structure of compact quantum groups au(q) and bu(q) and their …math.ecnu.edu.cn › rcfoa ›...
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![Page 1: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/1.jpg)
Structure of Compact Quantum GroupsAu(Q) and Bu(Q)
and their Isomorphism Classification
Shuzhou Wang
Department of MathematicsUniversity of Georgia
![Page 2: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/2.jpg)
1. The Notion of Quantum Groups
I G = Simple compact Lie group, e.g.
G = SU(2) = {
α −γ
γ α
: αα + γγ = 1, α, γ ∈ C }.
A = C(G)
I Lie group G⇐⇒ Hopf algebra (A,∆, ε,S).
∆ : A→ A⊗ A, ∆(f )(s, t) = f (st).
ε : A→ C, ε(f ) = f (e).
S : A→ A, S(f )(t) = f (t−1).
![Page 3: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/3.jpg)
1. The Notion of Quantum Groups
I G = Simple compact Lie group, e.g.
G = SU(2) = {
α −γ
γ α
: αα + γγ = 1, α, γ ∈ C }.
A = C(G)
I Lie group G⇐⇒ Hopf algebra (A,∆, ε,S).
∆ : A→ A⊗ A, ∆(f )(s, t) = f (st).
ε : A→ C, ε(f ) = f (e).
S : A→ A, S(f )(t) = f (t−1).
![Page 4: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/4.jpg)
1. The Notion of Quantum Groups
I G = Simple compact Lie group, e.g.
G = SU(2) = {
α −γ
γ α
: αα + γγ = 1, α, γ ∈ C }.
A = C(G)
I Lie group G⇐⇒ Hopf algebra (A,∆, ε,S).
∆ : A→ A⊗ A, ∆(f )(s, t) = f (st).
ε : A→ C, ε(f ) = f (e).
S : A→ A, S(f )(t) = f (t−1).
![Page 5: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/5.jpg)
1. The Notion of Quantum Groups (cont.)
Spectacular development in Mid-1980’s:
Drinfeld, Jimbo, Woronowicz,
Idea of Quantization:
Commuting functions on G, e.g. α, γ,
⇓
Non-commuting operators, e.g. α, γ,
Commutative C(G) =⇒ Noncommutative C(Gq).
Gq = quantum group
![Page 6: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/6.jpg)
1. The Notion of Quantum Groups (cont.)
Spectacular development in Mid-1980’s:
Drinfeld, Jimbo, Woronowicz,
Idea of Quantization:
Commuting functions on G, e.g. α, γ,
⇓
Non-commuting operators, e.g. α, γ,
Commutative C(G) =⇒ Noncommutative C(Gq).
Gq = quantum group
![Page 7: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/7.jpg)
1. The Notion of Quantum Groups (cont.)
I Recall Hilbert 5th Problem:
Characterization of Lie groups among topological groups.
I New Problem:
Characterization of quantum groups among Hopf algebras.
I Lesson:
Quantums groups = “nice Hopf algebras”
Restrict to such Hopf algebras to obtain nice and deep
theory.
![Page 8: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/8.jpg)
1. The Notion of Quantum Groups (cont.)
I Recall Hilbert 5th Problem:
Characterization of Lie groups among topological groups.
I New Problem:
Characterization of quantum groups among Hopf algebras.
I Lesson:
Quantums groups = “nice Hopf algebras”
Restrict to such Hopf algebras to obtain nice and deep
theory.
![Page 9: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/9.jpg)
1. The Notion of Quantum Groups (cont.)
I Recall Hilbert 5th Problem:
Characterization of Lie groups among topological groups.
I New Problem:
Characterization of quantum groups among Hopf algebras.
I Lesson:
Quantums groups = “nice Hopf algebras”
Restrict to such Hopf algebras to obtain nice and deep
theory.
![Page 10: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/10.jpg)
1. The Notion of Quantum Groups (cont.)
I DEFINITION: A compact matrix quantum group (CMQG) is
a pair G = (A,u) of a unital C∗-algebra A and
u = (uij)ni,j=1 ∈ Mn(A) satisfying
(1) ∃ ∆ : A −→ A⊗ A with
∆(uij) =n∑
k=1
uik ⊗ ukj , i , j = 1, · · · ,n;
(2) ∃ u−1 in Mn(A) and anti-morphism S on
A = ∗ − alg(uij) with
S(S(a∗)∗) = a, a ∈ A; S(u) = u−1.
