YangianY (glm|n) this result reads as follows.
Proposition 3.2.1 ([22]). The matrix T (u) defined in (2.1) has the following decomposi-tion in terms of quasideterminants:
T (u) = F (u)D(u)E(u)
where
D(u) =
d1(u) · · · 0
d2(u)...
.... . .
0 · · · dm+n(u)
,
F (u) =
1 · · · 0
f21(u). . .
......
. . .fm+n,1(u) fm+n,2(u) · · · 1
, E(u) =
1 e12(u) · · · e1,m+n(u)
. . . e2,m+n(u). . .
...0 1
,
are unique matrices with entries given by the following formulae:
di(u) =
∣∣∣∣∣∣∣t11(u) · · · t1,i−1(u) t1i(u)
... . . . ...ti1(u) · · · ti,i−1(u) tii(u)
∣∣∣∣∣∣∣ ,
eij(u) = di(u)−1
∣∣∣∣∣∣∣∣∣t11(u) · · · t1,i−1(u) t1j(u)
... . . . ......
ti−1,i(u) · · · ti−1,i−1(u) ti−1,j(u)
ti1(u) · · · ti,i−1(u) tij(u)
∣∣∣∣∣∣∣∣∣ ,
fji(u) =
∣∣∣∣∣∣∣∣∣t11(u) · · · t1,i−1(u) t1i(u)
... . . . ......
ti−1,1(u) · · · ti−1,i−1(u) ti−1,i(u)
tji(u) · · · tj,i−1(u) tji(u)
∣∣∣∣∣∣∣∣∣ di(u)−1.
This decomposition is called the Gauss decomposition of the matrix T (u). Weuse the following notation for the coefficients of the quasideterminants.
di(u) =∑r≥0
d(r)i u−r; (di(u))
−1 =∑r≥0
d′(r)i u−r; (3.4)
eij(v) =∑r≥1
e(r)ij v
−r; fji(v) =∑r≥1
f(r)ji v
−r. (3.5)
21
It is easy to recover each generating series tij(u) by multiplying together andtaking commutators of the series di(u), ej(u) := ej,j+1(u), and fj(v) := fj+1,j(u)for 1 ≤ i ≤ m + n, 1 ≤ j ≤ m + n − 1. Indeed, for each pair i, j such that1 < i+ 1 < j ≤ m+ n− 1, we have:
e(r)ij = (−1)j−1[e
(r)i,j−1, e
(1)j−1]; f
(r)ji = (−1)j−1[f
(1)j−1, f
(r)i,j−1]. (3.6)
Thus the Yangian Y (glm|n) is generated by the coefficients of the series
di(u), ej(u), fj(u) | 1 ≤ i ≤ m+ n; 1 ≤ j ≤ m+ n− 1 .
The main goal of this chapter is to describe a presentation for the Yangian Y (glm|n)using the coefficients of the quasideterminants (3.4) as generators.
3.3 Relations between quasideterminants
3.3.1 Two maps between Yangians
For Yangians Y (glm|n) with small m and n, such as Y (gl1|1) and Y (gl2|1), it is fea-sible to use this matrix relationship T (u) = F (u)D(u)E(u) directly to translate thedefining relations (2.3) into relations between the generating series di(u), ej(u) andfj(u). However, in order to transfer these results to the general case of Y (glm|n) wemust define various homomorphisms between Yangians.
Lemma 3.3.1. The map ρm|n : Y (glm|n) → Y (gln|m) defined by
ρm|n(tij(u)) = tm+n+1−i,m+n+1−j(−u).
is an associative algebra isomorphism.
Note where we have swapped m and n in the above. We use the same sym-bols for the generators of both Y (glm|n) and Y (gln|m). It should be clear from thecontext to which algebra tij(u) belongs.
Proof. We check that the map ρm|n preserves the defining relation (2.3). In order tomake this clearer, we will write p(i) for the parity function on Y (glm|n), so that
p(i) =
0 if 1 ≤ i ≤ m1 if m+ 1 ≤ i ≤ m+ n
22
We also write i = m+ n+ 1− i. Then,
[ρ(tij(u)), ρ(tkl(v))] = [ti,j(−u), tk,l(−v)]
=(−1)(p(i)+1)(p(j)+1)+(p(j)+1)(p(k)+1)+(p(i)+1)(p(k)+1)
(−u + v)(tk,j(−u)ti,l(−v)− tk,j(−v)ti,l(−u))
=
((−1)p(i)p(j)+p(j)p(k)+p(i)p(k)
(u− v)(ρ(tkj(u))ρ(til(v))− ρ(tkj(v))ρ(til(u)))
)
as required.
This map ρm|n is not interesting to us at all, except that it allows us to definethe following map ζm|n.
Proposition 3.3.2. Let ζm|n : Y (glm|n) → Y (gln|m) be the associative algebra isomor-phism given by ζm|n = ρm|n ωm|n, where ωm|n is the Y (glm|n) automorphism given by
ωm|n : T (u) 7→ T (−u)−1.
That is,ζm|n : tij(u) 7→ t′m+n+1−i,m+n+1−j(u).
Then:
ζm|n :
di(u) 7→ (dm+n−i+1(u))−1 ,
ek(u) 7→ −fm+n−k(u),fk(u) 7→ −em+n−k(u),
(3.7)
for 1 ≤ i ≤ m+ n and 1 ≤ k ≤ m+ n− 1.
Proof. We multiply out the matrix products
T (u) = F (u)D(u)E(u)
andT (u)−1 = E(u)−1D(u)−1F (u)−1.
These show that for all 1 ≤ i < j ≤ m+ n,
tii(u) = di(u) +∑k<i
fik(u)dk(u)eki(u),
tij(u) = di(u)eij(u) +∑k<i
fik(u)dk(u)ekj(u), (3.8)
tji(u) = fji(u)di(u) +∑k<i
fjk(u)dk(u)eki(u),
23
and
t′ii(u) = di(u)−1 +
∑k>i
e′ik(u)dk(u)−1f ′ki(u),
t′ij(u) = e′ij(u)dj(u)−1 +
∑k>j
e′ik(u)dk(u)−1f ′kj(u), (3.9)
t′ji(u) = dj(u)−1f ′ji(u) +
∑k>j
e′jk(u)dk(u)−1f ′ki(u),
wheree′ij(u) =
∑i=i0<i1<...<is=j
(−1)sei0i1(u)ei1i2(u) · · · eis−1is(u)
andf ′ji(u) =
∑i=i0<i1<...<is=j
(−1)sfisis−1(u) · · · fi2i1(u)fi1i0(u).
Then immediately we have
ζm|n(d1(u)) = dm+n,m+n(u)−1,
ζm|n(e1j(u)) = f ′m+n,m+n+1−j(u),
ζm|n(fj1(u)) = e′m+n+1−j,m+n(u).
By induction on i, we derive:
ζm|n(di(u)) = (dm+n+1−i(u))−1 ,
ζm|n(eij(u)) = f ′m+n+1−i,m+n+1−j(u),
ζm|n(fji(u)) = e′m+n+1−j,m+n+1−i(u).
The result stated in the proposition is the special case of this where j = i+ 1.
When it is reasonable we will write simply ζ for the map ζm|n. The map ζm|nrestricts to the isomorphism U(glm|n) → U(gln|m) defined by
Eij 7→ Em+n+1−i,m+n+1−j.
It can be calculated explicitly (using induction and basic properties of quasideter-minants) for any 1 ≤ i, j ≤ m+ n to give the following result:
ζ(t(r)m+n+1−i,m+n+1−j) =
∑r1+...+rp=rr1,...,rp>0
(−1)p
m+n∑k1,...,kp−1=1
t(r1)ik1t(r2)k1k2
. . . t(rp−1)kp−1j .
Also, ζ is not a Hopf algebra map between the two Yangians, but instead has thefollowing property.
24
Proposition 3.3.3. Let τ : Y (gln|m)⊗Y (gln|m) → Y (gln|m)⊗Y (gln|m) be the map givenby
τ(y1 ⊗ y2) = y2 ⊗ y1 (−1)y1y2
for all homogeneous elements y1, y2 ∈ Y (gln|m). Then:
(ζ ⊗ ζ) ∆ = τ ∆ ζ.
Proof. Recall that∆ : T (u) 7→ T[1](u)T[2](u),
where following [38] we write
T[1](u) =m+n∑i,j=1
tij(u)⊗ 1⊗ Eij(−1)j(i+1),
T[2](u) =m+n∑i,j=1
1⊗ tij(u)⊗ Eij(−1)j(i+1).
Then since ∆ is an algebra homomorphism and we must have that
∆ : T (u)−1 7→ T[2](u)−1T[1](u)
−1,
which gives explicitly:
∆(t′ij(u)) =m+n∑k=1
t′kj(u)⊗ t′ik(u)(−1)(i+k)(j+k).
It is easy to see that this coincides with ((ζ ⊗ ζ) τ ∆ ζ) (t′ij(u)).
Let ϕm|n : Y (glm|n) → Y (glm+k|n) be the inclusion which sends each t(r)ij in
Y (glm|n) to the generator t(r)k+i,k+j in Y (glm+k|n); and let ψk : Y (glm|n) → Y (glm+k|n)be the injective homomorphism defined by
ψk = ωm+k|n ϕm|n ωm|n. (3.10)
Then, for any 1 ≤ i, j ≤ m+ n (see Lemma 4.2 of [6]) we have:
ψk(tij(u)) =
∣∣∣∣∣∣∣∣∣t11(u) · · · t1k(u) t1,k+j(u)
... . . . ......
tk1(u) · · · tkk(u) tk,k+j(u)
tk+i,1(u) · · · tk+i,k(u) tk+i,k+j(u)
∣∣∣∣∣∣∣∣∣ .As an immediate consequence we have the following lemma.
25
Lemma 3.3.4. For k, l ≥ 1, we have
ψk(dl(u)) = dk+l(u),
ψk(el(u)) = ek+l(u),
ψk(fl(u)) = fk+l(u).
Remark 3.3.1. The map ψk sends the element t′ (r)ij in Y (glm|n) to the element t′ (r)k+i,k+j
in Y (glm+k|n). Then the subalgebra ψk
(Y (glm|n)
)is generated by the elements
t′ (r)k+s,k+tns,t=1. This implies, by (2.1.2), that all elements of this subalgebra com-
mute with those of the subalgebra generated by the elements t(r)ij ki,j=1. This im-
plies in particular that for any i, j ≥ 1, the quasideterminants di(u) and dj(v) com-mute.
3.3.2 Relations in the Yangian Y (glN)
Brundan and Kleshchev [6] found a presentation for the Yangian Y (glN) usingquasideterminant generators. In the process they established the following rela-tions.
Lemma 3.3.5. The following identities hold in Y (glN)((u−1, v−1)) for i, j = 1, . . . , N−1:
(u− v)[di(u), ej(v)] = (δi,j − δi,j+1)di(u)(ej(v)− ej(u)), (3.11)(u− v)[di(u), fj(v)] = (δi,j+1 − δi,j)(fj(v)− fj(u))di(u), (3.12)(u− v)[ei(u), fj(v)] = δij(di(u)
−1di+1(u)− di(v)−1di+1(v)), (3.13)
(u− v)[ei(u), ei(v)] = (ei(v)− ei(u))2, (3.14)
(u− v)[fi(u), fi(v)] = −(fi(v)− fi(u))2, (3.15)
(u− v)[ei(u), ei+1(v)] = ei(u)ei+1(v)− ei(v)ei+1(v)
−ei,i+2(u) + ei,i+2(v), (3.16)(u− v)[fi(u), fi+1(v)] = −fi+1(v)fi(u) + fi+1(v)fi(v)
+fi+2,i(u)− fi+2,i(v). (3.17)
Furthermore, if i 6= j then:
[[ei(u), ej(v)], ej(v)] = 0, [[fi(u), fj(v)], fj(v)] = 0
[[ei(u), ej(v)], ej(w)] = − [[ei(u), ej(w)], ej(w)], and[[fi(u), fj(v)], fj(w)] = − [[fi(u), fj(w)], fj(w)].
26
3.3.3 Relations in the Yangian Y (gl1|1)
Lemma 3.3.6. The following identities hold in Y (gl1|1)((u−1, v−1)):
(u− v)[d1(u), e1(v)] = d1(u)(e1(v)− e1(u)), (3.18)(u− v)[d1(u), f1(v)] = (f1(u)− f1(v)) d1(u), (3.19)(u− v)[d2(u), e1(v)] = d2(u)(e1(v)− e1(u)), (3.20)(u− v)[d2(u), f1(v)] = (f1(u)− f1(v)) d2(u), (3.21)(u− v)[e1(u), f1(v)] = d1(v)
−1d2(v)− d1(u)−1d2(u), (3.22)
[e1(u), e1(v)] = 0, [f1(u), f1(v)] = 0. (3.23)
Proof. We calculate by (2.1.2) the commutator
(u− v)[t11(u), t′12(v)] = t11(u)t′12(v) + t12(u)t′22(v).
Substituting in the expressions from (3.8) and (3.9) (for the case where m = 1 andn = 1) then cancelling d2(v), gives (3.18). Similarly, by considering the commuta-tors [t11(u), t
′21(v)], [t12(v), t
′22(u)], and [t21(v), t
′22(u)] we derive the relations (3.19),
(3.20) and (3.21) respectively.
To prove (3.22), we calculate the commutator [t12(u), t′21(v)] to find
d1(u)d1(v)−1 + d1(u)(e1(v)d2(v)
−1 − (u− v + 1)e1(u)d2(v)−1)f1(v)
= d2(v)−1d2(u) + d2(v)
−1(f1(u)d1(u) + (u− v − 1)f1(v)d1(u))e1(u)
then substitute into this the following expressions derived from (3.19) and (3.20):
f1(u)d1(u) + (u− v − 1)f1(v)d1(u) = (u− v)d1(u)f1(v)
e1(v)d2(v)−1 − (u− v + 1)e1(u)d2(v)
−1 = −(u− v)d2(v)−1e1(u)
and multiply on the left by d1(u)−1d2(v).
To prove (3.23), we note that since [t12(u), t′12(v)] = 0, we have
d1(u)e1(u)e1(v)d2(v)−1 = −e1(v)d2(v)
−1d1(u)e1(u).
Then we use the identities (3.18) and (3.20) to move d2(v)−1 to the right and d1(u)
to the left, so that they may be cancelled. For this we use the substitutions:
(u− v)e1(v)d1(u) = (u− v − 1)d1(u)e1(v) + d1(u)e1(u)
(u− v)d2(v)−1e1(u) = (u− v + 1)e1(u)d2(v)
−1 − e1(v)d2(v)−1
This gives the relation
(u− v + 1)[e1(u), e1(v)] = e1(v)2 − (u− v + 1)(u− v − 1)
e1(u)2.
