Superintegrable SystemsAuthor(s): Boris KupershmidtSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 81, No. 20, [Part 2: Physical Sciences] (Oct. 15, 1984), pp. 6562-6563Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/24775 .

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Proc. Natl. Acad. Sci. USA Vol. 81, pp. 6562-6563, October 1984 Mathematics

Superintegrable systems (supersymmetry/Lax equations/conservation laws)

BORIS KUPERSHMIDT

The University of Tennessee Space Institute, Tullahoma, TN 37388; and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545

Communicated by Peter D. Lax, June 15, 1984

ABSTRACT Families of matrix differential superintegra- ble systems of Lax type are constructed. Each family is a com- mutative Lie superalgebra with an infinite common set of con- servation laws.

Introduction. Since the recent discovery of integrability of the Korteweg-de Vries equation, there has been a rapid growth of what is nowadays called the theory of integrable systems. This theory, still in active development, has ac- quired by now a clear mathematical foundation (1) and a number of deep results (2, 3). Meanwhile, the physical part of the theory-i.e., the study of particle-like behavior of nonlinear classical fields (4)-has branched out into a new and exciting area of "supersymmetry" (see, e.g., refs. 5 and 6), where anticommutative variables of the Grassmann type are treated on an equal footing with commutative variables of the usual mathematics. This development has influenced an initial opening in the area that, for want of a better name, I shall call superintegrable systems. Among them are super- symmetric two-dimensional a-models (7) and supersymme- tric two-dimensional Toda lattices (8) associated with classi- cal Lie superalgebras (9). Although none of these systems are of evolution type, one knows from the commutative the- ory that the Toda-type equations are intimately connected with the modified Lax equations, which are of evolution type (2, 3, 10). This suggests that evolutionary superintegra- ble families exist, and I show below that this is indeed the case. The plan is as follows. In the first section we adjust the basic notions of commutative differential algebra to incorpo- rate anticommuting variables. In the second section I de- scribe the structure of the centralizer of a nondegenerate even semisimple matrix differential operator. In the next section the Lax equations are introduced and their basic properties are stated. In the last section a construction is given of an infinite set of common conservation laws for each commuting set of Lax equations, variational deriva- tives of these conservation laws are computed, and the non- Hamiltonian character of the theory is explained. Superinte- grable systems connected with the basic classical simple Lie superalgebras are left out of the paper; as is clear from ref. 3, for their construction one needs only to use the basic struc- tural facts of the theory (which are given in Sections 2-4 below) together with the properties of the root space decom- position (which are given in ref. 9). I have assumed known, and tried to stay as close as possible to, the basic construc- tions of the commutative theory of ref. 1. The ideas of ref. 11 for the residue calculus of pseudodifferential operators in the form given in ref. 12 are adapted in supersymmetric setting in Section 4. Finally, a few elementary facts about superalge- bras and modules over them can be found in refs. 9 and 13.

1. The Objects. Let k' be a commutative Q-algebra with unity, k a commutative superalgebra over k', also with unity, C a commutative superalgebra and a k-bimodule, C' an asso-

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ciative superalgebra and a C-bimodule, Q a Z2-graded C-bi- module, and Ql' a Z2-graded C'-bimodule. We denote p: (Q) -* Z2 the parity of Z2-homogeneous elements in superalge- bras and modules over them. These homogeneous elements are called p-homogeneous. All formulae written for p-homo- geneous elements are extended by additivity to all elements.

We fix two nonnegative integers t0 and El, at least one of which is positive, set t = to + fl, and for a Z2-graded vector space V denote by Ov the supermatrix (to, el)-structure con- sisting of e x e matrices with entries in V and the grading p on Ov given as p(X) = p(Xap) + p(a) + p(13), X E Ov, where p(a) = 0 for a c to and p(a) = 1 for a > t0. In these nota- tions, Oc, becomes an associative superalgebra and a Oc- bimodule, and On, becomes a Z2-graded Oc-bimodule.