![Page 11: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/11.jpg)
1. The Notion of Quantum Groups (cont.)
I DEFINITION: A compact matrix quantum group (CMQG) is
a pair G = (A,u) of a unital C∗-algebra A and
u = (uij)ni,j=1 ∈ Mn(A) satisfying
(1) ∃ ∆ : A −→ A⊗ A with
∆(uij) =n∑
k=1
uik ⊗ ukj , i , j = 1, · · · ,n;
(2) ∃ u−1 in Mn(A) and anti-morphism S on
A = ∗ − alg(uij) with
S(S(a∗)∗) = a, a ∈ A; S(u) = u−1.
![Page 12: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/12.jpg)
1. The Notion of Quantum Groups (cont.)
I DEFINITION: A compact matrix quantum group (CMQG) is
a pair G = (A,u) of a unital C∗-algebra A and
u = (uij)ni,j=1 ∈ Mn(A) satisfying
(1) ∃ ∆ : A −→ A⊗ A with
∆(uij) =n∑
k=1
uik ⊗ ukj , i , j = 1, · · · ,n;
(2) ∃ u−1 in Mn(A) and anti-morphism S on
A = ∗ − alg(uij) with
S(S(a∗)∗) = a, a ∈ A; S(u) = u−1.
![Page 13: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/13.jpg)
1. The Notion of Quantum Groups (cont.)
I Note: Equivalent definition of CMQG is obtained if
condition (2) is replaced with
(2′) ∃ u−1 and (ut )−1 in Mn(A)
I There are other equivalent definitions of CMQG
![Page 14: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/14.jpg)
1. The Notion of Quantum Groups (cont.)
I Note: Equivalent definition of CMQG is obtained if
condition (2) is replaced with
(2′) ∃ u−1 and (ut )−1 in Mn(A)
I There are other equivalent definitions of CMQG
![Page 15: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/15.jpg)
1. The Notion of Quantum Groups (cont.)
Famous Example: C(SUq(2)), q ∈ R, q 6= 0
Generators: α, γ.
Relations: the 2× 2 matrix u :=
[α −qγ∗
γ α∗
]is unitary,
i.e.
α∗α + γ∗γ = 1, αα∗ + q2γγ∗ = 1,
γγ∗ = γ∗γ, αγ = qγα, αγ∗ = qγ∗α.
![Page 16: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/16.jpg)
1. The Notion of Quantum Groups (cont.)
Famous Example: C(SUq(2)), q ∈ R, q 6= 0
Generators: α, γ.
Relations: the 2× 2 matrix u :=
[α −qγ∗
γ α∗
]is unitary,
i.e.
α∗α + γ∗γ = 1, αα∗ + q2γγ∗ = 1,
γγ∗ = γ∗γ, αγ = qγα, αγ∗ = qγ∗α.
![Page 17: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/17.jpg)
1. The Notion of Quantum Groups (cont.)
Famous Example: C(SUq(2)), q ∈ R, q 6= 0
Generators: α, γ.
Relations: the 2× 2 matrix u :=
[α −qγ∗
γ α∗
]is unitary,
i.e.
α∗α + γ∗γ = 1, αα∗ + q2γγ∗ = 1,
γγ∗ = γ∗γ, αγ = qγα, αγ∗ = qγ∗α.
![Page 18: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/18.jpg)
1. The Notion of Quantum Groups (cont.)
Famous Example: C(SUq(2)), q ∈ R, q 6= 0
Generators: α, γ.
Relations: the 2× 2 matrix u :=
[α −qγ∗
γ α∗
]is unitary,
i.e.
α∗α + γ∗γ = 1, αα∗ + q2γγ∗ = 1,
γγ∗ = γ∗γ, αγ = qγα, αγ∗ = qγ∗α.
![Page 19: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/19.jpg)
1. The Notion of Quantum Groups (cont.)
Facts:
I (ut )−1 =
α∗ −q−1γ
q2γ∗ α
I Hopf algebra structure same as C(SU(2)):
∆(uij) =2∑
k=1
uik ⊗ ukj , i , j = 1,2;
ε(uij) = δij , i , j = 1,2;
S(u) = u−1.
![Page 20: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/20.jpg)
1. The Notion of Quantum Groups (cont.)
Facts:
I (ut )−1 =
α∗ −q−1γ
q2γ∗ α
I Hopf algebra structure same as C(SU(2)):
∆(uij) =2∑
k=1
uik ⊗ ukj , i , j = 1,2;
ε(uij) = δij , i , j = 1,2;
S(u) = u−1.