27
Now substituting into this v = u + 1, we find e1(v)2 = 0, and thus our relation
follows. The second relation in (3.23) is derived in the same way by consideringthe commutator [t21(u), t
′21(v)].
3.3.4 Relations in the Yangian Y (gl2|1)
Lemma 3.3.7. The following identities hold in Y (gl2|1)((u−1, v−1)):
[e1(u), f2(v)] = 0, , (3.24)(u− v)[e1(u), e2(v)] = e1(u)e2(v)− e1(v)e2(v)
−e13(u) + e13(v), (3.25)(u− v)[f1(u), f2(v)] = −f2(v)f1(u) + f2(v)f1(v)
+f31(u)− f31(v), (3.26)[e13(u), e2(v)] = e2(v)[e1(u), e2(v)], (3.27)
[e1(u), e1(v)e2(v)− e13(v)] = [e1(u), e2(v)]e1(u), (3.28)
Proof. To prove (3.24), we calculate the commutator [t12(u), t′32(v)], and substitute
in the appropriate expressions from (3.8) and (3.9). Note that f2(v) = ψ1(f1(v))commutes with d1(u), and d3(v) commutes with e1(u).
Next, to show (3.25), we calculate [t12(u), t′23(v)] and cancel d1(u) on the left, and
d3(v)−1 on the right. The proof of (3.26) is similar, but we calculate [t21(u), t
′32(v)].
To prove (3.27), we need to derive another identity first. By 3.20, we have
(u− v)[d3(v)−1, e2(u)] = (e2(u)− e2(v))d3(v)
−1
Taking u0 coefficients gives
[d3(v)−1, e
(1)2 ] = −e2(v)d3(v)
−1
Hence,
[e13(u), d3(v)−1] = [[e1(u), e
(1)2 ], d3(v)
−1]
= [e1(u), [e(1)2 , d3(v)
−1]]
= [e1(u), e2(v)d3(v)−1]
= [e1(u), e2(v)]d3(v)−1.
Now, since [t13(u), t23(v)] = 0, we have
[d1(u)e13(u), e2(v)d3(v)−1] = d1(u)[e13(u), e2(v)d3(v)
−1] = 0.
28
Cancelling d1(u) ,
0 = [e13(u), e2(v)d3(v)−1] = [e13(u), e2(v)] d3(v)
−1 − e2(v)[e13(u), d3(v)−1]
= ([e13(u), e2(v)]− e2(v)[e1(u), e2(v)]) d3(v)−1.
Now we prove (3.28). Considering u0- coefficients of 2 gives:
[e(1)1 , e2(v)] = e13(v)− e1(v)e2(v).
By Lemma 5.4(i) in [6] we have
[d1(u), e(1)1 ] = d1(u)e1(u).
Now we can calculate:
[d1(u), e1(v)e2(v)− e13(v)] = [d1(u), [e(1)1 , e2(v)]]
= [[d1(u), e(1)1 ], e2(v)]
= [d1(u)e1(u), e2(v)]
= d1(u)[e1(u), e2(v)]
The proof continues the same way as in [6].
Lemma 3.3.8. In the algebra Y (gl2|1((u−1, v−1))
(u− v)[[e1(u), e2(v)], e2(v)] = 0 (3.29)(u− v)[[e2(u), e1(v)], e1(v)] = 0 (3.30)
[[e1(u), e2(v)], e2(w)] + [[e1(u), e2(w)], e2(v)] = 0 (3.31)[e1(u), [e1(v), e1(w)] ] + [e1(v), [e1(u), e1(w)] ] = 0. (3.32)
Proof. To prove (3.29) we just expand out the right hand and note that e2(v)2 = 0.We expand out the left-hand side of (3.30) using (3.25) and then (3.28) to get:
(u− v)[e2(u), e1(v)], e1(v)]] = [e1(v)e2(u)− e1(u)e2(u) + e13(u)− e13(v), e1(v)]
= e1(v)e2(u)e1(v)− e1(v)e1(v)e2(u)
+[e1(v), e2(u)]e1(v)− [e13(v), e1(v)]
= e1(v) [e2(u), e1(v)]− [e2(u), e1(v)] e1(v)
−[e13(v), e1(v)]
= − [[e2(u), e1(v)].e1(v)]− [e13(v), e1(v)].
So we have(u− v + 1)[e2(u), e1(v)], e1(v)]] = −[e13(v), e1(v)].
29
Setting u = v − 1 in the left-hand side of this equation shows that the right-handside is identically zero. Then we have (3.30).
Expanding out the left-hand side of (3.31) in a straightforward manner andcancelling terms, we find that it equals
e1(u)[e2(v), e2(w)]− [e2(v), e2(w)]e1(u).
By Lemma 3.3.6(4), this is 0.
Finally, we prove (3.32) by showing that
(u− v)(v − w)(u− w)[e1(u), [e1(v), e2(w)]] (3.33)
is symmetric in u and v. We expand out (3.33) using (3.24) and (3.30). We find thatit equals:
(u− v)(u− w) [e1(u), e1(v)e2(w)− e1(w)e2(w) + e13(w)− e13(v)]
= (u− v)(u− w) [e1(u), e1(v)]e2(w) + (u− v)(u− w) e1(v)[e1(u), e2(w)]
−(u− v)(u− w) e1(u)[e1(u), e2(w)] + (u− v)(u− w) e1(u)[e1(u), e2(v)]
−(u− v)(u− w) [e1(u), e1(v)]e2(v)− (u− v)(u− w) e1(v)[e1(v)e2(v)].
Now using (3.24),(3.14) this becomes:
(u− v) (e1(u)− e1(v))2 e2(w)
+(u− v) e1(v) (e1(u)e2(w)− e1(w)e2(w) + e13(w)− e13(u))
−(u− v) e1(u) (e1(u)e2(w)− e1(w)e2(w) + e13(w)− e13(u))
+(u− w) e1(u) (e1(u)e2(v)− e1(v)e2(v) + e13(v)− e13(u))
−(u− w) (e1(u)− e1(v))2 e2(v)
−(u− w) e1(v) (e1(u)e2(v)− e1(v)e2(v) + e13(v)− e13(u)) ,
which we can show is indeed symmetric in u and v by expanding out all bracketsand collecting like terms.
30
3.3.5 Relations in the Yangian Y (glm|n)
Lemma 3.3.9. The following relations hold in the algebra Y (glm|n)((u−1, v−1)).
[di(u), dj(v)] = 0 for all 1 ≤ i, j ≤ m+ n (3.34)
(u− v)[di(u), ej(v)] =
(δij − δi,j+1)di(u)(ej(v)− ej(u)), if j < m,(δij + δi.j+1)di(u)(ej(v)− ej(u)), if j = m,−(δij − δi,j+1)di(u)(ej(v)− ej(u)), if j > m,
(3.35)
(u− v)[di(u), fj(v)] =
−(δij − δi,j+1)(fj(v)− fj(u))di(u), if j < m,−(δij + δi,j+1)(fj(v)− fj(u))di(u), if j = m,(δij − δi,j+1)(fj(v)− fj(u))di(u), if j > m,
(3.36)
(u− v)[ei(u), fj(v)] = (−1)j+1δij(di(u)
−1di+1(u)− di(v)−1di+1(v)
), (3.37)
(u− v)[ej(u), ej(v)] =
(−1)j+1 (ej(v)− ej(u))
2 , if j 6= m,0, if j = m,
(3.38)
(u− v)[fj(u), fj(v)] =
−(−1)j+1 (fj(v)− fj(u))
2 , if j 6= m,0, if j = m,
(3.39)
(u− v)[ej(u), ej+1(v)] = (−1)j+1 (ej(u)ej+1(v)− ej(v)ej+1(v) (3.40)− ej,j+2(u) + ej,j+2(v)) ,
(u− v)[fj(u), fj+1(v)] = −(−1)j+1 (fj+1(v)fj(u)− fj+1(v)fj(v) (3.41)− fj+2,j(u) + fj+2,j(v)) ,
[ei(u), ej(v)] = 0 for |i− j| > 1; (3.42)[fi(u), fj(v)] = 0 for |i− j| > 1; (3.43)
Furthermore, if i 6= j then:
[[ei(u), ej(v)], ej(v)] = 0, [[fi(u), fj(v)], fj(v)] = 0 (3.44)[[ei(u), ej(v)], ej(w)] = − [[ei(u), ej(w)], ej(w)], (3.45)[[fi(u), fj(v)], fj(w)] = − [[fi(u), fj(w)], fj(w)]. (3.46)
Proof. The relations for i, j between 1 and m are an easy consequence of thosealready found for the Yangians Y (glm) in [6] and for Y (gl1|1) and Y (gl2|1) in Sec-tions 3.3.3 and 3.3.4, and the fact that the natural inclusions Y (glm) → Y (glm|n)and Y (gl2|1) → Y (glm|n) are homomorphisms. The remaining relations follow byapplying the map ζn|m to the corresponding relations in Y (gln|m).
Lemma 3.3.10. In addition, we have the following relations in Y (glm|n) when m > 1 andn > 1. For any r, s ≥ 1,
[ [e(r)m−1, e
(1)m ] , [e(1)
m , e(s)m+1] ] = 0; and [ [f
(r)m−1, f
(1)m ] , [f (1)
m , f(s)m+1] ] = 0. (3.47)
31
Proof. We prove the result in Y (gl2|2), and then map this result into the YangianY (glm|n) via the map ψm−2. First we show the following relation:
[e13(u) , e2(z)e3(z)− e2,4(z)] = 0. (3.48)
Indeed, we have:
[e13(u), e2(z)e3(z)− e24(z)] = [e13(u), e′24(w)]
= [t11(u)−1t13(u) , −t′24(w)t′44(w)−1]
= 0
Now we find the commutator
(u− v)(w − z)[[e1(u), e2(v)] , [e2(w), e3(z)]].
By (3.25), this is
[e1(u)e2(v)− e1(v)e2(v)− e13(u) + e13(v), −e2(w)e3(z) + e24(w) + e2(z)e3(z)− e24(z)].
Taking the coefficient of u−rz−s and using (3.48) we find the first relation in (3.47).The other part follows from this with the use of the map ζ .
3.4 New Presentation of Y (glm|n)
Now we can state the presentation result that we have been working towards inthis chapter. The proof is very closely based on the proof of Theorem 5.2 in [6] .
Theorem 3.4.1. The Yangian Y (glm|n) is isomorphic as an associative superalgebra tothe algebra with even generators d(r)
i , d ′ (r)i , f (r)j , e(r)j , (for 1 ≤ i ≤ m + n, 1 ≤ j ≤
m + n − 1, j 6= m, r ≥ 1) and odd generators e(r)m , f (r)m (where again r ≥ 1) and the
following defining relations:
d(0)i = 1;
r∑t=0
d(t)i d
′ (r−t)i = δr,0;
[d(r)i , d
(s)l ] = 0;
32
[d(r)i , e
(s)j ] =
(δi,j − δi,j+1)
∑r−1t=0 d
(t)i e
(r+s−1−t)j , for j < m,
(δi,j + δi,j+1)∑r−1
t=0 d(t)i e
(r+s−1−t)j , for j = m,
−(δi,j − δi,j+1)∑r−1
t=0 d(t)i e
(r+s−1−t)j , for j > m,
(3.49)
[d(r)i , f
(r)j ] =
−(δi,j − δi,j+1)
∑r−1t=0 f
(r+s−1−t)j d
(t)i , for j < m;
−(δi,j + δi,j+1)∑r−1
t=0 f(r+s−1−t)j d
(t)i , for j = m;
(δi,j − δi,j+1)∑r−1
t=0 f(r+s−1−t)j d
(t)i , for j > m;
(3.50)
[e(r)j , f
(s)k ] =
−δj,k
∑r+s−1t=0 d
′ (t)j d
(r+s−1−t)j+1 , for j < m;
+δj,k∑r+s−1
t=0 d′ (t)j d
(r+s−1−t)j+1 , for j ≥ m;
(3.51)
[e(r)m , e(s)m ] = 0, [f (r)m , f (s)
m ] = 0; (3.52)
[e(r)j , e
(s+1)j ]− [e
(r+1)j , e
(s)j ] = (−1)j
(e(r)j e
(s)j + e
(s)j e
(r)j
), for j 6= m; (3.53)
[f(r+1)j , f
(s)j ]− [f
(r)j , f
(s+1)j ] = (−1)j
(f
(r)j f
(s)j + f
(s)j f
(r)j
), for j 6= m; (3.54)
[e(r)j , e
(s+1)j+1 ]− [e
(r+1)j , e
(s)j+1] = −(−1)je
(r)j e
(s)j+1 (3.55)
[f(r+1)j , f
(s)j+1]− [f
(r)j , f
(s+1)j+1 ] = −(−1)jf
(s)j+1f
(r)j ; (3.56)
[e(r)j , e
(s)k ] = 0; and [f
(r)j , f
(s)k ] = 0, if |j − k| > 1; (3.57)
[[e(r)j , e
(s)k ], e
(t)k ] + [[e
(r)j , e
(t)k ], e
(s)k ] = 0, if j 6= k; (3.58)
[[f(r)j , f
(s)k ], f
(t)k ] + [[f
(r)j , f
(t)k ], f
(s)k ] = 0, if j 6= k; (3.59)
[ [e(r)m−1, e
(1)m ] , [e(1)
m , e(s)m+1] ] = 0 (3.60)
[ [f(r)m−1, f
(1)m ] , [f (1)
m , f(s)m+1] ] = 0 (3.61)
for all r, s, t ≥ 1. and all admissible i, j, k.
Remark 3.4.1. Relations (3.53) and (3.54) are equivalent to the following relations:
[e(r)j , e
(s)j ] = (−1)j
(s−1∑t=1
e(t)j e
(r+s−1−t)j −
r−1∑t=1
e(t)j e
(r+s−1−t)j
), for j 6= m;
[f(r)j , f
(s)j ] = (−1)j
(r−1∑t=1
f(t)j f
(r+s−1−t)j −
s−1∑t=1
f(t)j f
(r+s−1−t)j
), for j 6= m;
Proof. Let Ym|n be the associative algebra given by the relations in the theorem. ByLemma 3.3.9 and Lemma 3.3.10 the map from Ym|n to the Yangian Y (glm|n) thatsends every element of Ym|n to the element of the same name in the Yangian is a
33
homomorphism. We have already stated in Section 3.2 that Y (glm|n) is generatedby the elements:
d
(r)i , e
(r)j , f
(r)j
∣∣∣ 1 ≤ i ≤ m+ n, 1 ≤ j ≤ m+ n− 1, r ≥ 1.