If C" is an associative superalgebra, we denote (CU,)Lie the corresponding Lie superalgebra with the bracket [a, b] = ab - (-l(a)p(b)ba (see ref. 9). We now specialize C = k[ui], m E Z+, 1 c a, c?4

where the range for {(i, axx)} will be specified later on. The grading p on C is defined as p(u,fma) = p(a) + p(,B). Let a: C -* C be an even derivation of C over k acting on generators of C as a(uf')) = u(j, ') (all derivations in this paper are left derivations). Set fl = f(C) = {Fdult I'p>p E C, finite sums}, and let d: C >f l(C) be the universal odd derivation over k acting on generators of C as d(kui,a) ()k kdu4ma, k E k; thus, p(dujtmaj) = 1 + p(ul7ap) = 1 + p(a) + p(,8). We uniquely extend a on fl l(C) by requiring it to com- mute with d. Set C' = C((f)): = {c' = i_ C rjI N < oo, Cr E C}, Q ' = Q(( -)), with the associative multiplication in C', C-actions on C', and C'-actions on Ql' made possible by the relations mf = E;;= (m)(arf)m-r, W-mlf =f (-1)r(7+) (arf)-mrl, m 2 0, f E C or fl (see ref. 11). The p-grading on C' and fl' is defined as p(ffl = p(f), f E C or fl. The derivations d and a are naturally extended on C', Oc, and Oc while a also acts on On and OQ (see ref. 11). We write a - b if (a - b) E Im a.

We introduce w-grading on C, C', Ql, fQ',Oc', and On, by w(ujfma) = w(dum) = n - i + m, w(O) = 1, w(k) = 0, and w(X) = W(Xap) for X E Ov, Xa E V. For Q = Yqr r E C' or Oc, or fl' or On,, we define Q+ = Ir?o qr tr, Q_ = Q - Q+, and Res Q = q-1.

A derivation X of C over k is called evolutionary if it com- mutes with a. Such an evolution derivation is uniquely de- fined by its action on the generators ui,ag: X = E[dm(X (Ui,aP))] */auiMa), where a/au$ma is the natural derivation of C over k of parity p(i). The Lie superalgebra of evolution derivations is denoted Dev(C); the Z2-grading on D`V(C) is given by p(X) = p(X(Ui,aP)) + p(a) + p(/3), X E D ev(C).

2. The Centralizer. We set L = Y.=O ujfi E ?Oc, where ui = (Ui,ap) are matrices satisfying the following conditions: (i) Un is even, a-constant (i.e., U,, C Ok), invertible, and diago- nalizable, which we shall consider already diagonal: Un = diag(kl, ..., ke), ka E ko. Let ir: k -* ko/(kl) denote the natu- ral homomorphism (see ref. 13). We require that if ir(ka) = ir(kp) then ka, = ko [i.e., the dimension of the center of the centralizer is the same for Un and r(Un)]. (ia) aUn1 Cz Im ad Un in (Oc)Lie i.e., Un_1^a = 0 when k,, = kp.

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Mathematics: Kupershmidt Proc. NatL. Acad. Sci. USA 81 (1984) 6563

Accordingly, we specify C = k[u(m, ] such that 0 < i ? n - 1 and un1,aI3 is absent whenever ka = kp.

Denote Z(L) the centralizer of L in Oc'* Let Q E Z(L), Q = y2aq,re and suppose Q is p-homogeneous. Comparing the (n+a-x-coefficients in the equality QL = LQ, for x = 0, 1, 2, we find that qa is a-constant, diagonal, and (- 1P(a)p(qj qaaa = (-1)P(n)P(~)qa,, whenever ka = k,. The elements q,,' with such qa are called admissible.

THEOREM 1. Let qa6' be an admissible p-homogeneous element. Then (i) Z(L) contains unique p-homogeneous ele- ment of the form Q = qXa + (lower order in 6 terms) which is w-homogeneous of degree a; (ii) Z(L), as k-bimodule, con- sist precisely of the sums of such elements; (iii) (Z(L))Lie is trivial, that is, any two elements in Oc that commute with L supercommute between themselves.

3. The Equations. For each w- and p-homogeneous Q E Z(L) we define an evolution derivation aQ in C by

aQ(L) = [Q+, L] = [-Q-, L], [1]

with the bracket [, ] taken in (OC,)Lie. Thus, aQ(u,-1) E Im ad u", and p(3Q) = p(Q). Eqs. 1 are of the Lax type.