![Page 21: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/21.jpg)
1. The Notion of Quantum Groups (cont.)
Facts:
I (ut )−1 =
α∗ −q−1γ
q2γ∗ α
I Hopf algebra structure same as C(SU(2)):
∆(uij) =2∑
k=1
uik ⊗ ukj , i , j = 1,2;
ε(uij) = δij , i , j = 1,2;
S(u) = u−1.
![Page 22: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/22.jpg)
1. The Notion of Quantum Groups (cont.)
I Gq = Deformation of arbitrary simple compact Lie groups:
Soibelman et al. based on Drinfeld-Jimbo’s Uq(g)
Guq = Twisting of Gq
I Woronowicz’s theory: Haar measure, Peter-Weyl,
Tannka-Krein, etc.
I Any other classses of examples besides Gq, Guq?
I Yes: Universal quantum groups Au(Q) and Bu(Q),
quantum permutation groups Aaut (Xn), etc.
![Page 23: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/23.jpg)
1. The Notion of Quantum Groups (cont.)
I Gq = Deformation of arbitrary simple compact Lie groups:
Soibelman et al. based on Drinfeld-Jimbo’s Uq(g)
Guq = Twisting of Gq
I Woronowicz’s theory: Haar measure, Peter-Weyl,
Tannka-Krein, etc.
I Any other classses of examples besides Gq, Guq?
I Yes: Universal quantum groups Au(Q) and Bu(Q),
quantum permutation groups Aaut (Xn), etc.
![Page 24: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/24.jpg)
1. The Notion of Quantum Groups (cont.)
I Gq = Deformation of arbitrary simple compact Lie groups:
Soibelman et al. based on Drinfeld-Jimbo’s Uq(g)
Guq = Twisting of Gq
I Woronowicz’s theory: Haar measure, Peter-Weyl,
Tannka-Krein, etc.
I Any other classses of examples besides Gq, Guq?
I Yes: Universal quantum groups Au(Q) and Bu(Q),
quantum permutation groups Aaut (Xn), etc.
![Page 25: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/25.jpg)
1. The Notion of Quantum Groups (cont.)
I Gq = Deformation of arbitrary simple compact Lie groups:
Soibelman et al. based on Drinfeld-Jimbo’s Uq(g)
Guq = Twisting of Gq
I Woronowicz’s theory: Haar measure, Peter-Weyl,
Tannka-Krein, etc.
I Any other classses of examples besides Gq, Guq?
I Yes: Universal quantum groups Au(Q) and Bu(Q),
quantum permutation groups Aaut (Xn), etc.
![Page 26: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/26.jpg)
1. The Notion of Quantum Groups (cont.)
I Gq = Deformation of arbitrary simple compact Lie groups:
Soibelman et al. based on Drinfeld-Jimbo’s Uq(g)
Guq = Twisting of Gq
I Woronowicz’s theory: Haar measure, Peter-Weyl,
Tannka-Krein, etc.
I Any other classses of examples besides Gq, Guq?
I Yes: Universal quantum groups Au(Q) and Bu(Q),
quantum permutation groups Aaut (Xn), etc.
![Page 27: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/27.jpg)
1. The Notion of Quantum Groups (cont.)
I Gq = Deformation of arbitrary simple compact Lie groups:
Soibelman et al. based on Drinfeld-Jimbo’s Uq(g)
Guq = Twisting of Gq
I Woronowicz’s theory: Haar measure, Peter-Weyl,
Tannka-Krein, etc.
I Any other classses of examples besides Gq, Guq?
I Yes: Universal quantum groups Au(Q) and Bu(Q),
quantum permutation groups Aaut (Xn), etc.
![Page 28: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/28.jpg)
2. Universal CMQGs Au(Q) and Bu(Q)
For u = (uij), u := (u∗ij ), u∗ := ut ; Q is an n × n non-singular
complex scalar matrix.
Au(Q) := C∗{uij : u∗ = u−1, (ut )−1 = QuQ−1}
Bu(Q) := C∗{uij : u∗ = u−1, (ut )−1 = QuQ−1}
![Page 29: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/29.jpg)
2. Universal CMQGs Au(Q) and Bu(Q)
For u = (uij), u := (u∗ij ), u∗ := ut ; Q is an n × n non-singular
complex scalar matrix.