Thus this homomorphism is surjective. We need to show that it is injective. Ourmethod is as follows: we show that the algebra Ym|n is spanned as a vector spaceby the monomials in the elements f (r)
ji , d(r)i , e
(r)ij with 1 ≤ i < j ≤ m + n, r ≥ 1,
taken in some fixed order so that the f ’s come before d’s and d’s come before e’s.(These elements are defined inductively by f (r)
i+1,i = f(r)i ; e
(r)i,i+1 = e
(r)i and
f(r)j,i = [ f
(1)j,j−1 , f
(r)j−1,i ] (−1)j−1; e
(r)i,j = [ e
(r)i,j−1 , e
(1)j−1,j ] (−1)j−1, for j > i+ 1).
Since the image of these monomials in the Yangian form a basis for Y (glm|n), itfollows that the map is an isomorphism.
Let Y +m|n, Y −
m|n and Y 0m|n be the subalgebras of Ym|n generated by all elements of
the form e(r)i , f (r)
i and d(r)i , respectively. By the defining relations (3.49), (3.50) and
(3.51), we know that Ym|n is spanned by the monomials where all f ’s come beforeall d’s and all d’s come before all e’s. Also, since the d’s commute, we may assumethat they are written in some fixed order. If we can show that the subalgebraY +
m|n is spanned by the monomials in e(r)ij written in some fixed order, then by
applying the map ζ we can show that the subalgebra Y −m|n is similarly spanned
by the monomials in f (r)ji written in some fixed order. This will then complete the
proof.
Define an ascending filtration on Y +m|n by setting deg(e
(r)i ) = r−1, and denote
by grL Y +m|n the corresponding graded algebra. Let e(r)ij be the image of e(r)ij in the
(r − 1)-th component of the graded algebra grL Y +m|n. We claim that these images
satisfy:
[e(r)ij , e
(s)kl ] = (−1)j δkj e
(r+s−1)il − (−1)i j+j k+i kδil e
(r+s−1)kj . (3.62)
From this relation it follows that the graded algebra grL Y +m|n is spanned by the
monomials in e(r)ij taken in some fixed order. Hence Y +
m|n is itself spanned by the
monomials in e(r)ij taken in some fixed order.
So now it remains only to prove the claim (3.62). We begin by noting the
34
following relations.
[e(r)i,i+1, e
(s)k,k+1] = 0, if |i− k| 6= 1. (3.63)
[e(r+1)i,i+1 , e
(s)k,k+1] = [e
(r)i,i+1, e
(s+1)k,k+1], if |i− k| = 1, (3.64)
[e(r)i,i+1 , [e
(s)i,i+1, e
(t)k,k+1] ] = −[e
(s)i,i+1 , [e
(r)i,i+1, e
(t)k,k+1] ], if |i− k| = 1, (3.65)
e(r)ij = [e
(r)i,j−1, e
(1)j−1,j] (−1)j−1 = [e
(1)i,i+1, e
(r)i+1,j] (−1)i+1, for j > i+ 1. (3.66)
Here, (3.63) is a consequence of (3.57); (3.64) is a consequence of (3.55); and (3.65)is a consequence of (3.58). The first part of the last relation (3.66) follows from thedefinition of the elements e(r)ij . The second part of (3.66) follows from the first partusing (3.64) and induction on the difference j − i.
Now we break up the problem of showing (3.62) into cases. We assume with-out loss of generality that i ≤ k. If j < k, then [e
(r)ij , e
(s)kl ] = 0 by (3.63) and (3.66).
Consider the case where j = k. By (3.64) and (3.66) we have
[e(r)j−1,j , e
(s)j,j+1] = (−1)je
(r+s−1)j−1,j+1.
We bracket both sides of this with e(1)j+1,j+2, e(1)j+2,j+3, . . ., e
(1)l−1,l in turn to obtain:
[e(r)j−1,j, e
(s)jl ] = (−1)j e
(r+s−1)j−1,l ,
then bracket both sides of this new equation with e(1)j−2,j−1, . . . , e(1)i,i+1 to get the rela-
tion:[e
(r)i,j , e
(s)j,l ] = (−1)j e
(r+s−1)i,l .
Before we consider the case j > k in detail, we prove the following specialcases:
[e(r)i,i+2, e
(s)i+1,i+2] = 0, for 1 ≤ i ≤ m+ n− 2, (3.67)
[e(r)i,i+1, e
(s)i,i+2] = 0, for 1 ≤ i ≤ m+ n− 2, (3.68)
[e(r)i,i+2, e
(s)i+1,i+3] = 0 for 1 ≤ i ≤ m+ n− 3. (3.69)
[e(r)ij , e
(s)k,k+1] = 0 for 1 ≤ i < k < j ≤ m+ n. (3.70)
Indeed, for (3.67), we have:
(−1)i+1 [e(r)i,i+2, e
(s)i+1,i+2]
= [[e(r)i,i+1, e
(1)i+1,i+2], e
(s)i+1,i+2] by (3.66)
= −[[e(r)i,i+1, e
(s)i+1,i+2], e
(1)i+1,i+2] by (3.65)
= −[[e(r+s−1)i,i+1 , e
(1)i+1,i+2], e
(1)i+1,i+2] by (3.64),
35
which is 0 by (3.65). The relation (3.68) is shown in a very similar way.
When i + 1 = m, the relation (3.69) follows directly from (3.60). On the otherhand, when i+ 1 6= m, the left-hand side of (3.69) equals
(−1)(i+1 + i+2) [[e(r)i,i+1, e
(1)i+1,i+2] , [e
(1)i+1,i+2, e
(s)i+2,i+3] ]
= (−1)(i+1 + i+2) [e(1)i+1,i+2, [e
(r)i,i+1, e
(1)i+1,i+2], e
(s)i+2,i+3]]
= (−1)(i+1 + i+2) [e(1)i+1,i+2, [e
(r)i,i+1, [e
(1)i+1,i+2, e
(s)i+2,i+3]]]
= (−1)(i+1 + i+2)[[e(1)i+1,i+2, e
(r)i,i+1], [e
(1)i+1,i+2, e
(s)i+2,i+3]],
= − (−1)(i+1 + i+2) [[e(r)i,i+1, e
(1)i+1,i+2] , [e
(1)i+1,i+2, e
(s)i+2,i+3]].
Hence the commutator is zero. Here we have used (3.58) and the super-Jacobiidentity, and the fact that since i+1 6= m, no two of the elements we are concernedwith are odd.
Finally, we use (3.66) relation to reduce the problem of showing (3.70) to thatof showing
[e(r)i,k+1, e
(s)k,k+2] = 0, and
[e(r)i,k+1, e
(s)k,k+1] = 0,
for all i ≤ k. The first of these relations follows from (3.68) and (3.69) by inductionon the difference k − i, using (3.66). The second follows from (3.67), again byinduction on k − i, using the relation (3.66).
Now we properly begin the case j > k. We break this into the followingsubcases:
Case 1: i < k, j = l. Expanding e(s)kj by (3.66) and then using the super-Jacobi identityand (3.70), we have:
[e(r)ij , e
(s)kj ] = ±[ e
(1)k,k+1, [e
(r)i,j , e
(s)k+1,j] ].
Continuing on in this fashion, we find:
[e(r)ij , e
(s)kj ] = ±
[e(1)k,k+1, . . . [e
(r)ij , e
(s)j−1,j] . . .
],
so our problem reduces to showing that [e(r)ij , e
(s)j−1,j] = 0. We now expand out
the e(r)ij in this using (3.66) and apply the super-Jacobi identity to reduce thisproblem to that of showing that [e
(r)j−2,j, e
(s)j−1,j] = 0. Then we have the result
in this case by (3.67).
36
Case 2: i < k, j > l. We expand out e(s)kl using (3.66) and then apply the super-Jacobiidentity and (3.70) to find:
[e(r)ij , e
(s)kl ] = ±[e
(1)k,k+1, [e
(r)ij , e
(s)k+1,l]].
Repeating this process as many times as is necessary we eventually get
[e(r)ij , e
(s)kl ] = ±[e
(1)k,k+1, . . . , [e
(r)ij , e
(s)l−1,l] . . .].
which is 0 by (3.70).
Case 3: i < k, j < l. We prove this case by induction on the difference l − j. Whenl−j = 1, we have by expanding out e(s)k,j+1 and using the super-Jacobi identitythat
[e(r)ij , e
(s)k,j+1] = [[e
(r)ij , e
(s)kj ], e
(1)j,j+1] (−1)j + [e
(r)kj , [e
(s)ij , e
(1)j,j+1]] (−1)i j+j k+i k
= [[e(r)ij , e
(s)kj ], e
(1)j,j+1] (−1)j + [e
(s)i,j+1, e
(r)kj ](−1)(j+j+1)(j+k).
The first term is 0 by the Case 1 and the second term is 0 by Case 2. Whenl − j > 1,
[e(r)ij , e
(s)kl ] = [[e
(r)ij , e
(s)k,l−1], e
(1)l−1,l](−1)l−1,
which is 0 by the induction hypothesis.
Case 4: i = k, j < l. We use (3.66) (and (3.63) and Case 2) to reduce this case to (3.68).
Case 5: i = k, j = l. If j = i + 1, then this is (3.63). Otherwise, we can expand outone term with (3.66) to find:
[e(r)ij , e
(s)ij ] = ±[[e
(r)i,j−1, e
(s)ij ], e
(1)j−1,j] +±[e
(r)i,j−1, [e
(1)j−1,j, e
(s)ij ] ].
The first term is 0 by Case 4 and the second term is 0 by Case 1.
Case 6: i = k, j > l. This follows immediately from Case 4.
This completes the proof of the claim (3.62), which completes the proof of thetheorem.
37
38
Chapter 4
The Centre of Y (glm|n)
In his pioneering paper, Nazarov [40] identified two families of central elementsof the Yangian Y (glm|n). The first family was the set of coefficients of a formalpower series called the quantum contraction z(u) , and the second was the set of co-efficients of another formal power series, called the quantum Berezinian . Nazarovproved a relationship between the two series, which is a quantum analogue ofthe Liouville theorem [3], and also conjectured that coefficients of the quantumBerezinian generate the centre of the Yangian Y (glm|n). (It follows from the Liou-ville formula that the coefficients of the quantum contraction then also generatethe centre).
In this chapter, we give an alternative proof of the fact that the coefficientsof the quantum Berezinian are central. This proof relies heavily on the Gaussdecomposition of the Yangian presented in Chapter 3, using a method inspired bythe work of Brundan and Kleshchev [6]. We also use the Poincare-Birkhoff-Witttheorem to prove Nazarov’s conjecture.
4.1 The Quantum Berezinian
The quantum Berezinian is a super-analogue of the quantum determinant. Recallthat Berezin generalized the concept of the commutative determinant to the caseof supercommmutative rings [4], and that this super-determinant (1.3) is namedthe Berezinian in his honour.
Definition 4.1.1. The quantum Berezinian is the following formal power series with
39
coefficients in the Yangian Y (glm|n):
bm|n(u) :=∑τ∈Sm
sgn(τ) tτ(1)1(u)tτ(2)2(u− 1) · · · tτ(m)m(u−m+ 1)
×∑σ∈Sn
sgn(σ) t′m+1,m+σ(1)(u−m+ 1) · · · t′m+n,m+σ(n)(u−m+ n)
The Yangian Y (glm|n) is a quantization of the super-commutative algebrawith generators t(r)ij . In this context, the quantum Berezinian bm|n(u) has as itsclassical limit the usual Berezinian (see [40]) .
For convenience, let us write:
Cm(u) :=∑
τ∈Sm
sgn(τ)tτ(1)1(u)tτ(2)2(u− 1) · · · tτ(m)m(u−m + 1).
It is clear that Cm(u) is an element of the subalgebra of Y (glm|n) generated by theset t(r)ij 1≤i,j≤m;r≥0. This subalgebra is isomorphic to the Yangian Y (glm) of the Liealgebra glm by the inclusion Y (glm) → Y (glm|n) which sends each generator t(r)ij
in Y (glm) to the generator of the same name in Y (glm|n). Moreover, Cm(u) is theimage under this map of the quantum determinant of Y (glm) [6, 38]. Then it is wellknown (see Theorem 2.32 in [39]) that we have:
Cm(u) = d1(u)d2(u− 1) · · · dm(u−m+ 1).
We can extend this observation as follows:
Theorem 4.1.1. We can write the quantum Berezinian as follows:
bm|n(u) = d1(u) d2(u− 1) · · · dm(u−m+ 1)
× dm+1(u−m+ 1)−1 · · · dm+n(u−m+ n)−1.
Proof. Note that the second part of the expression for bm|n(u) in Definition 4.1.1 isthe image under the isomorphism ζ : Y (gln|m) → Y (glm|n) of∑
σ∈Sn
sgn(σ) tn,σ(n)(u−m+ 1) · · · t2,σ(2)(u−m+ n− 1) t1,σ(1)(u+m− n), (4.1)
where in this expression (4.1) we follow the convention for denoting generatorsin the Yangian Y (gln|m). We recognise (by comparing with (8.3) of [6]) that this isCn(u−m+ n), the image of the quantum determinant of Y (gln) under the naturalinclusion Y (gln) → Y (gln|m). So to verify the claim we calculate the image ofCn(u −m + n) under this map explicitly in terms of the quasideterminants di(v).By Proposition 3.3.2, the image of di(v) in Y (gln|m) is (dm+n+1−i(v))
−1 in Y (glm|n),which gives the desired result.
40
Lemma 4.1.1. Let glm|n[x] be the polynomial current algebra and I = E11 + . . . +Em+n,m+n. The centre of U(glm|n[x]) is generated by I, Ix, Ix2, . . ..
Proof. We reduce the problem to that of the well-known even case considered forexample in Lemma 7.1 of [6]. First note that the supersymmetrization map givesan isomorphism between the glm|n[x]-modules U(glm|n[x]) and S(glm|n[x]), whereS(glm|n[x]) denotes the supersymmetric algebra of glm|n[x]. The natural action ofglm|n[x] on S(glm|n[x]) is obtained by extending the adjoint action. The Lie algebraglm|n has the root space decomposition:
glm|n = h⊕k⊕
i=1
gαi
where h is the Cartan subalgebra, α1, . . . , αk is the set of roots relative to h, andgαi
is the root space corresponding the root αi. Let eαibe a root vector correspond-
ing to root αi. Suppose P ∈ S(glm|n[x]) is an arbitrary glm|n-invariant element andM is the maximal integer such that eαi
xM occurs in P for some root αi. Then wemay write:
P =∑
s
As
(eα1x
M)s1
. . .(eαk
xM)sk , (4.2)
where we sum over tuples of positive integers s = (s1, . . . , sk), and for each suchs, the As is a monomial in elements hxr for h ∈ h, r ≥ 0, and eαi
xr for r < M .
