Pre-Workshop School Topological Quantum …hhlee/Franz-Slides.pdf1 Coalgebras, bialgebras, Hopf algebras 2 Compact quantum groups: De nition 3 Two important theorems The Haar state

Pre-Workshop School

Topological Quantum Groupps and HarmonicAnalysis

From Hopf Algebras to Compact Quantum Groups

Uwe Franz

LMB, Universite Bourgogne Franche-Comte

13/05/2017 (School) and 15-19/05/2017 (Workshop)

Uwe Franz (UBFC) 1st introductory lecture May 2017 1 / 38

Today’s Program

Introductory Lectures

9:00-11:00 Uwe Franz: From Hopf algebras toCompact Quantum Groups

11:30-12:30 Adam Skalski: From Analysis to Algebra& 14:00-15:00 and back via Representations

15:30-17:30 Christian Voigt: Tannaka-Krein Duality


Outline of this Lecture1

1 Coalgebras, bialgebras, Hopf algebras

2 Compact quantum groups: Definition

3 Two important theoremsThe Haar stateThe ∗-Hopf algebra of a compact quantum group

4 Compact quantum groups: Examples“Classical” examplesThe free permutation quantum group S+

n

The Woronowicz quantum group SUq(2)

1This work was supported by the French “Investissements d’Avenir” program, projectISITE-BFC (contract ANR-15-IDEX-03).


(Algebraic) tensor product

V1, . . . ,Vn vector spaces (over C). There exists a vector spaceV1 ⊗ · · · ⊗ Vn and a multi-linear map ı : V1 × · · · × Vn → V1 ⊗ · · · ⊗ Vn

such that for any multi-linear map f : V1 × · · · × Vn →W there exists aunique linear map f : V1 ⊗ · · · ⊗ Vn →W such that

V1 × · · · × Vnı //

f��

V1 ⊗ · · · ⊗ Vn

fvvW

commutes, i.e. f = f ◦ ı.


(Algebraic) tensor product

The tensor product ⊗ is a functor:For linear maps f1 : V1 →W1, . . . , fn : Vn →Wn, there exists a linear map

f1 ⊗ · · · ⊗ fn : V1 ⊗ · · · ⊗ Vn →W1 ⊗ · · · ⊗Wn

such thatV1 × · · · × Vn

ı //

f1×···×fn��

V1 ⊗ · · · ⊗ Vn

f1⊗···⊗fn��

W1 × · · · ×Wnı //W1 ⊗ · · · ⊗Wn

commutes.

Remark

The category of vector spaces becomes in this way a monoidal category.More about this in the lecture by Christian Voigt.


Algebras

Definition

An algebra (=unital associative algebra over C) is a triple (A,m, e) with Aa vector space, m : A⊗ A→ A, e : C→ A linear maps, such that

A⊗ A⊗ Aid⊗m //

m⊗id��

A⊗ A

m��

A⊗ A m// A

and

A⊗ C

id⊗e��

A

id��

∼= //∼=oo C⊗ A

e⊗id��

A⊗ A m// A A⊗ Amoo

commute.


Coalgebras: Dualize = revert all arrows

Definition

An coalgebra is a triple (C ,∆, ε), with C a vector space, ∆ : C → C ⊗ C ,ε : C → C linear maps, such that

C ⊗ C ⊗ C C ⊗ Cid⊗∆oo

C ⊗ C

∆⊗id

OO

C∆

oo

∆

OO

and

C ⊗ C∼= // C C⊗ C

∼=oo

C ⊗ C

id⊗ε

OO

C

id

OO

∆oo

∆// C ⊗ C

ε⊗id

OO

commute.


Bialgebras

Definition

f : A1 → A2 (f : C1 → C2, resp.) is a morphism of (co-)algebras, if

A1 ⊗ A1

m��

f⊗f // A2 ⊗ A2

m��

A1f

// A2

(or C1 ⊗ C1f⊗f // C2 ⊗ C2

C1

∆

OO

f// C2

∆

OO resp.)

and

Ce��

∼= // Ce��

A1f // A2

(or C∼= // C

C1

ε

OO

f // C2

ε

OO resp.)

commute.


Bialgebras

Denote by τV ,W : V ⊗W →W ⊗ V the flip, τ(v ⊗ w) = w ⊗ v .