Au(Q) := C∗{uij : u∗ = u−1, (ut )−1 = QuQ−1}
Bu(Q) := C∗{uij : u∗ = u−1, (ut )−1 = QuQ−1}
![Page 30: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/30.jpg)
2. Universal CMQGs Au(Q) and Bu(Q)
For u = (uij), u := (u∗ij ), u∗ := ut ; Q is an n × n non-singular
complex scalar matrix.
Au(Q) := C∗{uij : u∗ = u−1, (ut )−1 = QuQ−1}
Bu(Q) := C∗{uij : u∗ = u−1, (ut )−1 = QuQ−1}
![Page 31: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/31.jpg)
2. Au(Q) and Bu(Q) (cont.)
THEOREM 1.
(1) Au(Q) and Bu(Q) are CMQGs (in fact quantum
automorphism groups).
(2) Every CMQG is a quantum subgroup of Au(Q) for some
Q > 0;
Every CMQG with self conjugate fundamental representation is
a quantum subgroup of Bu(Q) for some Q.
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2. Au(Q) and Bu(Q) (cont.)
THEOREM 1.
(1) Au(Q) and Bu(Q) are CMQGs (in fact quantum
automorphism groups).
(2) Every CMQG is a quantum subgroup of Au(Q) for some
Q > 0;
Every CMQG with self conjugate fundamental representation is
a quantum subgroup of Bu(Q) for some Q.
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2. Au(Q) and Bu(Q) (cont.)
THEOREM 1.
(1) Au(Q) and Bu(Q) are CMQGs (in fact quantum
automorphism groups).
(2) Every CMQG is a quantum subgroup of Au(Q) for some
Q > 0;
Every CMQG with self conjugate fundamental representation is
a quantum subgroup of Bu(Q) for some Q.
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2. Au(Q) and Bu(Q) (cont.)
I The C∗-algebras Au(Q) and Bu(Q) are non-nuclear (even
non-exact) for generic Q’s.
e.g. C∗(Fn) is a quotient of Au(Q) for Q > 0.
I Bu(Q) = C(SUq(2)) for
Q =
0 −1
q 0
, q ∈ R, q 6= 0.
I Banica’s computed the fusion rings of Au(Q) (for Q > 0)
and Bu(Q) (for QQ) in his deep thesis.
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2. Au(Q) and Bu(Q) (cont.)
I The C∗-algebras Au(Q) and Bu(Q) are non-nuclear (even
non-exact) for generic Q’s.
e.g. C∗(Fn) is a quotient of Au(Q) for Q > 0.
I Bu(Q) = C(SUq(2)) for
Q =
0 −1
q 0
, q ∈ R, q 6= 0.
I Banica’s computed the fusion rings of Au(Q) (for Q > 0)
and Bu(Q) (for QQ) in his deep thesis.
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2. Au(Q) and Bu(Q) (cont.)
I The C∗-algebras Au(Q) and Bu(Q) are non-nuclear (even
non-exact) for generic Q’s.
e.g. C∗(Fn) is a quotient of Au(Q) for Q > 0.
I Bu(Q) = C(SUq(2)) for
Q =
0 −1
q 0
, q ∈ R, q 6= 0.
I Banica’s computed the fusion rings of Au(Q) (for Q > 0)
and Bu(Q) (for QQ) in his deep thesis.
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3. Au(Q) for Positive Q
Q > 0 is called normalized if Tr(Q) = Tr(Q−1).
THEOREM 2. Let Q ∈ GL(n,C) and Q′ ∈ GL(n′,C) be positive,
normalized, with eigen values q1 ≥ q2 ≥ · · · ≥ qn and
q′1 ≥ q′2 ≥ · · · ≥ q′n′ respectively.
Then, Au(Q) is isomorphic to Au(Q′) iff,
(i) n = n′, and
(ii) (q1,q2, · · · ,qn) = (q′1,q′2, · · · ,q′n) or
(q−1n ,q−1
n−1, · · · ,q−11 ) = (q′1,q
′2, · · · ,q′n).
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3. Au(Q) for Positive Q
Q > 0 is called normalized if Tr(Q) = Tr(Q−1).
THEOREM 2. Let Q ∈ GL(n,C) and Q′ ∈ GL(n′,C) be positive,
normalized, with eigen values q1 ≥ q2 ≥ · · · ≥ qn and
q′1 ≥ q′2 ≥ · · · ≥ q′n′ respectively.