For any h ∈ h, we have by assumption that:
0 = [hx, P ]
=∑
s
[hx,As](eα1x
M)s1
. . .(eαk
xM)sk
+k∑
i=1
siαi(h)∑
s
As
(eα1x
M)s1
. . .(eαixM)si−1
. . .(eαk
xM)sk(eαixM+1
).
Then taking the coefficient of(eαixM+1
)we find that for all h ∈ h, and for all roots
αi that:siαi(h)
∑s
As
(eα1x
M)s1
. . .(eαixM)si−1
. . .(eαk
xM)sk = 0.
Since αi(h) is not zero for all h ∈ h, and the monomials corresponding to different sare linearly independent, we must have that si = 0. Thus P is a sum of monomialsin hxr, where h ∈ h and r ≥ 0. The Cartan subalgebra h contains only evenelements, and so the action of glm|n[x] on invariant elements P is the same as theaction of glm+n[x]. Then we may use Lemma 7.1 of [6] to obtain our desired result.
41
Theorem 4.1.2. The coefficients of the quantum Berezinian generate the centre of Y (glm|n).
Proof. First we show that the coefficients are central. By Remark 3.3.1, the quan-tum Berezinian bm|n(u) commutes with di(v) for 1 ≤ i ≤ m+n. So we need to showthat bm|n(u) commutes with ei(v) and fi(v) for each i between 1 and m+ n− 1. Webreak this problem into three cases:
Case 1: 1 ≤ i ≤ m − 1. By Theorem 7.2 in [6], ei(v) commutes with Cm(u).On the other hand, ei(v) is an element of the subalgebra generated by t(r)jk 1≤j,k≤m
and so by Remark 3.3.4 commutes with dm+s(u−m + s)−1 = t′m+s,m+s(u−m + s)for 1 ≤ s ≤ n. Similarly, bm|n(u) commutes with fi(v) for 1 ≤ i ≤ m− 1.
Case 2: m+ 1 ≤ i ≤ m+ n− 1. By Proposition 3.3.2, we have
ei(v) = ζ(−fm+n−i(v)), fi(v) = ζ(−em+n−i(v)),
and ζ(bn|m(u)) = bm|n(u). So we simply apply the isomorphism ζ to the results ofCase 1 in the Yangian Y (gln|m).
Case 3: i=m. Consider the Yangian Y (gl1|1). For this algebra we have
b1|1(u) = d1(u)d2(u)−1
and we would like to show it commutes with e1(v). So it will suffice to show
d1(u)e1(v)d2(u) = d2(u)e1(v)d1(u). (4.3)
We have(t11(u) t12(u)t21(u) t22(u)
)=
(d1(u) d1(u) e1(u)f1(u)d1(u) f1(u)d1(u)e1(u) + d2(u)
), (4.4)(
t′11(v) t′12(v)t′21(v) t′22(v)
)=
(d1(v)−1+ e1(v)d2(v)−1f1(v) −e1(v) d2(v)−1
−d2(v)−1f1(v) d2(v)−1
). (4.5)
An application of (2.1.2) gives
(u− v)[t11(u), t′12(v)] = t11(u)t
′12(v) + t12(u)t
′22(v).
We substitute in this the expressions from (4.4) and (4.5) then cancel d2(v) andrearrange to find:
(u− v)e1(v)d1(u) = (u− v − 1)d1(u)e1(v) + d1(u)e1(u)
Similarly, by considering the commutator [t12(u), t′22(v)], we derive the relation
(u− v)e1(v)d2(u) = (u− v − 1)d2(u)e1(v) + d2(u)e1(u).
42
From these relations it is clear that (4.3) holds.
Now we return our attention to the general Yangian Y (glm|n). By similarappeals to Remark 3.3.1 as in the first case, we see that em(v) commutes withd1(u) · · · dm−1(u−m+2) and with dm+2(u−m+2)−1 · · · dm+n(u− m + n)−1. So weneed only show that em(v) commutes with dm(u−m+ 1)dm+1(u−m+ 1)−1. Thisfollows immediately if we apply the homomorphism ψm−1 to the identity (4.3) inY (gl1|1). Thus the coefficients of the quantum Berezinian are central.
Now writebm|n(u) = 1 +
∑r≥1
bru−r.
Our proof that the coefficients bi generate the centre is based on that of Theorem2.13 in [38]. Recall from Corollary 2.5.1 that the graded algebra gr2Y (glm|n) isisomorphic to U(glm|n[x]). We show that for any r = 1, 2, . . . , the coefficient br hasdegree r−1 with respect to deg2(.) and that its image in the (r−1)th component ofgr2Y (glm|n) coincides with Ixr−1. Indeed, if we expand out the expression (4.1.1)for the quantum Berezinian, using the fact from [22] that
dj(u) = tjj(u)−∑k,l<j
tjk(u)(|T (u)1,2,...,j−1,1,2,...,j−1|lk)−1tlj(u),
we find
br =∑
l1+l2+...+lm+n=r
t(l1)11 t
(l2)22 · · · t(lm)
mm · (− t(lm+1)m+1,m+1) · · · (− t
(lm+n)m+n,m+n)
+ terms of lower degree.
Then it is clear that the terms with li = r for some i = 1, . . . ,m + n have degreer − 1, and all else have lower degree. Then
br = t(r)11 + . . .+ t(r)mm − t
(r)m+1,m+1 − . . .− t
(r)m+n,m+n + terms of lower degree.
The result follows when we evaluate the image of the graded part of this underthe isomorphism in Corollary (2.5.1).
43
44
Chapter 5
Stukopin’s Presentation of Y (slm|n)
Vladimir Stukopin [45, 46] defined a series of Yangians associated to the classicalLie superalgebras, and gave a presentation for the Yangian associated to slm|n inthe case where m 6= n. This chapter is devoted to reviewing his work. However,first we need to review some of the basic definitions relating to quantum groupsfrom the quantization point of view. Our presentation is based on the introduc-tions to Hopf algebras and quantization theory given in [41, 33] but we considerthe superanalogues of these structures, as studied in [2, 32].
5.1 Quantization of Super Lie Bialgebras
5.1.1 Super Lie Bialgebras
A super Lie bialgebra g is a Lie superalgebra with an additional structure thatdefines a Lie superbracket on the dual space g∗ in such a way that the Lie super-bracket on g and the Lie superbracket on g∗ are compatible in a certain way. Theconcept of super Lie bialgebra is a generalization of that of Lie bialgebra. (Theyare also called Lie super-bialgebras and Lie bisuperalgebras - there appears to beno consensus on where to put the “super”). Basic definitions and results regardingsuper Lie bialgebras are taken from [2, 32].
Definition 5.1.1. A super Lie bialgebra is a triple (g, [·, ·], δ) such that:
(i) g is a Lie superalgebra with bracket [·, ·].
(ii) δ is a linear map δ : g → g⊗ g whose dual δ∗ : g∗ ⊗ g∗ → g∗ defines a Lie su-
45
perbracket on g∗. In other words, we require that δ be skew-supersymmetric,
δ + τ δ = 0, (5.1)
and that δ satisfies the co-super-Jacobi identity :
(1 + P123 + P132)(1⊗ δ)δ = 0. (5.2)
(ii) δ is a 1-cocycle of g with coefficients in g⊗ g. This means that
δ([X, Y ]) = (adX ⊗ 1 + 1⊗ adX)δ(Y )− (−1)X Y (adY ⊗ 1 + 1⊗ adY )δ(X),
for all X, Y ∈ g.
Example 5.1.1. If g is a basic simple classical Lie superalgebra with invariant innerproduct then we can define a super Lie bialgebra structure on g[w] = g⊗C[[w]] by
δ : g[w] → (g⊗ g)[w1, w2] = g[w]⊗ g[w]
δ(f(w)) = (ad(f(w1))⊗ 1 + 1⊗ ad(f(w2)))t
(w1 − w2),
where t is the Casimir element defined in Section A.9. We write rij =tij
(wi − wj).
Indeed, that δ is super-skew-symmetric follows from the fact that r12 + r21is invariant under the adjoint action of g; and that δ satisfies the co-super-Jacobiidentity follows from the fact that rij satisfies the classical Yang-Baxter equation:
[r12, r13] + [r12, r23] + [r13, r23] = 0.
That δ is a 1-cocycle follows from a straightforward calculation using the super-Jacobi identity.
5.1.2 Co-Poisson Hopf Superalgebras
The universal enveloping algebra of any Lie superalgebra g is naturally a Hopfsuperalgebra (see Section 2.2) with coproduct defined by:
∆(a) = a⊗ 1 + 1⊗ a, for all a ∈ g,
and extended to a unique algebra homomorphism. If g is also a super Lie bialge-bra, then U(g) becomes what is called a co-Poisson Hopf superalgebra.
Definition 5.1.2. A co-Poisson Hopf superalgebra over a field k is a co-commutativeHopf superalgebra (A,m,∆, ι, S) with an even linear map δ : A → A ⊗ A, calledthe Poisson cobracket, satisfying the following conditions:
46
(i) δ is skew-symmetric;
(ii) δ satisfies the co-Jacobi identity (5.2).
(iii) δ and ∆ together satisfy the co-super-Leibniz identity:
(∆⊗ id)δ = (id⊗ δ)∆ + (id⊗ τ)(δ ⊗ id)∆. (5.3)
(iv) δ and ∆ are compatible in the sense that
δ(a1a2) = δ(a1)∆(a2) + ∆(a1)δ(a2), for all a1, a2 ∈ A. (5.4)
5.1.3 The h-adic topology
Drinfeld’s definition of quantizations of a co-Poisson Hopf superalgebra [15] wasinspired by the physical notion of quantization. To describe the definition, weneed to consider algebras over k[[h]], the ring of formal power series in a variablehwith coefficients in k. This variable h is meant to remind one of Planck’s constant(which is small) and so it makes sense to consider on our ring a metric spacetopology in which elements that differ by large powers of h are considered veryclose together. This topology is called the h-adic topology.
Definition 5.1.3. Let M be a k[[h]]-module. Define a map d : M ×M → R by
d(x, y) = e−ν(x−y),
for all x, y ∈M , where ν(x) is the largest non-negative integer such that x ∈ hnM .Then d is a metric on M . The corresponding metric space topology on M is calledthe h-adic topology.
The setsm+ hnM, where m ∈M,n ∈ N,
form a basis for the h-adic topology on M [41].
Given two Cauchy sequences xnn∈N and ynn∈N in a k[[h]]-module M , wesay that they are equivalent if the difference xn − yn tends to zero as n goes toinfinity. The completion M of the module M is the set of equivalence classes ofCauchy sequences in M , with operations given pointwise:
xn+ yn = xn + yn, axn = axn, for all a ∈ A.
We identify M with the submodule of M consisting of constant sequences in M .
47
5.1.4 Definition of Quantization
Definition 5.1.4. Let (A,m,∆, ι, ε, S) be a co-Poisson Hopf superalgebra over afield k, and let δ : A → A ⊗ A be the corresponding cobracket. A quantization ofA is a Hopf superalgebra (Ah,mh,∆h, ιh, εh, Sh) over k[[h]] satisfying the followingproperties.
(i) Ah is isomorphic to A[[h]] as a topological k[[h]]-module.
(ii) Ah is complete in the h-adic topology.
(iii) Ah/hAh is isomorphic to A as a Hopf superalgebra.
(iv) For any a ∈ Ah, we have.
(∆h(a)−∆opph (a))
hmodh = δ(amodh). (5.5)
When we say that Ah is a Hopf superalgebra over k[[h]] we mean that mh, ∆h,ιh, εh, and Sh are k-linear maps between the appropriate spaces (taking comple-tions of tensor products where necessary):
mh : A[[h]]⊗A[[h]] → A[[h]],
∆h : A[[h]] → A[[h]]⊗A[[h]],
ιh : k[[h]] → A[[h]],
εh : A[[h]] → k[[h]],
Sh : A[[h]] → A[[h]],
that they are continuous in the h-adic topology and satisfy the axioms for being aHopf algebra. By 5.1.4(iii) we mean that these maps can be written in the form:
mh = m+m1h+m2h2 + . . . ,
∆h = ∆ + ∆1h+ ∆h2 + . . . ,
ιh = ι+ ι1h+ ι2h2 + . . . ,
εh = ε+ ε1h+ ε2h2 + . . . ,
Sh = S + S1h+ S2h2 + . . . ,
where for each positive integer i,
mi : A⊗ A→ A, ∆i : A→ A⊗ A,
ιi : k → A, εi : A→ k, Si : A→ A.
48
are k-linear maps which are extended first k[[h]]-linearly and then to the h-adiccompletion.
The first three conditions in the definition of a quantization are those whichqualify A to be a deformation of A as a Hopf superalgebra [41]. The last is a specialcondition appropriate to the co-Poisson structure on A. Given any deformationAh of a Hopf superalgebra A, we may define a super Lie cobracket on A by theleft-hand side of (5.5). Thus a quantization is a deformation for which the inducedcobracket agrees with the original one. A quantization of a (super) Lie bialgebra g
is by definition a quantization of U(g) as a co-Poisson Hopf superalgebra [15].
Example 5.1.2. In Example 5.1.1 we defined a super Lie bialgebra on g[w] whereg is a basic simple Lie superalgebra such as slm|n or psln|n. The Lie bialgebra g[w]is naturally graded by setting the degree of w equal to 1, and the Lie cobracketdecreases the degree of an element awr by one. Thus it is natural to seek a quan-tization of U(g[w]) that is a graded Hopf superalgebra over k[[h]] where k[[h]] isgraded so that degree h is also equal to 1. Such a quantization is called homoge-neous.
In the case where g is a classical Lie algebra, this homogeneous quantiza-tion of U(g[w]) is unique up to isomorphism [15]. It is called the Yangian Y (g).For classical simple Lie superalgebras g, Stukopin [46] described a homogeneousquantization of U(g[w]) and called it the Yangian Y (g). It is not clear whether thisYangian is unique as an homogeneous quantization.
5.2 Stukopin’s Presentation
Recall that the Lie algebra slm|n is given by the following presentation: presenta-tion [25, 42]. We take generators
hi, x+j , x
−j | 1 ≤ i ≤ m+ n− 1.
The generators hi, x±j are declared even for all i and all j 6= m; and x±m are de-
clared odd. The defining relations are:
[hi, hj] = 0;
[hi, x±j ] = ±aijx
±j ;
[x+i , x
−j ] = δi,jhi;
[x±m, x±m] = 0;
[x±i , x±j ] = 0, if |i− j| > 1;
[x±i , [x±i , x±j ] ] = 0, if |i− j| = 1;
[ [x±m−1, x±m] , [x±m+1, x
±m]] = 0,
49
for all i, j between 1 and m+n− 1. Here A = (aij)m+n−1i,j=1 is the symmetric Cartan
matrix of the Lie superalgebra slm|n with entries aii = 2 for all i < m , amm = 0,aii = −2 for all i > m, ai+1,i = ai,i+1 = −1 for all i < m, ai+1,i = ai,i+1 = 1 for alli ≥ m, and all other entries 0.