Proposition

If (A,m, e) is an algebra, then (A⊗ A,m⊗, e ⊗ e) with

m⊗ = (m ⊗m) ◦ (id⊗ τA,A ⊗ id)

is also an algebra.

Proposition

If (C ,∆, ε) is a coalgebra, then (C ⊗ C ,∆⊗, ε⊗ ε) with

∆⊗ = (id⊗ τC ,C ⊗ id) ◦ (∆⊗∆)

is also a coalgebra.


Bialgebras

Remark

(C, id, id) is an algebra and a coalgebra.

Definition-Proposition

(B,m, e,∆, ε) is a bialgebra if

(B,m, e) is an algebra

(B,∆, ε) is a coalgebra

the following equivalent conditions are satisfied:

∆ and ε are morphisms of algebrasm and e are morphisms of coalgebras.


Bialgebras

The compatibility conditions mean for example∆ ◦m = (m⊗) ◦ (∆⊗∆) = (m ⊗m) ◦∆⊗,i.e.

B ⊗ B

m

��

∆⊗∆

''B ⊗ B ⊗ B ⊗ B

id⊗τB,B⊗id

��

B

∆

��

B ⊗ B ⊗ B ⊗ B

m⊗mwwB ⊗ B

commutes.Uwe Franz (UBFC) 1st introductory lecture May 2017 11 / 38

Bialgebras

Figure: Another representation of the same compatibility condition


Hopf algebras

Definition

For (A,m, e) an algebra and (C ,∆, ε) a coalgebra, we can define amultiplication (called convolution) on

Hom(C ,A) = {f : C → A linear}

byf1 ? f2 = m ◦ (f1 ⊗ f2) ◦∆.

The convolution ? turns Hom(C,A) into an algebra with unit e ◦ ε.


Hopf algebras

Definition

A bialgebra (B,m, e,∆, ε) is called a Hopf algebra, if there exists aninverse (w.r.t. ?) for id ∈ Hom(B,B).

S : B → B is the inverse of id w.r.t. ? if

B ⊗ B

S⊗id��

B∆oo ∆ //

e◦ε��

B ⊗ B

id⊗S��

B ⊗ B m// B B ⊗ Bmoo

commutes. S is unique (if it exists), it is called the antipode.

Proposition

S is an algebra and coalgebra anti-homomorphism, i.e.

S ◦m = m ◦ τB,B ◦ (S ⊗ S) ∆ ◦ S = (S ⊗ S) ◦ τB,B ◦∆


∗-Hopf algebras

Definition

A ∗-Hopf algebra (H,m, e,∆, ε,S , ∗) is a Hopf algebra (H,m, e,∆, ε,S)equipped with a conjugate-linear anti-multiplicative involution

∗ : H → H

such that ∆ : H → H ⊗ H is a ∗-morphism (the involution on H ⊗ H is(a⊗ b)∗ = a∗ ⊗ b∗).

Proposition

The counit ε in a ∗-Hopf algebra is a ∗-homomorphism.

The antipode in a ∗-Hopf algebra is invertible and satisfiesS ◦ ∗ ◦ S ◦ ∗ = id.


∗-Hopf algebras: Examples

(a) If G is a group, then the group ∗-algebra CG is a ∗-Hopf algebra withthe coproduct, counit, and antipode

∆(g) = g ⊗ g ε(g) = 1 S(g) = g−1

for g ∈ G .

(b) If H is a finite-dimensional ∗-Hopf algebra, then the dual space H ′ isa ∗-Hopf algebra with the dual operations,

mH′ = ∆′H , eH′ = ε′H ,∆H′ = m′H , εH′ = e ′H ,SH′ = S ′H ,

and the involution (f ∗)(a) = f(S(a)∗

)for f ∈ H ′, a ∈ H.

(c) If G is a finite group, then the algebra CG of functions on G is a∗-Hopf algebra, with

∆f (g1, g2) = f (g1g2) ε(f ) = f (e) S(f ) : g 7→ f (g−1)

for f ∈ CG , g1, g2, g ∈ G (identity CG×G ∼= CG ⊗ CG ).In this case (CG )′ ∼= CG .


Conclusion

Summary

The category of finite-dimensional ∗-Hopf algebras has a nice dualitytheory and includes finite groups.