Then, Au(Q) is isomorphic to Au(Q′) iff,
(i) n = n′, and
(ii) (q1,q2, · · · ,qn) = (q′1,q′2, · · · ,q′n) or
(q−1n ,q−1
n−1, · · · ,q−11 ) = (q′1,q
′2, · · · ,q′n).
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3. Au(Q) for Positive Q
Q > 0 is called normalized if Tr(Q) = Tr(Q−1).
THEOREM 2. Let Q ∈ GL(n,C) and Q′ ∈ GL(n′,C) be positive,
normalized, with eigen values q1 ≥ q2 ≥ · · · ≥ qn and
q′1 ≥ q′2 ≥ · · · ≥ q′n′ respectively.
Then, Au(Q) is isomorphic to Au(Q′) iff,
(i) n = n′, and
(ii) (q1,q2, · · · ,qn) = (q′1,q′2, · · · ,q′n) or
(q−1n ,q−1
n−1, · · · ,q−11 ) = (q′1,q
′2, · · · ,q′n).
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3. Au(Q) for Positive Q
Q > 0 is called normalized if Tr(Q) = Tr(Q−1).
THEOREM 2. Let Q ∈ GL(n,C) and Q′ ∈ GL(n′,C) be positive,
normalized, with eigen values q1 ≥ q2 ≥ · · · ≥ qn and
q′1 ≥ q′2 ≥ · · · ≥ q′n′ respectively.
Then, Au(Q) is isomorphic to Au(Q′) iff,
(i) n = n′, and
(ii) (q1,q2, · · · ,qn) = (q′1,q′2, · · · ,q′n) or
(q−1n ,q−1
n−1, · · · ,q−11 ) = (q′1,q
′2, · · · ,q′n).
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4. Bu(Q) with QQ ∈ RIn
THEOREM 3. Let Q ∈ GL(n,C) and Q′ ∈ GL(n′,C) be such
that QQ ∈ RIn, Q′Q′ ∈ RIn′ , respectively.
Then, Bu(Q) is
isomorphic to Bu(Q′) iff,
(i) n = n′, and
(ii) there exist S ∈ U(n) and z ∈ C∗ such that
Q = zStQ′S.
Note: The quantum groups Bu(Q) are simple in an appropriate
sense: cf. S. Wang: “Simple compact quantum groups I”, JFA,
256 (2009), 3313-3341.
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4. Bu(Q) with QQ ∈ RIn
THEOREM 3. Let Q ∈ GL(n,C) and Q′ ∈ GL(n′,C) be such
that QQ ∈ RIn, Q′Q′ ∈ RIn′ , respectively. Then, Bu(Q) is
isomorphic to Bu(Q′) iff,
(i) n = n′,
and
(ii) there exist S ∈ U(n) and z ∈ C∗ such that
Q = zStQ′S.
Note: The quantum groups Bu(Q) are simple in an appropriate
sense: cf. S. Wang: “Simple compact quantum groups I”, JFA,
256 (2009), 3313-3341.
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4. Bu(Q) with QQ ∈ RIn
THEOREM 3. Let Q ∈ GL(n,C) and Q′ ∈ GL(n′,C) be such
that QQ ∈ RIn, Q′Q′ ∈ RIn′ , respectively. Then, Bu(Q) is
isomorphic to Bu(Q′) iff,
(i) n = n′, and
(ii) there exist S ∈ U(n) and z ∈ C∗ such that
Q = zStQ′S.
Note: The quantum groups Bu(Q) are simple in an appropriate
sense: cf. S. Wang: “Simple compact quantum groups I”, JFA,
256 (2009), 3313-3341.
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4. Bu(Q) with QQ ∈ RIn
THEOREM 3. Let Q ∈ GL(n,C) and Q′ ∈ GL(n′,C) be such
that QQ ∈ RIn, Q′Q′ ∈ RIn′ , respectively. Then, Bu(Q) is
isomorphic to Bu(Q′) iff,
(i) n = n′, and
(ii) there exist S ∈ U(n) and z ∈ C∗ such that
Q = zStQ′S.
Note: The quantum groups Bu(Q) are simple in an appropriate
sense: cf. S. Wang: “Simple compact quantum groups I”, JFA,
256 (2009), 3313-3341.
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5. Au(Q) and Bu(Q) for Arbitrary Q
Note: Au(Q) = C(T), Bu(Q) = C∗(Z/2Z) for Q ∈ GL(1,C).
THEOREM 4. Let Q ∈ GL(n,C). Then there exists positive
matrices Pj such that
Au(Q) ∼= Au(P1) ∗ Au(P2) ∗ · · · ∗ Au(Pk ).