Stukopin gives two presentations of the Yangian associated with slm|n whenm 6= n. We will temporarily denote this Yangian by Y (slm|n) to distinguish it fromour new definition of the Yangian for slm|n in Chapter 6. It turns out that the twodefinitions give isomorphic associative Hopf superalgebras (see Section 6.2) andthus are equivalent.
Definition 5.2.1. The Yangian Y (slm|n) is the Hopf superalgebra over C with gen-erators x±i,0, hi,0, x±i,1, and h′i,1 with 1 ≤ i ≤ m + n − 1, and definiing relations
[hi,0, hj,0] = [hi,0, h′j,1] = [h′i,1, h
′j,1] = 0,
[hi,0, x±j,0] = ±aijx
±j,0,
[h′i,1, x±j,0] = ±aijx
±j,1,
[x+i,0, x
−j,0] = δijhi,0,
[x+i,1, x
−j,0] = δi,jhi,1 := h′i,1 + 1
2h2
i,0,
[x±i,1, x±j,0]− [x±i,0, x
±j,1] = ±aij
2
(x±i,0x
±j,0 + x±j,0x
±i,0
),
[x±m,1, x±m,0] = 0,
[x±i,0, [x±i,0, x
±j,0]] = 0, for i 6= j,
[[x±m−1,1, x±m,0], [x
±m+1,0, x
±m,0]] = 0,
[[h′j,1, x+i,1], x
−j,1] + [x+
i,1, [h′i,1, x
−j,1]] = 0, for i 6= m,
[[h′m−1,1, x+m,1], x
−m,1] + [x+
m,1, [h′m−1,1, x
−m,1]] = 0.
Here all generators are even, except for x±m,0 and x±m,1 which are odd.
The comultiplication is defined on the generators as follows :
∆(hi,0) = hi,0 ⊗ 1 + 1⊗ hi,0, (5.6)∆(x±i,0) = x±i,0 ⊗ 1 + 1⊗ x±i,0, (5.7)
∆(hi,1) = hi,1 ⊗ 1 + 1⊗ hi,1 + hi,0 ⊗ hi,0 + [hi,0 ⊗ 1,Ω], (5.8)
where Ω =∑
α∈∆+x−α ⊗ x+
α ∈ slm|n.
Remark 5.2.1. The coproduct here is slightly different to what is written in [45].I’ve changed the formula for (5.7) so that it agrees with the standard Hopf super-algebra structure on the universal enveloping algebra U(slm|n).
50
Proposition 5.2.1 (Stukopin). The subalgebra Y (slm|n) is isomorphic to the associativesuperalgebra over C defined by the generators x±i,s and hi,s for 1 ≤ i ≤ m + n − 1 ands ∈ Z+, and by the relations
[hi,r, hj,s] = 0,
[x+i,r, x
−j,s] = δijhi,r+s , (5.9)
[hi,0, x±j,s] = ±aijx
±j,s , (5.10)
[hi,r+1, x±j,s]−[hi,r, x
±j,s+1] =
±aij
2(hi,rx
±j,s + x±j,shi,r) for i, j not both m,(5.11)
[hm,r+1, x±m,s] = 0, (5.12)
[x±m,r , x±m,s] = 0, (5.13)
[x±i,r , x±j,s] = 0, if |i− j| > 1, (5.14)
[x±i,r+1, x±j,s]− [x±i,r, x
±j,s+1] =
±aij
2(x±i,rx
±j,s + x±j,sx
±i,r) for i, j not both m,(5.15)
[x±i,r, [x±i,s, x
±j,t]] = −[x±i,s, [x
±i,r, x
±j,t]], if |i− j| = 1, (5.16)
[ [x±m−1,r, x±m,0], [x
±m,0, x
±m+1,s] ] = 0, (5.17)
where r, s and t are arbitrary positive integers and aij are the elements of the Cartanmatrix above. The generators x±m,s are odd and all other generators are even.
Remark 5.2.2. The relation (5.17) is different to that given in [45]. Stukopin givesa more restrictive set of relations:
[ [x±m−1,r, x±m,q], [x
±m,t, x
±m+1,s] ] = 0, for all q, r, s, t ≥ 0,
but this more restrictive relation leads to the algebra thus defined being trivial.Therefore, it was necessary to correct it.
We use the following notation for the coefficients:
hi(u) := 1 +∑s≥0
hi,su−s−1,
x+i (u) :=
∑s≥0
x+i,su
−s−1, (5.18)
x−i (u) :=∑s≥0
x−i,su−s−1.
51
52
Chapter 6
New Presentation of Y (slm|n)
In this chapter we show how the Yangian Y (slm|n) described by Stukopin [45](and reviewed in Section 5.2) is isomorphic to a Hopf subalgebra of the Y (glm|n).We construct this isomorphism explicitly and examine the structure of the algebraY (glm|n). We also consider the case where m = n and define Yangians Y (sln|n) andY (psln|n). We justify our definition by showing that the Yangian Y (psln|n) is anhomogeneous quantization of U(psln|n[w]) with trivial centre.
6.1 New presentation of Y (slm|n)
We define the Yangian Y (slm|n) associated with the special linear Lie superalgebraas the following subalgebra of Y (glm|n):
Y (slm|n) := y ∈ Y (glm|n) | µf (y) = y for all f ,
where we take µf as defined as in [38]. In other words, for a formal power series
f = 1 + f1u−1 + f2u
−2 + . . . ∈ C[[u−1]],
the map µf is the automorphism of Y (glm|n) given by
µf : T (u) 7→ f(u)T (u).
This is justified by analogy with the definition of the Yangian Y (slN) as a subalge-bra of the Yangian Y (glN) in [38]. This definition is intended for both equal andinequal values of m and n.
53
Proposition 6.1.1. Let Zm|n denote the centre of the Yangian Y (glm|n). Then for m 6= n,we have
Y (glm|n) ∼= Zm|n ⊗ Y (slm|n).
Proof. We assume that m > n. (The result for n < m follows from this by theapplication of the map ζ). This proof is very similar to that of Proposition 2.16in [38]. We use the fact, stated there, that for any commutative associative algebraA and any formal series,
a(u) = 1 + a1u−1 + a2u
−2 + . . . ∈ A[[u−1]],
and any positive integer K, there exists a unique series
a(u) = 1 + a1u−1 + a2u
−2 + . . . ∈ A[[u−1]]
such thata(u) = a(u)a(u− 1) · · · a(u−K + 1). (6.1)
We set a(u) = bm|n(u) andK = m−n in the commutative subalgebra Y 0 ⊂ Y (glm|n)
generated by the elements d(r)i for i = 1, . . . ,m+ n and r ≥ 1. Write
bm|n(u) = b(u)b(u− 1) · · · b(u−m+ n+ 1).
By the definition of the map µf we have that
µf (bm|n(u)) = f(u)f(u− 1) · · · f(u−m+ n+ 1)bm|n(u).
It follows from the uniqueness of the expansion (6.1) that µf (b(u)) = f(u)b(u)
for all f . Also, the coefficients bk, (k ≥ 1) of the series b(u) generate the centreZm|n since we may recover the coefficients of the series bm|n(u) from them. Theremaining parts of the proof are exactly the same as in [38].
Lemma 6.1.2. For any m,n ≥ 0, the coefficients of the series
d1(u)−1di+1(u), ei(u), fi(u), for 1 ≤ i ≤ m+ n− 1, (6.2)
generate the subalgebra Y (slm|n).
Proof. It is clear that the coefficients of the series d1(u) together with those of theseries listed above generate the Yangian Y (glm|n). Also, or any f , the map µf leavesthe coefficients of the series in (6.1.2) fixed and maps µf (d1(u)) = f(u)d1(u). Bythe Poincare-Birkhoff-Witt theorem, any element P of Y (glm|n) is a polynomialin d
(1)1 , d
(2)1 , d
(3)1 , . . . and the other generators that are fixed by µf for all f . We
can assume further that in each monomial in P the generators are ordered so that
54
the f (r)i ’s come before the d(r)
i ’s, which come before the e(r)i ’s. Suppose that P ∈Y (slm|n) and that R is the maximum r such that d(r)
1 occurs in P , and K is themaximum power of d(r)
1 occurring in P for any r. Fix f = 1 + λu−R, where λ is anarbitrary nonzero complex number. Then we can write:
P =∑
a1,a2,...,aR
FaDa
(d
(1)1
)a1(d
(2)1
)a2
. . .(d
(R)1
)aR
Ea,
where Fa, Da and Ea are monomials in the generators fixed by µf , and we sumover all R-tuples a = (a1, a2, . . . , aR) of positive integers not exceeding K. Then
µf (P ) =∑
a1,a2,...,aR
FaDa
(d
(1)1
)a1(d
(2)1
)a2
. . .(d
(R)1 + λ
)aR
Ea = P.
By the linear independence of the different monomials and the fact that λ is anarbitrary complex number, we see that in fact d(R)
1 cannot occur in P ∈ Y (slm|n).
6.2 Isomorphism Between the Two Presentations
In the case where m 6= n we have two definitions for the Yangian associated to theLie superalgebra slm|n. Here we show that they are equivalent. We define a map
φ : Y (slm|n) → Y (slm|n)
by
φ(hi(u)) = di(u+ 12(−1)i(m− i) )−1di+1(u+ 1
2(−1)i(m− i) ),
φ(x+i (u)) = fi(u+ 1
2(−1)i(m− i) ) (6.3)
φ(x−i (u)) = (−1)i ei(u+ 12(−1)i(m− i) )
for 1 ≤ i ≤ m+n− 1. We verify that φ is a homomorphism of associative algebrasin Appendix B.
Proposition 6.2.1. The map φ : Y (slm|n) → Y (slm|n) is an associative algebra isomor-phism.
Proof. By Lemma 6.1.2 the homomorphism φ is surjective. We need to show φ isinjective. We do this by constructing a set of monomials that span Y (slm|n), andwhose image under φ is a basis for the Yangian Y (slm|n). Following [34, 45] weconstruct this basis as follows.
55
Let α be a positive root of slm|n and α = αi1 + . . . + αip a decomposition of αinto a sum of roots such that
x±α = [x±i1 , [x±i2, . . . , [x±ip−1
, x±ip ] . . .]]
is a nonzero root vector in slm|n. Suppose s > 0 and we have a decompositions = s1 + . . . + sp of s into p non-negative integers. Then define the root vectorx±α,s1+...+sp
in the Yangian Y (glm|n) by
x±α,s1+...+sp= φ([x±i1,s1
, [x±i2,s2, . . . , [x±ip−1,sp−1
, x±ip,sp] . . .]]). (6.4)
With respect to the second filtration defined in (2.3), the degree of an elementhi,s or x±i,s is equal to its second index s, and deg2(x
±α,s1+...+sp
) = s. If s = s′1+ . . .+s′pis another decomposition of s into non-negative integers, then (since the definingrelations in Proposition 5.2.1 are satisfied by the elements of the Yangian) we have
deg2(x±α,s′1+...+s′p
− x±α,s1+...+sp) ≤ s− 1. (6.5)
Now for each s > 0 fix the decomposition s = 0 + . . . + 0 + s to be used alwaysand write x±α,s = x±α,0+...+0+s. Also any positive root α is just α = εi − εj for some1 ≤ i ≤ j − 1 ≤ m + n − 1. We then write: x±i,j;s = x±α,0+...+0+s. Now choose anytotal ordering ≺ on the set
x−i,j;q, hi,r, x+i,j;s | 1 ≤ i ≤ j − 1 ≤ m+ n− 1, q, r, s > 0
and define Ω(≺) to be the set of ordered monomials in these elements, where theodd elements (x±i,j;r with i ≤ m but j > m) occur with power at most 1.
Define the length l(M) of a monomial in x−i,j;q, hi,r, x+i,j;s as the number of fac-
tors of M and note that by the relations in Proposition 5.2.1, if we rearrange thefactors ofM , then we obtain additional terms of either smaller degree, or the samedegree but smaller length. Then by induction on the degree d of a polynomial,and for fixed degree d, induction on the maximal length of its terms, we see thatY (slm|n) is spanned by the elements of Ω(≺). (This argument is given in [34] forthe Yangian Y (slN)).
Now suppose that some linear combination Σ of the monomials in Ω(≺) isequal to 0, and that the highest degree of a monomial term in Σ is r. The degreer part of Σ must be equal to zero. This will be the sum of products of the highestdegree parts of elements x−i,j;r, hi,r, x
+i,j;r, which by the isomorphism gr2Y (slm|n) ∼=
U(slm|n[x]) get mapped to the elements
ε−i,j Eijxr−1, (−1)iEii − (−1)i+1Ei+1,i+1; ε+
i,j Ejixr,
56
respectively, where ε±i,j is some power of −1. Together these elements form ba-sis for slm|n[x], and so by the Poincare-Birkhoff-Witt theorem for Lie superalge-bras [44], the set of ordered monomials in them, containing powers of at most oneof the odd elements, are linearly independent. This implies that the highest degreepart of Σ must in fact be trivial. Thus Ω(≺) is a basis for Y (slm|n).
Now, we define a set Ω(≺) in Y (slm|n) by the same formulas as in (6.4), ex-cept now we take the symbols to represent the elements of Y (slm|n). We definea filtration on Y (slm|n) by setting the degree of an element hi,s or x±i,s equal to itssecond index s. All the arguments required to show that Ω(≺) span the Yangiandepended only on the relations in Proposition 5.2.1, and thus hold true for Ω(≺)in Y (slm|n). Then Ω(≺) is a set of monomials that span Y (slm|n), and whose imageunder φ, Ω(≺), is a basis for Y (slm|n).
Now that we have established an isomorphism, we will identify the elementsin the presentation 5.2.1 with their image in Y (glm|n) under φ.
Remark 6.2.1. This proof does not rely upon the fact that slm|n is simple. Thus wemay generalize it to the case where m = n. In this case we have that the YangianY (sln|n) is still given by the presentation in Proposition 5.2.1.
Next we calculate the restriction of the coproduct defined in §2.2 in terms ofthe Stukopin generators, to see if it agrees with the coproduct given in §5.2. Wewrite λi = 1
2(−1)i(m− i).
Theorem 6.2.1. For m 6= n we have that ∆ = ∆.
