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Graduate School

Topological Quantum Groups

First introductory lecture

Uwe Franz (Universite de Franche-Comte)

Topological Quantum Groups, Banach Center, Bedlewo, 28. June - 11. July 2015

Monday, June 29, 2015

Uwe Franz (UFC) (TopQG2015) 1st introductory lecture Monday, June 29, 2015 1 / 29

This morning’s program

Introductory lectures

Uwe Franz: Hopf algebras, definition of compact quantum groups inthe C∗-setting, some examples

Adam Skalski: Haar state, representation theory of a compactquantum group, ways of completing Pol(G)

Piotr So ltan: discrete quantum groups, the multiplicative unitary,possibly relation between Kac property of G and unimodularity of G


Outline

1 Coalgebras, bialgebras, Hopf algebras

2 Compact Quantum Groups: Definition

3 Compact Quantum Groups: Examples


(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.


(Algebraic) tensor product

The tensor product ⊗ is a functor:For f1 : V1 →W1, . . . , f1 : Vn →Wn, there existsf1 ⊗ · · · ⊗ fn : V1 ⊗ · · · ⊗ Vn →W1 ⊗ · · · ⊗Wn such that

V1 × · · · × Vnı //

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 Sergey Neshveyev.


Algebras

Definition

An algebra (=unital associative algebra) is a triple (A,m, e) with A avector 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⊗) ◦ (∆⊗∆),i.e. the diagram

B ⊗ B

m

��

∆⊗∆

''B ⊗ B ⊗ B ⊗ B

id⊗τB,B⊗id

��

B

∆

��

B ⊗ B ⊗ B ⊗ B

m⊗mwwB ⊗ B

commutes.Uwe Franz (UFC) (TopQG2015) 1st introductory lecture Monday, June 29, 2015 11 / 29

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 → Hsuch 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

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 .

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.

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→ g−1

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


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 in the same way.


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.

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,

∆ : 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.


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

From 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: 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



Definition, bis

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

C ∗-algebra generated by the entries of a 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. More about this in the lecture byAdam 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.


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)};


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.

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.


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