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5. Au(Q) and Bu(Q) for Arbitrary Q
Note: Au(Q) = C(T), Bu(Q) = C∗(Z/2Z) for Q ∈ GL(1,C).
THEOREM 4. Let Q ∈ GL(n,C). Then there exists positive
matrices Pj such that
Au(Q) ∼= Au(P1) ∗ Au(P2) ∗ · · · ∗ Au(Pk ).
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5. Au(Q) and Bu(Q) for Arbitrary Q
Note: Au(Q) = C(T), Bu(Q) = C∗(Z/2Z) for Q ∈ GL(1,C).
THEOREM 4. Let Q ∈ GL(n,C). Then there exists positive
matrices Pj such that
Au(Q) ∼= Au(P1) ∗ Au(P2) ∗ · · · ∗ Au(Pk ).
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5. Au(Q) and Bu(Q) for Arbitrary Q
Note: Au(Q) = C(T), Bu(Q) = C∗(Z/2Z) for Q ∈ GL(1,C).
THEOREM 4. Let Q ∈ GL(n,C). Then there exists positive
matrices Pj such that
Au(Q) ∼= Au(P1) ∗ Au(P2) ∗ · · · ∗ Au(Pk ).
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5. Au(Q) and Bu(Q) for Arbitrary Q
THEOREM 5. Let Q ∈ GL(n,C). Then there exist positive
matrices Pi (i ≤ k ) and matrices Qj (j ≤ l) with QjQj ’s are
nonzero scalars
and that
Bu(Q) ∼= Au(P1) ∗ Au(P2) ∗ · · · ∗ Au(Pk )∗
∗Bu(Q1) ∗ Bu(Q2) ∗ · · · ∗ Bu(Ql).
Note: This is contrary to an earlier belief that Au(Pi)’s do not
appear in the decomp.!
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5. Au(Q) and Bu(Q) for Arbitrary Q
THEOREM 5. Let Q ∈ GL(n,C). Then there exist positive
matrices Pi (i ≤ k ) and matrices Qj (j ≤ l) with QjQj ’s are
nonzero scalars and that
Bu(Q) ∼= Au(P1) ∗ Au(P2) ∗ · · · ∗ Au(Pk )∗
∗Bu(Q1) ∗ Bu(Q2) ∗ · · · ∗ Bu(Ql).
Note: This is contrary to an earlier belief that Au(Pi)’s do not
appear in the decomp.!
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5. Au(Q) and Bu(Q) for Arbitrary Q
THEOREM 5. Let Q ∈ GL(n,C). Then there exist positive
matrices Pi (i ≤ k ) and matrices Qj (j ≤ l) with QjQj ’s are
nonzero scalars and that
Bu(Q) ∼= Au(P1) ∗ Au(P2) ∗ · · · ∗ Au(Pk )∗
∗Bu(Q1) ∗ Bu(Q2) ∗ · · · ∗ Bu(Ql).
Note: This is contrary to an earlier belief that Au(Pi)’s do not
appear in the decomp.!
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5. Au(Q) and Bu(Q) for Arbitrary Q
COROLLARY of THEOREM 4.
(1). Let Q = diag(eiθ1P1,eiθ2P2, · · · ,eiθk Pk ), with positive
matrices Pj and distinct angles 0 ≤ θj < 2π, j = 1, · · · , k , k ≥ 1
Then
Au(Q) ∼= Au(P1) ∗ Au(P2) ∗ · · · ∗ Au(Pk ).
(2). Let Q ∈ GL(2,C) be a non-normal matrix. Then
Au(Q) = C(T).
(3). For Q ∈ GL(2,C), Au(Q) is isomorphic to either C(T), or
C(T) ∗ C(T), or Au(diag(1,q)) with 0 < q ≤ 1.
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5. Au(Q) and Bu(Q) for Arbitrary Q
COROLLARY of THEOREM 4.
(1). Let Q = diag(eiθ1P1,eiθ2P2, · · · ,eiθk Pk ), with positive
matrices Pj and distinct angles 0 ≤ θj < 2π, j = 1, · · · , k , k ≥ 1
Then
Au(Q) ∼= Au(P1) ∗ Au(P2) ∗ · · · ∗ Au(Pk ).
(2). Let Q ∈ GL(2,C) be a non-normal matrix. Then
Au(Q) = C(T).