THEOREM 2. For Q, P E Z(L), (i) [dQ, ap] = 0, i.e. dp and aQ supercommute; (ii)

aQ(P) = [Q+, P] = [-Q-, P]. [2]

THEOREM 3. Let C D C be a Z2-graded extension of C, with a w-grading on C agreeing with that of C, with a deriva- tion d: C -* C being even, surjective, agreeing with a on C, having only elements of k in its kernel, and having w-weight one (see ref. 1). (i) Then there exists and unique an element K E OE,, w(K) = 0, p(K) = 0, of the form K = 1 + 1r>O Xr-r, Xr E OZ, such that K-1 LK = Un6n. (ii) Let aQ be uniquely extended on C by requiring it to commute with a and being w- and p-homogeneous. Then aQ(K) = -Q_K.

Remarks: (i) From Theorem 1 it follows that we can re- strict ourselves only to even elements Q of Z(Lj by requiring qa to have only ones and zeroes on the diagonal. (ii) To avoid the introduction of the extension C D C, one could start di- rectly with K, define C as k[XI(m) I and L as KUntn K-1, and thereafter define C as k[uW ]. Then Theorem 3 amounts to the statement that Kqa,a K- E Oc' for admissible elements qa*a. This point of view was suggested for the differential Lax equations in ref. 14 and was taken for the discrete Lax equations in chapter IX of ref. 15.

4. The Conservation Laws and Their Variational Deriva- tives. For q E Oc we define the supertrace str q = (-1)P(a)[l+p(q)Iqtev. Recall that str ([q, q']) = 0 for q, q' E (OC)Lie (ref. 13).

LEMMA 1. Define a k-bilinearform on Oc' (or (Oc)Lie) by

(X, Y) = str Res(XY). [3]

Then (i) (X, Y) - (Y, X)(-1)P(x)P(Y). Equivalently, str Res ([X, Y]) - 0. (ii) The form (. , .) is invariant (modulo Im 8) on (OC,)Lie: (X, [Y, Z]) - ([X, Y], Z), X, Y, Z E (OC,)Lie.

THEOREM 4. (i) Each super-Lax Eq. I has an infinite common set of conservation laws str Res (Q), Q E Z(L), w(Q) 2 0. (ii) Let 0 be the dimension of the center of the centralizer of Un in ?k (= the number of different elements amongst kl, ..., ke). Then, for each nonnegative integer a not divisible by w(L), there exists e linearly independent over k even conservation laws of the Eq. 1 with w-weight a + 1,

coming from 0 linearly independent, even, admissible ele- ments qafa;for a: n, the corresponding number is 0 - 1.

For w- and p-homogeneous Q E Z(L), with w(Q) > 0, de- note HQ = [n/w(Q)] str Res (Q).

LEMMA 2. For w- and p-homogeneous Q, P E Z(L),

w(P) str Res(PdQ) - (-1)P(P)+P(Q)+P(P)P(Q)w(Q) str Res(QdP). [4]

COROLLARY 1.

dHLQ - str Res(dL o Q). [5]

Set Q = yerqr. Writing 5 in longhand, we obtain Corollary 2.

COROLLARY 2.

1= ()P(P)P(Q) HQ

0 ? i S n - 1 [6]

where 5/8Ujfja = Em (-a)m a/auigm) and in Eq. 6 only those terms corresponding to i = n - 1 are present for which ka 7 k,.

Thus, for an even Q E Z(L), one can rewrite Eq. 6 in the form

Q_ = -1 A + H 5uo 5u',

+-n AH + *, H = HLQ. [7]

Substituting Eq. 6 into Eq. 1, we arrive at an expression of the form aQ(Ui,a3) = (_j1)P(H)[P(a)+p(PI3 jv Bj,aj6y,3jx (8H/8uj,,,,), H = HLQ, with some matrix B E Oc' In con- trast to the commutative case (12), the matrix B is not super- Hamiltonian. This loss of the first Hamiltonian structure makes a really significant difference between general super- and usual integrable systems.

This work was supportcd in part by the National Science Founda- tion and the Department of Energy.

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