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
Uwe Franz (UBFC) 1st introductory lecture May 2017 2 / 38
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).
Uwe Franz (UBFC) 1st introductory lecture May 2017 3 / 38
(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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 4 / 38
(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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 5 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 6 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 7 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 8 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 9 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 10 / 38
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
Uwe Franz (UBFC) 1st introductory lecture May 2017 12 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 13 / 38
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 ◦∆
Uwe Franz (UBFC) 1st introductory lecture May 2017 14 / 38
∗-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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 15 / 38
∗-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 .
Uwe Franz (UBFC) 1st introductory lecture May 2017 16 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 17 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 18 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 19 / 38
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).
Uwe Franz (UBFC) 1st introductory lecture May 2017 20 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 21 / 38
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).
Uwe Franz (UBFC) 1st introductory lecture May 2017 22 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 23 / 38
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!
Uwe Franz (UBFC) 1st introductory lecture May 2017 24 / 38
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 )∗.
Uwe Franz (UBFC) 1st introductory lecture May 2017 25 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 26 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 27 / 38
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).
Uwe Franz (UBFC) 1st introductory lecture May 2017 28 / 38
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).
Uwe Franz (UBFC) 1st introductory lecture May 2017 29 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 30 / 38
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 .
Uwe Franz (UBFC) 1st introductory lecture May 2017 31 / 38
Examples: the free permutation quantum groups
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 .
Uwe Franz (UBFC) 1st introductory lecture May 2017 32 / 38
Examples: the free permutation quantum groups
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 33 / 38
Examples: the free permutation quantum groups
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).
Uwe Franz (UBFC) 1st introductory lecture May 2017 34 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 35 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 36 / 38
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).
Uwe Franz (UBFC) 1st introductory lecture May 2017 37 / 38
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.
Uwe Franz (UBFC) 1st introductory lecture May 2017 38 / 38