(3). For Q ∈ GL(2,C), Au(Q) is isomorphic to either C(T), or
C(T) ∗ C(T), or Au(diag(1,q)) with 0 < q ≤ 1.
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5. Au(Q) and Bu(Q) for Arbitrary Q
COROLLARY of THEOREM 4.
(1). Let Q = diag(eiθ1P1,eiθ2P2, · · · ,eiθk Pk ), with positive
matrices Pj and distinct angles 0 ≤ θj < 2π, j = 1, · · · , k , k ≥ 1
Then
Au(Q) ∼= Au(P1) ∗ Au(P2) ∗ · · · ∗ Au(Pk ).
(2). Let Q ∈ GL(2,C) be a non-normal matrix. Then
Au(Q) = C(T).
(3). For Q ∈ GL(2,C), Au(Q) is isomorphic to either C(T), or
C(T) ∗ C(T), or Au(diag(1,q)) with 0 < q ≤ 1.
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5. Au(Q) and Bu(Q) for Arbitrary Q
COROLLARY of THEOREM 4.
(1). Let Q = diag(eiθ1P1,eiθ2P2, · · · ,eiθk Pk ), with positive
matrices Pj and distinct angles 0 ≤ θj < 2π, j = 1, · · · , k , k ≥ 1
Then
Au(Q) ∼= Au(P1) ∗ Au(P2) ∗ · · · ∗ Au(Pk ).
(2). Let Q ∈ GL(2,C) be a non-normal matrix. Then
Au(Q) = C(T).
(3). For Q ∈ GL(2,C), Au(Q) is isomorphic to either C(T), or
C(T) ∗ C(T), or Au(diag(1,q)) with 0 < q ≤ 1.
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5. Au(Q) and Bu(Q) for Arbitrary Q
COROLLARY of THEOREM 5.
(1). Let Q = diag(T1,T2, · · · ,Tk ) be such that Tj Tj = λj Inj ,
where the λj ’s are distinct non-zero real numbers (the nj ’s need
not be different), j = 1, · · · , k , k ≥ 1.
Then
Bu(Q) ∼= Bu(T1) ∗ Bu(T2) ∗ · · · ∗ Bu(Tk ).
(2). Let Q =
[0 T
qT−1 0
], where T ∈ GL(n,C) and q is a
complex but non-real number. Then Bu(Q) is isomorphic to
Au(|T |2).
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5. Au(Q) and Bu(Q) for Arbitrary Q
COROLLARY of THEOREM 5.
(1). Let Q = diag(T1,T2, · · · ,Tk ) be such that Tj Tj = λj Inj ,
where the λj ’s are distinct non-zero real numbers (the nj ’s need
not be different), j = 1, · · · , k , k ≥ 1. Then
Bu(Q) ∼= Bu(T1) ∗ Bu(T2) ∗ · · · ∗ Bu(Tk ).
(2). Let Q =
[0 T
qT−1 0
], where T ∈ GL(n,C) and q is a
complex but non-real number. Then Bu(Q) is isomorphic to
Au(|T |2).
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5. Au(Q) and Bu(Q) for Arbitrary Q
COROLLARY of THEOREM 5.
(1). Let Q = diag(T1,T2, · · · ,Tk ) be such that Tj Tj = λj Inj ,
where the λj ’s are distinct non-zero real numbers (the nj ’s need
not be different), j = 1, · · · , k , k ≥ 1. Then
Bu(Q) ∼= Bu(T1) ∗ Bu(T2) ∗ · · · ∗ Bu(Tk ).
(2). Let Q =
[0 T
qT−1 0
], where T ∈ GL(n,C) and q is a
complex but non-real number.
Then Bu(Q) is isomorphic to
Au(|T |2).
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5. Au(Q) and Bu(Q) for Arbitrary Q
COROLLARY of THEOREM 5.
(1). Let Q = diag(T1,T2, · · · ,Tk ) be such that Tj Tj = λj Inj ,
where the λj ’s are distinct non-zero real numbers (the nj ’s need
not be different), j = 1, · · · , k , k ≥ 1. Then
Bu(Q) ∼= Bu(T1) ∗ Bu(T2) ∗ · · · ∗ Bu(Tk ).
(2). Let Q =
[0 T
qT−1 0
], where T ∈ GL(n,C) and q is a
complex but non-real number. Then Bu(Q) is isomorphic to
Au(|T |2).
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6. Important Related Work
A Conceptual Breakthrough:
Debashish Goswami used Au(Q) to prove the existence of
quantum isometry groups for very general classes of
noncommutative spaces in the sense of Connes.