Proof. We easily derive the following:
hi,0 = t(1)i+1,i+1 − t
(1)ii , (6.6)
hi,1 = t(2)i+1,i+1 − t
(2)ii − λi(t
(1)i+1,i+1 − t
(1)ii ) (6.7)
+i∑
j=1
t(1)ij t
(1)ji −
i∑j=1
t(1)i+1,jt
(1)j,i+1 − t
(1)ii t
(1)i+1,i+1, (6.8)
x+i,0 = t
(1)i+1,i, (6.9)
x−i,0 = (−1)it(1)i,i+1, (6.10)
Then we immediately verify that (5.6) and (5.7) holds true for the coproduct de-fined in Section 2.2. Now we need only show that ∆(hi,1) = ∆(hi,1), and then thefact that ∆(x±i,1) = ∆(xi,1) will follow from the fact that ∆ is an algebra map. We
57
find:
∆(hi,1) = hi,1 ⊗ 1 + 1⊗ hi,1 + hi,0 ⊗ hi,0
−τ(i∑
k=1
t(1)i+1,k ⊗ t
(1)k,i+1) +
m+n∑i+2
t(1)i+1,k ⊗ t
(1)k,i+1
+τ(i−1∑k=1
t(1)i,k ⊗ t
(1)ki )−
m+n∑k=i+1
t(1)ik ⊗ t
(1)ki ;
= −((−1)i + (−1)i+1)x−i,i+1;0 ⊗ x+i,i+1;0]
−∑k<i
(−1)i+1x−k,i+1;0 ⊗ x+k,i+1;0
+∑
k>i+1
(−1)i+1x−i+1,k;0 ⊗ x+i+1,k;0
+∑k<i
(−1)ix−k,i;0 ⊗ x+k,i;0
−∑
k>i+1
(−1)ix−i,k;0 ⊗ x+i,k;0,
Here we have consistently defined x−i,j;0 := (−1)it(1)ij and x+
i,j;0 := t(1)ji for i < j. Now
it is clear that ∆(hi,1) = ∆(hi,1) and the result follows.
So long as slm|n is simple (i.e., so long as m 6= n), our coproduct (as does anyHopf algebra coproduct) induces a super Lie bialgebra structure
δ : slm|n → slm|n ⊗ slm|n
given byδ = (∆− τ ∆). (6.11)
6.3 The Yangian Y (psln|n)
We define the Yangian of the projective special linear Lie superalgebra psln|n as thefollowing quotient:
Y (psln|n) := Y (sln|n)/⟨bn|n(u) = 1
⟩= Y (sln|n)/B, (6.12)
where B is the ideal in Y (sln|n) generated by the coefficients b1, b2, . . . of the quan-tum Berezinian. This definition is justified to a certain extent by Proposition 6.3.2
58
below. The Yangian Y (psln|n) is given by the presentation [46], for the case wherem = n, but with the additional relation
bn|n(u) = 1,
where bn|n(u) is the quantum Berezinian.
Lemma 6.3.1. For n > 1 the centre of U(psln|n[x]) is trivial.
Proof. We follow the argument of Lemma 4.1.1 using the properties of the root-space decomposition given in [31].
Proposition 6.3.2. The centre of the Yangian Y (psln|n) is trivial.
Proof. We show that gr2Y (sln|n) ∼= U(sln|n[x]), and that
gr(Y (psln|n)) ∼= U(psln|n).
Then the result follows from Lemma 6.3.1. Here the filtration on Y (psln|n),
C = A−1 ⊂ A0 ⊂ A1 ⊂ . . . ⊂ Ai ⊂ . . . ,
is defined by setting Ai = Yi + B where Yi is the set of elements a ∈ Y (sln|n) withdeg2(a) ≤ i, and gr(Y (psln|n)) is the corresponding graded algebra.
The restriction of the map in Corollary 2.5.1 to gr2Y (sln|n) is injective ontoits image in U(sln|n). By Lemma 6.1.2, this is the image of the coefficients of theseries d1(u)
−1di+1(u), ei(u) and fi(u), for i = 1, . . . , 2n− 1. Now, for any r ≥ 1, thecoefficients of u−r these series are, respectively:
t(r)i+1,i+1 − t
(r)11 + elements of lower degree,
t(r)i,i+1 + elements of lower degree,
t(r)i+1,i + elements of lower degree.
The image of these elements in U(gln|n[x]) is:
(−1)i+1Ei+1,i+1xr−1 − E11x
r−1,
(−1)iEi,i+1xr−1,
(−1)i+1Ei+1,ixr−1.
These elements generate precisely the subalgebra U(sln|n[x]). Thus we find that
gr2Y (sln|n) ∼= U(sln|n[x]).
59
The natural projection map p : Y (sln|n) → Y (psln|n) satisfies p(Yi) ⊂ Ai, and thusgives a natural surjective mapping
gr2Y (sln|n) ∼= U(sln|n[x]) → grY (A(n− 1, n− 1)),
with kernel the ideal I = 〈I, Ix, Ix2, . . .〉 ⊂ U(sln|n[x]). Then
grY (psln|n) ∼= U(sln|n[x])/I ∼= U(psln|n[x]).
Corollary 6.3.3. For n > 1, the centre of the subalgebra Y (sln|n) is generated by thecoefficients of the quantum Berezinian bn|n(u).
6.3.1 The Hopf structure on Y (psln|n)
Lemma 6.3.4. [40] In the Yangian Y (glm|n),
∆(bm|n(u)) = bm|n(u)⊗ bm|n(u),
S(bm|n(u)) = bm|n(u)−1,
ε(bm|n(u)) = 1.
Proof. By Theorem 4.1
bm|n(u) = Cm(u) ζ(Cn(u−m+ n)) ,
where Cm(u) is the image of the quantum determinant under the natural inclusionY (glm) → Y (glm|n) and Cn(u) is the image of the quantum determinant under thenatural inclusion Y (gln) → Y (gln|m). It is well-known [10] that
∆(Cm(u)) = Cm(u)⊗ Cm(u).
Furthermore, since Cn(u−m+ n) is even and ζ is an even linear map we have byProposition 3.3.3 that
∆(ζ(Cn(u−m+ n))) = ζ(Cn(u−m+ n))⊗ ζ(Cn(u−m+ n)).
Then
∆(bm|n(u)) = ∆(Cm(u))∆(ζ(Cn(u−m+ n)))
= Cm(u)ζ(Cn(u−m+ n))⊗ Cm(u)ζ(Cn(u−m+ n))
= bm|n(u)⊗ bm|n(u).
60
As a consequence of Lemma 6.3.4 the quotient algebra Y (psln|n) has a well-defined Hopf superalgebra structure given by the Hopf superalgebra structure ofY (sln|n). Also, since bm|n(u) is even, we have that
(∆− τ ∆)(bm|n(u)) = 0.
This Hopf superalgebra coproduct induces a well-defined super Lie bialgebra co-product on U(psln|n[w]), defined by the same formula (6.11), but where by ∆ wemean the coproduct on Y (psln|n) acting on cosets of the ideal generated by thecoefficients of bn|n(u).
Proposition 6.3.5. With this Hopf superalgebra structure, the Yangian Y (psln|n) is anhomogeneous quantization of the super Lie bialgebra U(psln|n[w]) with the super Lie bial-gebra structure as defined in Example 5.1.1.
Proof. We need to show that (∆−∆opp) agrees with δ, where
δ(awr) = (ad(a)⊗ 1)(t)wr
1 − wr2
w1 − w2
,
and t is the Casimir element (see Section A.9). It is enough to show this on thegenerators hi,0, x
±i,0, and hi,1 because then it will follow for the rest of the algebra
by part (iv) of Definition 5.1.2. For hi,0 and x±i,0 both δ and (∆−∆opp) are identicallyzero. For hi,1 the result is easy to show using the calculation of ∆(hi,1) in the proofof Theorem 6.2.1.
Example 6.3.1. As an example we consider the Lie superalgebras sl2|2 and psl2|2.The Cartan matrix in this case where m = 2 and n = 2 is the following:
A =
2 −1 0−1 0 10 1 −2
.
We have a presentation of sl2|2 with generators
h1, h2, h3, x±1 , x
±2 , x
±3 ,
(where x±2 are the only odd generators) and the following relations:
[hi, hj] = 0;
[x+i , x
−j ] = δi,jhi;
[hi, x±j ] = ±aijx
±j ;
[x±1 , x±3 ] = 0;
[x±i , [x±i , x±j ] ] = 0, if |i− j| = 1;
[x±2 , x±2 ] = 0,
[[x±1 , x±2 ] , [x±2 , x
±3 ]] = 0, (6.13)
61
for i, j = 1, 2, 3. A presentation of psl2|2 is obtained from by adding the relation:
h1 + 2h2 + h3 = 0. (6.14)
We define
x±13 := ±[x±1 , x±2 ], x±14 := ±[x±13, x
±3 ], x±24 := ±[x±2 , x
±3 ].
The Casimir element is:
t = −h2 ⊗ h1 + (−h1 − 2h2)⊗ h2
+x−1 ⊗ x+1 + x+
1 ⊗ x−1 + x−2 ⊗ x+2 − x+
2 ⊗ x−2 + x−3 ⊗ x+3 + x+
3 ⊗ x−3+x−13 ⊗ x+
13 − x+13 ⊗ x−13 − x−14 ⊗ x+
14 + x+14 ⊗ x−14 − x−24 ⊗ x+
24 + x+24 ⊗ x−24.
Together the elementsh1, h2, x
±1 , x
±3 , x
±2 , x
±13, x
±14x
±24
form a basis for sl2|2. The natural representation ρ : sl2|2 → End(C2|2) may be givenas follows:
ρ(h1) = E11 − E22, ρ(h2) = E22 + E33, ρ(h3) = −E33 + E44,
ρ(x+1 ) = E12, ρ(x−1 ) = E21, ρ(x+
2 ) = E23, ρ(x−2 ) = E32,
ρ(x+3 ) = E34, ρ(x−3 ) = E43, ρ(x+
13) = E13, ρ(x−13) = E31,
ρ(x+14) = E14, ρ(x−14) = −E41, ρ(x+
24) = E24, ρ(x−24) = −E42.
Now the Yangian Y (sl2|2) may be considered either as the algebra Y (sl2|2)given by the presentation given by Proposition 5.2.1 or as the subalgebra Y (sl2|2)of Y (gl2|2) defined in Section 6.1, generated by the coefficients of the quasideter-minants di(u)
−1di+1(u), ei(u) and fi(u) where i = 1, 2, 3. The two presentations arerelated by the isomorphism φ : Y (sl2|2) → Y (sl2|2) given by:
φ(h1(u)) = d1(u+ 12)−1d2(u+ 1
2),
φ(h2(u)) = d2(u)−1d3(u),
φ(h3(u)) = d3(u+ 12)−1d4(u+ 1
2),
φ(x+1 (u)) = f1(u+ 1
2),
φ(x−1 (u)) = e1(u+ 12),
φ(x+2 (u)) = f2(u),
φ(x−2 (u)) = e2(u),
φ(x+3 (u)) = f3(u+ 1
2),
φ(x−3 (u)) = −e3(u+ 12).
62
We identify Y (sl2|2) with the subalgebra of Y (gl2|2) generated by these elements.The quantum Berezinian is the formal power series:
b2|2(u) = d1(u)d2(u− 1)d3(u− 1)−1d4(u− 1)−1
= h1(u− 12)−1h2(u)
−1h2(u− 1)−1h3(u− 12)−1.
A presentation for Y (psl2|2) is obtained from either of the two mentioned in Exam-ple 6.3.1 above by setting the quantum Berezinian b2|2(u) equal to 1. Note that byconsidering the coefficients of u−1 in the equation
b2|2(u)−1 = 1
we obtain (6.14). Further relations between the elements hi,s may be obtained byconsidering coefficients of higher powers.
We show explicitly that the Hopf algebra structure induced on Y (psl2|2) bythat on the larger Yangian Y (gl2|2) agrees with the natural co-Poisson structure onU(psl2|2[w]) (see (5.5)). We need to show that:
(∆− τ ∆)(h′i,1) = [hi,0 ⊗ 1, t] . (6.15)
The left-hand side of (6.15) depends only on the terms in h′i,1 with deg2 ≥ 1, andso it is enough to consider hi,1 instead of h′i,1. Now,
h1(u) = d1(u+ 12)−1d2(u+ 1
2)
= t11(u+ 12)−1
∣∣∣∣∣ t11(u+ 12) t12(u+ 1
2)
t21(u+ 12) t22(u+ 1
2)
∣∣∣∣∣ .
So the coefficient of u−2 is:
h1,1 = t(2)22 − t
(2)11 + (terms of lower degree),
63
(see also (6.8)). Then by (2.2),
(∆− τ ∆)(h1,1) = t(1)21 ⊗ t
(1)12 + t
(1)23 ⊗ t
(1)32 + t
(1)24 ⊗ t
(1)42
−t(1)12 ⊗ t
(1)21 + t
(1)32 ⊗ t
(1)23 + t
(1)42 ⊗ t
(1)24
−t(1)12 ⊗ t
(1)21 − t
(1)13 ⊗ t
(1)31 − t
(1)14 ⊗ t
(1)41
+t(1)21 ⊗ t
(1)12 − t
(1)31 ⊗ t
(1)13 − t
(1)41 ⊗ t
(1)14
= −2x−1,0 ⊗ x+1,0 + 2x+
1,0 ⊗ x−1,0 + x−2,0 ⊗ x+2,0 + x+
2,0 ⊗ x−2,0
−x−13,0 ⊗ x+13,0 − x+
13,0 ⊗ x−13,0 + x−14,0 ⊗ x+14,0 + x+
14,0 ⊗ x−14,0
−x−24,0 ⊗ x+24,0 − x+
24,0 ⊗ x−24,0,
= [h1,0, x−1,0]⊗ x+
1,0 + [h1,0, x+1,0]⊗ x−1,0
+[h1,0, x−2,0]⊗ x+
2,0 − [h1,0, x+2,0]⊗ x−2,0
+[h1,0, x−13,0]⊗ x+
13,0 − [h1,0, x+13,0]⊗ x−13,0
+[h1,0, x−14,0]⊗ x+
14,0 − [h1,0, x+14,0]⊗ x−14,0
+[h1,0, x−24,0]⊗ x+
24,0 − [h1,0, x+24,0]⊗ x−24,0,
= [h1,0 ⊗ 1, t],
where here we have identified the universal enveloping algebra U(sl2|2) with asubalgebra of Y (gl2|2). Thus we have verified in this particular case that the Liesuper bialgebra coproduct induced by ∆ is the same as that given in Example5.1.1. The general case is a similar but more involved.
64
Chapter 7
Conclusion
Now we have completed our study of the Yangian Y (glm|n). I hope that this thesisprovides a helpful description of its algebraic structure.