Question

How can we extend this to infinite groups?

Answer

Yes! This leads to the theory of compact and locally compact quantumgroups.Let us have a look at compact (or discrete) quantum groups.


C ∗-algebras

Definition

A = (A,m, e, ∗) a ∗-algebra, ‖ · ‖ a norm on A such that (A, ‖ · ‖) is aBanach space. (A, ‖ · ‖) is a C ∗-algebra, if

‖ · ‖ is submultiplicative, i.e.

‖ab‖ ≤ ‖a‖ ‖b‖ ∀a, b ∈ A

‖ · ‖ satisfies the C ∗-identity

‖a∗a‖ = ‖a‖2 ∀a ∈ A

Remark

Our definition of algebras included the existence of a unit, but non-unitalC ∗-algebras are defined as above with A not unital. If A is unital, then wehave 1∗ = 1 and ‖1‖ = 1.


C ∗-algebras

Examples (commutative)

If X is a compact Hausdorff space, then

C (X ) = {f : X → C continuous}

is a unital C ∗-algebra with the norm

‖f ‖∞ = supx∈X|f (x)|.

By a theorem of Gelfand, all commutative unital C ∗-algebras are ofthis form (up to isometric ∗-isomorphism).

Remark: Non-unital C ∗-algebras correspond to locally compactspaces.


C ∗-algebras

Examples (noncommutative)

If H is a Hilbert space, then

B(H) = {A : H → H linear, bounded}

is a unital C ∗-algebra with the operator norm

‖A‖ = supx∈H,‖x‖≤1

‖Ax‖.

Any norm-closed involutive subalgebra of B(H) is also a C ∗-algebra.

By a theorem of Gelfand and Naimark, all C ∗-algebras are of thisform (up to isometric ∗-isomorphism).


Minimal tensor product

Let A and B be two C ∗-algebras. In general A⊗ B is not a C ∗-algebra, if⊗ is the (algebra) tensor product.

Definition

Let‖c‖min = sup

ρA,ρB

∥∥∥∑ ρA(ai )⊗ ρB(bi )∥∥∥

for c =∑

ai ⊗ bi ∈ A⊗ B, where the sup runs over all representations(ρA,HA) and (ρB ,HB) of A and B, and the norm on the right-hand-side isthe operator norm on HA ⊗ HB .The completion

A⊗min B = A⊗ B‖·‖min

of A⊗ B with this norm is a C ∗-algebra, it is called the minimal (orspatial) tensor product of A and B.

Example: C (X )⊗min C (Y ) ∼= C (X × Y ) for X ,Y compact spaces.


Compact quantum groups: Definition

Definition (Woronowicz)

A compact quantum group is a pair G = (A,∆), where A is a unitalC ∗-algebra, and

∆ : A→ A⊗min A

is a unital ∗-homomorphism such that

∆ is coassociative, i.e. (∆⊗ id) ◦∆ = (id⊗∆) ◦∆

the quantum cancellation rules are satisfied

Lin((1⊗ A)∆(A)

)= A⊗min A = Lin

((A⊗ 1)∆(A)

).

A is called the algebra of “continuous functions” on G and also denotedby C (G).


Compact quantum groups: Definition

Definition

A morphism of compact quantum groups between compact quantumgroups G1 = (A1,∆1) and G2 = (A2,∆2) is a unital ∗-homomorphismπ : A1 → A2 such that

∆2 ◦ π = (π ⊗ π) ◦∆1.


The Haar state

Convolutions: for a ∈ A and ξ, ξ′ ∈ A∗ we define

ξ ? ξ′(a) = (ξ ⊗ ξ′)∆(a)

ξ ? a = (id⊗ ξ)∆(a)

a ? ξ = (ξ ⊗ id)∆(a)

Theorem (Woronowicz)

Let G = (A,∆) be a compact quantum group. There exists unique state(called the Haar measure) h on A such that

a ? h = h ? a = h(a)I , a ∈ A.

Note that, in general, h is neither faithful nor a trace!


The ∗-Hopf algebra Pol(G)

An n-dimensional unitary corepresentation of A :U = (ujk)1≤j ,k≤n ∈ Mn(A) a unitary such that for all j , k = 1, . . . , nwe have

∆(ujk) =n∑

p=1

ujp ⊗ upk .