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6. Important Related Work
A Conceptual Breakthrough:
Debashish Goswami used Au(Q) to prove the existence of
quantum isometry groups for very general classes of
noncommutative spaces in the sense of Connes.
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6. Important Related Work (cont.)
Banica’s Fusion Theorem 1:
The irreducible representations πx of the quantum group Au(Q)
are parameterized by x ∈ N ∗ N, where N ∗ N is the free monoid
with generators α and β and anti-multiplicative involution α = β,
and the classes of u, u are α, β respectively. With this
parametrization, one has the fusion rules
πx ⊗ πy =∑
x=ag,gb=y
πab
![Page 63: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/63.jpg)
6. Important Related Work (cont.)
Banica’s Fusion Theorem 1:
The irreducible representations πx of the quantum group Au(Q)
are parameterized by x ∈ N ∗ N,
where N ∗ N is the free monoid
with generators α and β and anti-multiplicative involution α = β,
and the classes of u, u are α, β respectively. With this
parametrization, one has the fusion rules
πx ⊗ πy =∑
x=ag,gb=y
πab
![Page 64: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/64.jpg)
6. Important Related Work (cont.)
Banica’s Fusion Theorem 1:
The irreducible representations πx of the quantum group Au(Q)
are parameterized by x ∈ N ∗ N, where N ∗ N is the free monoid
with generators α and β and anti-multiplicative involution α = β,
and the classes of u, u are α, β respectively.
With this
parametrization, one has the fusion rules
πx ⊗ πy =∑
x=ag,gb=y
πab
![Page 65: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/65.jpg)
6. Important Related Work (cont.)
Banica’s Fusion Theorem 1:
The irreducible representations πx of the quantum group Au(Q)
are parameterized by x ∈ N ∗ N, where N ∗ N is the free monoid
with generators α and β and anti-multiplicative involution α = β,
and the classes of u, u are α, β respectively. With this
parametrization, one has the fusion rules
πx ⊗ πy =∑
x=ag,gb=y
πab
![Page 66: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/66.jpg)
6. Important Related Work (cont.)
Banica’s Fusion Theorem 2:
The irreducible representations πk of the quantum group Bu(Q)
are parameterized by k ∈ Z+ = {0,1,2, · · · }, and one has the
fusion rules
πk ⊗ πl = π|k−l| ⊕ π|k−l|+2 ⊕ · · · ⊕ πk+l−2 ⊕ πk+l
![Page 67: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/67.jpg)
6. Important Related Work (cont.)
Banica’s Fusion Theorem 2:
The irreducible representations πk of the quantum group Bu(Q)
are parameterized by k ∈ Z+ = {0,1,2, · · · },
and one has the
fusion rules
πk ⊗ πl = π|k−l| ⊕ π|k−l|+2 ⊕ · · · ⊕ πk+l−2 ⊕ πk+l
![Page 68: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/68.jpg)
6. Important Related Work (cont.)
Banica’s Fusion Theorem 2:
The irreducible representations πk of the quantum group Bu(Q)
are parameterized by k ∈ Z+ = {0,1,2, · · · }, and one has the
fusion rules
πk ⊗ πl = π|k−l| ⊕ π|k−l|+2 ⊕ · · · ⊕ πk+l−2 ⊕ πk+l
![Page 69: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/69.jpg)
7. Open Problems
I (1) Construct explicit models of irreducible representations
πx (x ∈ N ∗ N) of the quantum groups Au(Q) for positive
Q > 0.
I (2) Construct explicit models of irreducible representations
πk (k ∈ Z+) of the quantum groups Bu(Q) for Q with
QQ = ±1.
![Page 70: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/70.jpg)
7. Open Problems
I (1) Construct explicit models of irreducible representations
πx (x ∈ N ∗ N) of the quantum groups Au(Q) for positive
Q > 0.
I (2) Construct explicit models of irreducible representations
πk (k ∈ Z+) of the quantum groups Bu(Q) for Q with
QQ = ±1.
![Page 71: Structure of Compact Quantum Groups Au(Q) and Bu(Q) and their …math.ecnu.edu.cn › RCFOA › visitors › Suzhou Wang › aubu-slides... · 2014-01-27 · 1. The Notion of Quantum](https://reader030.vdocuments.net/reader030/viewer/2022041111/5f10e3247e708231d44b4a22/html5/thumbnails/71.jpg)
THANKS FOR YOUR ATTENTION!