Following the work of Brundan and Kleshchev and others [6, 39] we haveproved its Poincare-Birkhoff-Witt theorem and constructed a new presentationusing the Gauss decomposition of the matrix T (u). We have also confirmed thedescription of its centre conjectured by Maxim Nazarov and clarified the proof ofthe fact that the quantum Berezinian is central.
Prior to this thesis, there were two approaches to Yangians of Lie superalge-bras in the literature - that involving the RTT presentation, and that constructedby Vladimir Stukopin [46, 45]. Now we have established the connection betweenthese two approaches. In particular, for m 6= n we have the result that
Y (glm|n) ∼= Zm|n ⊗ Y (slm|n).
This implies that the representation theory of Y (slm|n) can be developed from therepresentation theory of Y (glm|n) developed in [52]. Finally, we have consideredthe case where m = n and defined Yangians Y (sln|n) and Y (psln|n).
Nevertheless, we have left some questions unanswered. In particular it washoped that the new form of the quantum Berezinian given in Theorem 4.1 mightlead to a more straightforward proof of the quantum analogue of the Liouvilleformula [40]. However, no such proof was found. Also, the connection with ourresults and the Yangian in [14] remains to be determined.
65
66
Appendix A
Superalgebras
Superalgebras are generalizations of algebras associated with the category of su-per vector spaces rather than vector spaces. Good introductions to the theory ofLie superalgebras are given by [12, 31, 44, 49], and there is also a dictionary of Liesuperalgebras [19]. In this section we summarise the facts about Lie superalgebrasthat will be needed in this thesis.
A.1 Basic definitions
A super vector space is a Z2-graded vector space, i.e., a vector space
V = V0 ⊕ V1,
such that Vi Vj ⊆ Vi+j(mod2). The vectors in V0 are called even and the vectors inV1 are called odd. An homogeneous vector in V is one that is either even or odd(rather than a linear combination of both even and odd vectors). We define aparity function on the set of homogeneous vectors of V ,
V0 ∪ V1 → Z2
a 7→ a.
by letting a = 0 if a is even and a = 1 if a is odd.
Definition A.1.1. The tensor product of two vector spaces V and W is the tensorproduct of the underlying vector spaces, with the Z2-grading given by:
(V ⊗W )k =⊕
i+j=k(mod 2)
Vi ⊗Wj.
67
Maps between Z2-graded objects are called even if they preserve parity, andare called odd if they reverse parity. Thus, we call a linear map A : V → V even ifit maps V0 into V0 and V1 into V1. We call A odd if it maps V0 into V1 and V1 into V0.Then the set of all linear maps on V is itself a Z2-graded vector space, which maybe denoted End V .
Definition A.1.2. An associative superalgebra is a super vector space V , with aneven linear map m : V ⊗ V → V such that m is associative and the algebra has aone or unit element. The associative property means that the following diagramcommutes:
A⊗ A⊗ A
A⊗ A
A⊗ A
A
µ⊗id<<xxxxxxx
id⊗µ ##FFFF
FFF
µ
!!CCCC
CCC
µ
<<yyyyyyy
.
The unit element can be interpreted as saying that there is an even linear mapι : C → A such that the following diagrams commute
A
A⊗ C
A
A⊗ A
A
C⊗ A
A
A⊗ AOO∼=
OO
µ
id //
id⊗ι //OO
∼=
OO
µ
ι⊗id //
id //
.
The field C is regarded as Z2-graded in the trivial way, where every element iseven.
The super vector space EndV is an associative superalgebra with multiplica-tion given by composing linear maps. The identity map is the unit element.
Definition A.1.3. A Lie superalgebra is a super vector space g = g0 ⊕ g1 with aneven linear map [ , ] : g⊗ g → g satisfying the following axioms:
[a, b] = −(−1)a b[b, a]
[a, [b, c]] = [[a, b], c] + (−1)ab[b, [a, c]]
for all a, b, c ∈ g.
We can define a Lie super-bracket on the vector space EndV by setting:
[A,B] = AB − (−1)A BBA,
for all homogeneous A,B ∈ EndV . This Lie superalgebra is denoted gl(V ) andcalled the general linear Lie superalgebra on V (see §A.3).
68
Definition A.1.4. A representation of a Lie superalgebra g is a super vector spaceV with a Lie algebra homomorphism (that is, an even linear map that respects thesuper-bracket) ρ : g → gl(V ).
The adjoint representation of a Lie superalgebra g on its own underlying vectorspace is defined by
ad(A) ·B = [A,B]
for all A,B ∈ g.
A.2 The Rule of Signs
The basic rule of thumb when performing calculations with super-objects is thatwhenever the order in which two odd things appear is changed, a minus signappears [12]. We have seen examples of this in the definition of Lie superalgebrasabove, and now we give two more.
If A and B are two associative superalgebras, then we may define an associa-tive multiplication on the super vector space A⊗B by:
(a1 ⊗ b1)(a2 ⊗ b2) = (−1)b1a2 a1a2 ⊗ b1b2,
for all a1, a2 ∈ A and all b1, b2 ∈ B. Notice how in going from the left to theright-hand side, the order in which the elements a2 and b1 appear is swapped.
IfA is an associative superalgebra, then anA-module is a vector space V witha linear map π : A→ End V such that
π(ab)v = π(a)(π(b)v) and π(1)v = v
for all a, b ∈ A and all v ∈ V . As usual we use the notation a · v to denoteπ(a)(v). Let A and B be two associative superalgebras, and suppose we havean A-module V and a B-module W . Then the tensor product V ⊗W is an A⊗ B-module with the action given by:
(a⊗ b) · (v ⊗ w) = (−1)b v a · v ⊗ b · w,
for a ∈ A, b ∈ B, v ∈ V, and w ∈ W .
A.3 The Lie Superalgebra glm|n
Let Cm|n be the complex super vector superspace with even part of dimension mand odd part of dimension n.
69
Definition A.3.1. The general linear Lie algebra glm|m is the Z2-graded vector spaceof all linear maps Cm|n → Cm|n, with the Lie super-bracket:
[A,B] = AB − (−1)A BBA.
We fix an homogeneous basis v1, . . . , vm, vm+1, . . . , vm+n of V , where we takevi to be even and if i ≤ m, and take vi to be odd if i ≥ m + 1. It is convenient otassign a parity to the indices themselves: let i = 0 if i ≤ m and i = 1 if i ≥ m + 1.Then the Lie superalgebra glm|n is generated by the unit matricesEij , with definingrelations:
[Eij, Ekl] = δkjEil − (−1)(i+j)(k+l) δilEkj. (A.1)
A.4 The Lie Superalgebras slm|n and psln|n
Now let X be an arbitrary matrix in glm|n that breaks into homogeneous blocksaccording to the decomposition:
X =
(A BC D
);
(so A is m×m, B is m× n, C is n×m and D is n× n). We define the supertrace ofX to be:
str(X) := tr(A)− tr(D).
Since we know from linear algebra that the trace of a linear operator does not de-pend on the basis with respect to which it is expressed, and A and D are evenlinear operators, we have that the supertrace also does not depend on the homo-geneous basis chosen for V , and is thus a well-defined linear map str : glm|n → C.We also have that:
str([A,B]) = 0
for all A,B in glm|n, and so the subspace
slm|n = X ∈ glm|n| str(X) = 0
is a Lie sub-superalgebra of glm|n with codimension 1. It is called the special linearLie superalgebra.
When m 6= n, the Lie superalgebra slm|n is a simple Lie superalgebra. Onthe other hand, sln|n contains a one-dimensional ideal, namely that spanned bythe identity matrix. The Lie superalgebra obtained by quotienting out this ideal issimple so long as n ≥ 1. We write:
psln|n := sln|n/CI,
70
for this Lie algebra. It is called the projective special linear Lie superalgebra.
A simple Lie superalgebra g = g0 ⊕ g1 is called classical if the representationof the even subalgebra g0 on the odd part g1 is completely reducible. A simple Liesuperalgebra is classical if and only if the even part g0 is reductive [19]. A classicalLie superalgebra is called basic if there is a non-degenerate invariant bilinear formon g.
Kac [31] classified all the finite-dimensional classical Lie superalgebras, byproving that they all belong to the following list: An, Bn, . . . , E8 (the Lie super-algebras with no odd part), A(m,n), B(m,n), C(n), D(m,n), D(2, 1;α), F (4), G(3),P (n), Q(n). The family A(m,n), is defined by:
A(m,n) := slm+1|n+1 when m 6= n, and m,n ≥ 0
A(n, n) := psln+1|n+1 for n ≥ 0.
Thus it is the family containing the classical special linear and projective speciallinear Lie superalgebras. The family also includes ‘real forms’ of the Lie superal-gebra of type A(m,n) which are denoted sum|n. The Lie superalgebras A(m,n) areall basic (§8 [19]).
A.5 Root Space Decomposition
A Cartan subalgebra of a Lie superalgebra g = g0 ⊕ g1 is by definition a Cartansubalgebra of the even Lie algebra g0 [31]. If h is a Cartan subalgebra of the Liesuperalgebra g, then we can simultaneously diagonalize all adH for H ∈ h. Thisgives the root space decomposition:
g =⊕α∈h∗
gα
wheregα = A ∈ g | [H,A] = α(H)A for all H ∈ h
Elements of the set ∆ := α ∈ h∗ | gα 6= 0 are called roots. Important properties ofthe root space decomposition are given in §40 of [19] and §2.5.3 of [31].
A.6 Cartan Matrices
The Lie superalgebra slm|n may be defined explicitly by the following presentation([25, 43]). We have generators hi, x+
j , x−j | 1 ≤ i ≤ m + n − 1, where the
71
generators x±m are declared odd and all other generators are even. The definingrelations are:
[hi, hj] = 0;
[x+i , x
−j ] = δi,jhi;
[hi, x±j ] = ±aijx
±j ;
[x±i , x±j ] = 0, if |i− j| > 1;
[x±i , [x±i , x±j ] ] = 0, if |i− j| = 1;
[x±m, x±m] = 0,
[ [x±m−1, x±m] , [x±m+1, x
±m]] = 0, (A.2)
for all possible i, j, where A = (aij)m+n−1i,j=1 is the (m+ n+ 1)× (m+ n+ 1)-matrix:
A =
mth column↓
mthrow →
2 −1 0...
−1 2. . . . . .
...
0. . . . . .
...2 −1
· · · · · · · · · −1 0 1
1 −2. . .
. . . . . . 11 −2
·
.
Notice that the presentation of the Lie superalgebra slm|n has an unexpected extrarelation (A.2) that has no counterpart in the theory of Lie algebras.
In the case where m = n, a presentation for psln|n may be obtained by addingthe relation
n∑j=1
jhj +n−1∑j=1
jhn+j = 0. (A.3)
The matrix A is called the Cartan matrix of the Lie superalgebras slm|n andpsln|n. It is possible to define the Lie superalgebras slm|n and psln|n using the samepresentation but with slightly different Cartan matrices (see §2 [19]) but in thisthesis, we always use this symmetric Cartan matrix shown above.
72
A.7 The Killing Form
Definition A.7.1. The Killing form of a Lie superalgebra g is the bilinear form
κ : g× g → C
defined byκ(a, b) = str(ad(a)ad(b)), (A.4)
for all a, b ∈ g.
The Killing form κ of g = slm|n (for m 6= n, m,n ≥ 1) has the followingproperties ([19, 31]):
1. κ is consistent:κ(a, b) = 0 for all a ∈ g0 and b ∈ g1.
2. κ is supersymmetric:
κ(a, b) = (−1)a b κ(b, a).
3. κ is invariant:κ([a, b], c) = κ(a, [b, c]), for all a, b, c ∈ g.
4. κ is non-degenerate, which means that if for some a ∈ g, we have κ(a, b) = 0for all b ∈ g, then a = 0.
The Killing form on psln|n vanishes. However, for all basic Lie superalgebras g onecan define a consistent, supersymmetric, invariant, non-degenerate bilinear form( , ) : g× g → C by:
(eαi, e−αj
) = δij (eαi, e−αi
), (A.5)(hi, hj) = aij (eαi
, e−αj), (A.6)
where aij are the entries of the Cartan matrix, αi, αj are the corresponding simpleroots, and eαi
, eαjare the corresponding root vectors (see [19]). It is easy to show
that any two invariant bilinear forms on a Lie superalgebra are proportional. Thus( , ) and κ are the same up to scalar multiples except in cases such as psln|n wherethe Killing form vanishes.
73
A.8 Universal Enveloping Algebras
If g is a Lie superalgebra, then its universal enveloping algebra U(g) (see [44]) is theunique pair (U, i), such that U is an associative algebra with 1 and i : g → U is ahomomorphism satisfying
i([x, y]) = i(x) i(y)− (−1)x yi(y) i(x) (A.7)
for any homogeneous elements x, y ∈ g, as well as the following universal prop-erty. For any associative algebraAwith 1 and linear mapping φ : g → A satisfying
φ([x, y]) = φ(x)φ(y)− (−1)x yφ(y)φ(x)
for all homogeneous elements x , y ∈ g, there exists a unique homomorphism
ψ : U → A
(mapping 1 to 1) such that φ = ψ i.
We make use of the Poincare- Birkhoff-Witt theorem for Lie superalgebras[44, 49], which may be stated as follows:
Theorem A.8.1. Let g be a Lie superalgebra and let σ be the canonical mapping of g
into its universal enveloping algebra U(g). Suppose we are given a basis Xii∈I for g ofhomogeneous elements such that the index set I is totally ordered. Then the set of products
σ(Xi1)σ(Xi2) · · ·σ(Xir)
where (i1, . . . , ir) runs through all finite sequences in I such that
i1 ≤ i2 ≤ . . . ≤ ir
and ip < ip+1 when Xip and Xip+1 are both odd, is a basis for the vector space U(g). Inparticular,
U(g) ∼= U(g0)⊗ Λ(g1).
A.9 Casimir Elements
Let g be a Lie superalgebra with an essentially unique consistent, invariant, super-symmetric, non-degenerate bilinear form ( , ) as defined for the Lie superalgebrasslm|n and psln|n in Section A.7. Suppose we have a basis v1, . . . , vk. The corre-sponding dual basis v1, . . . , vk with respect to ( , ) satisfies (vi, vj) = δij . Thenthe Casimir element t is defined in the same way as for Lie algebras by:
t =k∑
i=1
xixi ∈ U(g).
74
The Casimir element is a central element of U(g), by essentially the same proof asin the Lie algebra case (see §6.2 of [28]).