Fix (U(s))s∈I a complete family of mutually inequivalent irreducibleunitary correpresentations (U(s))s∈I . The algebra Pol(G) is defined as

Pol(G) = Lin{u(s)jk ; s ∈ I, 1 ≤ j , k ≤ ns},

where ns is the dimension of U(s).

Theorem

Pol(G) is a dense ∗-subalgebra of A = C (G), which is a ∗-Hopf algebrawith

ε(u(s)jk ) = δjk and S(u

(s)jk ) = (u

(s)kj )∗.


The ∗-Hopf algebra Pol(G)

The Haar state is faithful on Pol(G).Its action on the coefficients of the irreducible corepresentations is given by

h(u(s)jk ) =

{1 if U(s) = (1) = trivial corep0 else.


Classical examples

Example

A compact group G can be viewed as a compact quantum group, withA = C (G ) and

∆G : C (G )→ C (G × G ) ∼= C (G )⊗min C (G ),

∆G f (g1, g2) = f (g1g2).

Remark

A continuous group homomorphism ϕ : G1 → G2 induces a morphism ofcompact quantum groups

πϕ : C (G2)→ C (G1)

πϕ(f ) = f ◦ ϕ

in the opposite direction.


Classical Examples

Theorem

If G = (A,∆) is a commutative compact quantum group (i.e. A iscommutative), then there exists a compact group G such that G isisomorphic to (C (G ),∆), i.e. there exist an ∗-isomorphism

π : A→ C (G )

such that∆G ◦ π = (π ⊗ π) ◦∆

(i.e. an isomorphism of compact quantum groups).


Classical examples, bis

Example

For a discrete group Γ we can turn the group C ∗-algebras C ∗r (Γ) andC ∗u (Γ) into compact quantum groups, denoted by Γ, if we set

∆γ = γ ⊗ γ

for γ ∈ Γ.

Theorem

If G = (A,∆) is a cocommutative compact quantum group (i.e.τA,A ◦∆ = ∆), then there exists a discrete group Γ and surjectivemorphisms of compact quantum groups

C ∗u (Γ)π1 // A

π2 // C ∗r (Γ).


Examples: functions on the permutation group Sn

Let n ≥ 1. The permuation group Sn is a finite group, so the algebraC (Sn) of functions on Sn is a finite-dimensional ∗-Hopf algebra andtherefore also a compact quantum group.Define fij : Sn → C by

fjk(π) = δj ,π(k)

Thefjk , 1 ≤ j , k ≤ n, generate C (Sn) as an algebra.They satisfay the relations

f ∗jk = fjk = f 2jk ∀1 ≤ j , k ≤ n

n∑j=1

fjk = 1 =n∑

j=1

fkj ∀1 ≤ k ≤ n

Furthermore, C (Sn) is the universal commutative C ∗-algebra generated bythese relations.


Examples: the free permutation quantum groups

Let A be a C ∗-algebra over C and n ∈ N.

Definition

(a) A square matrix u ∈ Mn(A) is called magic, if all its entries areprojections and each row or column sums up to 1.

(b) Let us denote by Pol(S+n ) the unital ∗-algebra generated by n2

elements ujk , 1 ≤ j , k ≤ n with the relations

u∗jk = ujk = u2jk ∀1 ≤ j , k ≤ n

n∑j=1

ujk = 1 =n∑

j=1

ukj ∀1 ≤ k ≤ n

Remark: We could add the relations ujkuj` = δk`ujk , ukju`j = δk`ukj .



Definition, bis

(c) The free permutation quantum group C (S+n ) is the universal

C ∗-algebra generated by the entries of an n × n magic square matrixu = (ujk), i.e. the completion of Pol(S+

n ) w.r.t. the (semi-)norm

‖c‖ = supρ‖ρ(c)‖

where the sup runs over all ∗-representations of Pol(S+n ) on some

Hilbert space (prove that this sup is finite!). It is a compact quantumgroup with the coproduct

∆ : C (S+n )→ C (S+

n )⊗min C (S+n )

determined by ∆(ujk) =∑n

`=1 uj` ⊗ u`k .



Remark

Other completions can be considered. You will probably learn more aboutthis in the lecture by Adam Skalski.

For n = 1, 2, 3, C (S+n ) is commutative and C (S+

n ) ∼= C (Sn), i.e. S+n is

isomorphic to the permutation group Sn.

For n ≥ 4, C (S+n ) is noncommutative and dimC (S+

n ) =∞, i.e. thereexist (infinitely many!) genuine “quantum permutations”. E.g.,

1− p p 0 0p 1− p 0 00 0 1− q q0 0 q 1− q

with p, q two projections.



S+n is a also called a liberation of Sn.

Idea: We “freed” the functions on the permutation group from theircommutativity constraint.

A matrix H = (hjk) ∈ Mn(C) is called a complex Hadamard matrix, if

1√nH is unitary and |hjk | = 1 ∀1 ≤ j , k ≤ n.

To any n × n Hadamard matrix we can associate an n-dimensionalrepresentation of C (S+

n ) by setting

ρH(ujk) = Projξjk with ξjk =

(hjihki

)1≤i≤n

∈ Cn,

and a quantum subgroup of S+n (the “Hopf image” of ρH).


Example: functions on SU(2)

LetSU(2) = {U ∈ M2(C) : U∗U = I = UU∗, det(U) = 1}.

SU(2) is a compact group, so C (SU(2)) is a commutative compactquantum group. The coordinate functions α, β, γ, δ : SU(2)→ C,

α(U) = u11 β(U) = u12

γ(U) = u21 δ(U) = u22

for U =

(u11 u12

u21 u22

)∈ SU(2) separate points and therefore generate

C (SU(2)) as a C ∗-algebra.


Example: functions on SU(2)

α, βγ, δ commute and they satisfy the relations

det(U) = 1 i.e. αδ − βγ = 1

UU∗ = I i.e.αα∗ + ββ∗ = 1 αγ∗ + βδ∗ = 0γα∗ + δβ∗ = 0 γγ∗ + δδ∗ = 1

U∗U = I i.e.α∗α + γ∗γ = 1 α∗β + γ∗γ = 0β∗α + δ∗γ = 0 β∗β + δ∗δ = 1

C (SU(2)) is the universal commutative C ∗-algebra generated by α = δ∗

and γ = −β∗ with the relation

α∗α + γ∗γ = 1.

The coproduct is given by

∆(α) = α⊗ α− γ∗ ⊗ γ ∆(γ) = γ ⊗ α− α∗ ⊗ γ.


Another example: SUq(2)

For q ∈ R\{0} the universal C∗-algebra generated by α, γ and the relations

α∗α + γ∗γ = 1 αα∗ + q2γγ∗ = 1

γγ∗ = γ∗γ αγ = qγα αγ∗ = qγ∗α

can be turned into a compact quantum group, with the comultiplication

∆

(α −qγ∗γ α∗

)=

(α −qγ∗γ α∗

)⊗(α −qγ∗γ α∗

),

i.e. ∆(α) = α⊗ α− qγ∗ ⊗ γ, etc.

For q = 1: C (SU1(2)) = C (SU(2)) = {continuous functions on thespecial unitary group SU(2)};See Jacek Krajczok, Piotr M. So ltan, “Center of the algebra offunctions on the quantum group SUq(2) and related topics,arXiv:1612.00996, to appear in Commentationes Mathematicae, andthe references therein for more information on SUq(2).


Selected references

Ann Maes, Alfons Van Daele, Notes on Compact Quantum Groups,arXiv:math/9803122, 1998.

Sergey Neshveyev, Lars Tuset, Compact Quantum Groups and TheirRepresentation Categories, SMF Specialized Courses, Vol. 20, 2013.

Anatoli Klimyk, Konrad Schmudgen, Quantum Groups and TheirRepresentations, Texts and Monographs in Physics, 1997.

Thomas Timmermann, An Invitation to Quantum Groups andDuality: From Hopf Algebras to Multiplicative Unitaries and Beyond,EMS Textbooks in Mathematics, 2008.

Stanis law L. Woronowicz, Compact Quantum Groups, Les Houches,Session LXIV, 1995, Quantum Symmetries, Elsevier 1998.


Documents

Pre-Workshop School Topological Quantum …hhlee/Franz-Slides.pdf1 Coalgebras, bialgebras, Hopf algebras 2 Compact quantum groups: De nition 3 Two important theorems The Haar state