It may prove helpful to note the following. For the Lie superalgebra slm|n,with m 6= n the dual vectors hj for 1 ≤ j ≤ m are given by:
h1 = (1− 1m−n
)h1 + (1− 2m−n
)h2 + (1− 3m−n
)h3 + . . .+ (1− mm−n
)hm
+(1− m−1m−n
)hm+1 + (1− m−2m−n
)hm+2 + . . .− 1m−n
hm+n−1,
h2 = 2h1 − h1,
h3 = 3h1 − 2h1 − h2,...
hm = mh1 − (m− 1)h1 − (m− 2)h2 − . . .− hm−1,
and the dual vectors hj for m ≤ j ≤ m+ n− 1 are given by:
hm+n−1 = h1 − (h1 + h2 + . . . hm+n−1);
hm+n−2 = 2hm+n−1 + hm+n−1,
hm+n−3 = 3hm+n−1 + 2hm+n−1 + hm+n−2,...
hm = nhm+n−1 + (n− 1)hm+n−1 + . . .+ hm+1.
For the Lie superalgebra psln|n the dual basis of the Cartan subalgebra is given by:
h1 = − (h2 + 2h3 + . . .+ (n− 1)hn + (n− 2)hn+1 + (n− 3)hn+2 + . . .+ h2n−2) ,
h2 = 2h1 − h1,
h3 = 3h1 − 2h1 − h2,...
hn = nh1 − (n− 1)h1 − . . .− hn−1,
and
h2n−2 = h1 − (h1 + . . .+ h2n−2),
h2n−3 = 2h2n−2 + h2n−2,
h2n−4 = 3h2n−2 + 2h2n−2 + h2n−1,...
hn = (n− 1)h2n−2 + (n− 2)h2n−2 + . . .+ hn+1.
75
A.10 The Symmetric Group Acts on Cm|n ⊗ . . .⊗ Cm|n
Recall that the symmetric group Sk has a presentation with k−1 generators (i, i+1) : 1 ≤ i ≤ k − 1 and the braid relations
(i, i+ 1)2 = 1
((i, i+ 1)(i+ 1, i+ 2))3 = 1
((i, i+ 1)(j, j + 1))2 = 1, for i+ 1 < j. (A.8)
We define an action of Sk on the tensor product Cm|n⊗ . . .⊗Cm|n of k copies of theZ2-graded vector space Cm|n, by
(i, i+ 1) 7→ Pi,i+1.
We verify that this is a well-defined action by showing that the permutation op-erators also obey the braid relations (A.8). For example, letting ej be a set ofhomogeneous basis vectors for Cm|n,
P 2i,i+1(ej1 ⊗ . . .⊗ eji
⊗ eji+1⊗ . . .⊗ ejk
)
= Pi,i+1(ej1 ⊗ . . .⊗ eji+1⊗ eji
⊗ . . .⊗ ejk)(−1)i(i+1)
= (ej1 ⊗ . . .⊗ eji⊗ eji+1
⊗ . . .⊗ ejk).
(Pi,i+1Pi+1,i+2)3(ej1 ⊗ . . .⊗ eji ⊗ eji+1 ⊗ eji+2 . . .⊗ ejk)
= (Pi,i+1Pi+1,i+2)2(ej1 ⊗ . . .⊗ eji+2 ⊗ eji ⊗ eji+1 . . .⊗ ejk)(−1)(i+1)(i+2)+i(i+2)
= Pi,i+1Pi+1,i+2(ej1 ⊗ . . .⊗ eji+1 ⊗ eji+2 ⊗ eji . . .⊗ ejk)(−1)i(i+2)+i(i+1)
= ej1 ⊗ . . .⊗ eji ⊗ eji+1 ⊗ eji+2 . . .⊗ ejk.
Thus we can consider the action of any element of σ ∈ Sk on (Cm|n)⊗k and use thenotation Pσ.
76
Appendix B
Proof that φ is a homomorphism
Here we check that the map φ : Y (slm|n) → Y (slm|n) defined in (6.3) is a homomor-phism. The proof that φ is a homomorphism consists of a set of straightforwardcalculations based on Lemma 3.3.9, but we give it for completeness.
For convenience we write λi = 12(−1)i(m − i), and recall that for any formal
power seriesf(u) = f−1 + f0u
−1 + f1u−2 + . . . ,
we have the identity
1
(u− v)(fi(v)− fi(v)) =
∑r,s≥0
fr+su−r−1v−s−1.
We also define elements in the Yangian Y (slm|n) by setting hi,r = φ(hi,r), x±i,r =
φ(x±i,r) and the corresponding formal power series by (5.18). This should not causeany confusion, since in the calculations that follow we shall be working exclu-sively in the Yangian Y (slm|n). Our aim is to show that these elements satisfy therelations given in Proposition 5.2.1. We take each relation in turn. We have alreadyestablished that the elements hi,r commute with eachother.
(5.9) For i 6= j, this relation is obvious from relation (3.37). Now consider the casewhere i = j 6= m. Then:
[x+i (u), x−i (v)]
= (−1)i[fi(u + λi), ei(v + λi)],
=(−1)i+i+1
(u− v)(di(v + λi)−1di+1(v + λi)− di(u + λi)−1di+1(u + λi)), by (3.37)
=1
(u− v)(hi(v)− hi(u)),
77
and equating coefficients of u−r−1v−s−1 on both sides gives the desired re-sult. (Here we use the fact that since i 6= m the two parities are equal:i = i+ 1). On the other hand, when i = j = m,
[x+m(u), x−m(v)] = [fm(u), em(v)],
= [em(v), fm(u)], since this is the anti-commutator
=(−1)m+1
(v − u)(dm(v)−1dm+1(v)− dm(u)−1dm+1(u)), by (3.37) ,
=1
(u− v)(hm(v)− hm(u)),
which again gives the desired relation.
(5.10-5.12) We consider the positive root vectors first. By relation (3.36), [hi(u), x±j (v)] =
0 unless j = i− 1, j = i or j = i+ 1. When j = i we have:
(u− v)[hi(u), x+
i (v)]
= (u− v)[di+1(u + λi)di(u + λi)−1 , fi(v + λi)
],
= (u− v)di+1(u + λi)[di(u + λi)−1 , fi(v + λi)
]+ (u− v) [di+1(u + λi) , fi(v + λi)] di(u + λi)−1,
= −(u− v)di+1(u + λi)di(u + λi)−1[di(u + λi), fi(v + λi)]di(u + λi)−1,
+(−1)i+1(fi(v + λi)− fi(u + λi))di+1(u + λi)di(u + λi)−1,
= (−1)i+1di+1(u + λi)di(u + λi)−1(fi(v + λi)− fi(u + λi))
+ (−1)i(fi(v + λi)− fi(u + λi))di+1(u + λi)di(u + λi)−1,
= (−1)i+1hi(u)(x+i (v)− x+
i (u)) + (−1)i(x+i (v)− x+
i (u))hi(u).
Now, taking coefficients of u0v−s−1 gives:
[hi,0, x+i,s] = ((−1)i+1 + (−1)i)x+
i,s,
=
2x+
i,s, for 1 ≤ i ≤ m− 1,0, for i = m,
−2x+i,s, for m+ 1 ≤ i ≤ m+ n− 1.
Thus we have verified (5.10)and (5.12) in the case where i = j. Taking coef-ficients of u−r−1v−s−1 for r, s ≥ 0 gives:
[hi,r+1, x+i,s]− [hi,r, x
+i,s+1]
= (−1)i+1hi,rx+i,s + (−1)ix+
i,shi,r,
=
hi,rx
+i,s + x+
i,shi,r for 1 ≤ i ≤ m− 1,0, for i = m,
−(hi,rx+i,s + x+
i,shi,r), for m+ 1 ≤ i ≤ m+ n− 1.
78
From this we see (5.11) for the case i = j 6= m directly. For i = j = m we use(5.15) and apply the identity above recursively.
Note for any 2 ≤ i ≤ m+n−1 we have λi−1 = λi +12(−1)i. Then for j = i−1,
(u− v − 12(−1)i)
[hi(u), x+
i−1(v)]
= (u− v − 12(−1)i)di+1(u + λi)
[di(u + λi)−1 , fi−1(v + λi + 1
2(−1)i)],
= −(−1)idi+1(u + λi)di(u + λi)−1(fi−1(v + λi + 12(−1)i)− fi−1(u + λi)),
= −(−1)ihi(u)(x+i−1(v)− x+
i−1(u−12(−1)i)).
Rearranging this gives
(u− v)[hi(u), x+
i−1(v)]
= −12 (−1)i(hi(u)x+
i−1(v) + x+i−1(v)hi(u))
+ (−1)ihi(u)x+i−1(u−
12(−1)i).
Now, taking appropriate coefficients of on both sides we have:
[hi,0, x+i−1,s] = −(−1)ix+
i−1,s,
[hi,r+1, x+i−1,s]− [hi,r, x
+i−1,s+1] = −1
2(−1)i(hi,rx
+i−1,s + x+
i−1,shi,r),
which agrees with (5.10) and (5.11).The case where j = i+ 1 is similar. We find:
(u− v + 12(−1)i+1)
[hi(u), x+
i+1(v)]
= (u− v + 12(−1)i+1)
[di+1(u + λi), fi+1(v + λi − 1
2(−1)i+1)]di(u + λi)−1
= −(−1)i+1(x+i+1(v)− x+
i+1(u−12(−1)i+1))hi(u).
Then
(u− v)[hi(u), x
+i+1(v)
]= −1
2(−1)i+1(x+
i+1(v)hi(u) + hi(u)x+i+1(v))
−(−1)i+1x+i+1(u− 1
2(−1)i+1)hi(u),
and by considering appropriate coefficients we see that in this case our re-sults agree with (5.10) and (5.11).Now we consider the negative root vectors. Again the commutator in ques-tion is zero unless j = i− 1, j = i or j = i+ 1. When i = j we have:
(u− v)[hi(u), x−i (v)]
= (−1)i(u− v)[di(u + λi)−1di+1(u + λi) , ei(v + λi)
]= (−1)i(u− v)di(u + λi)−1 [di+1(u + λi), ei(v + λi)]
− (−1)i(u− v)di(u + λi)−1 [di(u + λi), ei(v + λi)] di(u + λi)−1di+1(u + λi)
= −(−1)i+1hi(u)(x−i (v)− x−i (u))− (−1)i(x−i (v)− x−i (u))hi(u), by (3.35).
79
Taking coefficients of u0v−s−1 and u−r−1v−s−1 verifies (5.10) and (5.11) re-spectively.When j = i− 1 we have:
(u− v − 12(−1)i)[hi(u), x−i−1(v)]
= (−1)i−1(u− v − 12(−1)i)
[di(u + λi)−1di+1(u + λi), ei−1(v + λi + 1
2(−1)i)]
= −(−1)i−1(u− v − 12(−1)i)di(u + λi)−1
[di(u + λi), ei−1(v + λi + 1
2(−1)i)]hi(u)
= (−1)i+i−1(ei−1(v + λi−1)− eλi(u + λi))hi(u)
= (−1)i(x−i−1(v)− x−i−1(u−12(−1)i))hi(u).
Rearranging this gives:
(u− v)[hi(u), x
−i−1(v)
]= 1
2(−1)i(hi(u)x
−i−1(v) + x−i−1(v)hi(u)) + (−1)ix−i−1(u− 1
2(−1)i)hi(u).
Then taking appropriate coefficients gives the desired result.Finally, we consider odd root vectors with j = i+ 1. In this case we find:
(u− v + 12(−1)i+1)[hi(u), x−i+1(v)]
= (−1)i+1(u− v + 12(−1)i+1)di(u + λi)−1
[di+1(u + λi), ei+1(v + λi− 1
2(−1)i+1)]
= di(u + λi)−1di+1(u + λi)(ei+1(v + λi+1)− ei+1(u + λi))
= (−1)i+1hi(u)(x−i (v)− x−i (u + 12(−1)i+1)).
This gives:
(u− v)[hi(u), x
−i+1(v)
]= 1
2(−1)i(hi(u)x
−i−1(v) + x−i−1(v)hi(u))− (−1)ix−i−1(u+ 1
2(−1)i)hi(u).
Taking coefficients in this case also gives the desired result.
(5.13) This relation is clear from (3.38) and (3.39).
(5.14) This relation is clear from (3.42) and (3.43).
(5.15) The case where |i− j| > 1 is covered by (5.13)and (5.14). Also, the i = j caseis just a direct translation of (3.53) and (3.54). Now suppose j = i+ 1. Then
(u− v + 12(−1)i+1)
[x+
i (u), x+i+1(v)
]= (u− v + 1
2(−1)i+1)[fi(u + λi), fi+1(v + λi − 1
2(−1)i+1)]
= −(−1)i+1 (fi+1(v + λi+1)fi(u + λi) − fi+1(v + λi+1)fi(v + λi+1)+ fi+2,i(u + λi) + fi+2,i(v + λi+1)
= −(−1)i+1(x+
i+1(v)x+i (u))
)+ terms of entirely u or entirely v.
80
Rearranging this we find:
(u− v)[x+i (u), x+
i+1(v)] = −12(−1)i+1)(x+
i (u)x+i+1(v) + x+
i+1(v)x+i (u))
+ terms of entirely u or entirely v.
Then taking coefficients of u−r−1v−s−1 gives the desired result. To verify thecase with negative root vectors we make a similar calculation.
(5.16, 5.17) These are a straightforward consequence of (3.44), (3.45) and (3.46).
81
82
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87
Index
U(glm|n), 14∆, 12, 57µf , 53φ, 55ρm|n, 22∆, 50, 57ϕm|n, 25ζ , 23h-adic topology, 47
applications in physics, 4
Berezin, 4Berezinian, 3Brundan, Jon, 26, 32
Cartan matrix, 71Cartan matrix of slm|n, 50Casimir element, 74co-Poisson Hopf superalgebra, 46co-super-Jacobi identity, 46coproduct, 12, 13coproduct on the super-Yangian, 50
deformation, 49deg1, 13deg2, 13Drinfeld, Vladimir, 2, 47
filtrations, 13
Gauss decomposition, 21Gauss presentation of Y (glm|n), 32Gelfand, Israel, 19general linear Lie superalgebra glm|n, 68
homogeneous quantization, 49
homological relations between quaside-terminants, 20
Hopf superalgebras, 11
Killing form, 72Kleshchev, Alexander, 26, 32
Nazarov, Maxim, 4, 39, 40, 65
PBW theorem, 2PBW theorem for Y (glm|n), 15projective special linear Lie superalge-
bra psln|n, 58, 70
quantization, 48quantum Berezinian, 39quantum contraction, 39quantum determinant, 2quasideterminants, 19
Retakh, Vladimir, 19
special linear Lie superalgebra slm|n, 49,70
Stukopin’s presentation of Y (slm|n), 50Stukopin, Vladimir, 4, 45, 49super Lie bialgebra, 45super vector space, 67supertrace, 3, 70
Tolstoy, V. N., 4
Yangian of glm|n - definition, 7
Zhang, Ruibin, 4
88
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