Structure of Compact Quantum Groups A u (Q) and B u (Q) and their Isomorphism Classification

Structure of Compact Quantum Groups A u (Q) and B u (Q) and their Isomorphism Classification Shuzhou Wang Department of Mathematics University of Georgia

1. The Notion of Quantum Groups G = Simple compact Lie group, e.g. G = SU(2) = { α γ : αᾱ + γ γ = 1, α, γ C }. γ ᾱ A = C(G)

1. The Notion of Quantum Groups G = Simple compact Lie group, e.g. G = SU(2) = { α γ : αᾱ + γ γ = 1, α, γ C }. γ ᾱ A = C(G) Lie group G Hopf algebra (A,, ε, S).

1. The Notion of Quantum Groups G = Simple compact Lie group, e.g. G = SU(2) = { α γ : αᾱ + γ γ = 1, α, γ C }. γ ᾱ A = C(G) 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 ).

1. The Notion of Quantum Groups (cont.) Spectacular development in Mid-1980 s: Drinfeld, Jimbo, Woronowicz,

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(G q ). G q = quantum group

1. The Notion of Quantum Groups (cont.) Recall Hilbert 5th Problem: Characterization of Lie groups among topological groups.

1. The Notion of Quantum Groups (cont.) Recall Hilbert 5th Problem: Characterization of Lie groups among topological groups. New Problem: Characterization of quantum groups among Hopf algebras.

1. The Notion of Quantum Groups (cont.) Recall Hilbert 5th Problem: Characterization of Lie groups among topological groups. New Problem: Characterization of quantum groups among Hopf algebras. Lesson: Quantums groups = nice Hopf algebras Restrict to such Hopf algebras to obtain nice and deep theory.

1. The Notion of Quantum Groups (cont.) DEFINITION: A compact matrix quantum group (CMQG) is a pair G = (A, u) of a unital C -algebra A and u = (u ij ) n i,j=1 M n(a) satisfying

1. The Notion of Quantum Groups (cont.) DEFINITION: A compact matrix quantum group (CMQG) is a pair G = (A, u) of a unital C -algebra A and u = (u ij ) n i,j=1 M n(a) satisfying (1) : A A A with n (u ij ) = u ik u kj, i, j = 1,, n; k=1

1. The Notion of Quantum Groups (cont.) Note: Equivalent definition of CMQG is obtained if condition (2) is replaced with (2 ) u 1 and (u t ) 1 in M n (A)

1. The Notion of Quantum Groups (cont.) Note: Equivalent definition of CMQG is obtained if condition (2) is replaced with (2 ) u 1 and (u t ) 1 in M n (A) There are other equivalent definitions of CMQG

1. The Notion of Quantum Groups (cont.) Famous Example: C(SU q (2)), q R, q 0

1. The Notion of Quantum Groups (cont.) Famous Example: C(SU q (2)), q R, q 0 Generators: α, γ.

1. The Notion of Quantum Groups (cont.) Famous Example: C(SU q (2)), q R, q 0 Generators: α, γ. [ α qγ Relations: the 2 2 matrix u := γ α ] is unitary,

1. The Notion of Quantum Groups (cont.) Famous Example: C(SU q (2)), q R, q 0 Generators: α, γ. [ α qγ Relations: the 2 2 matrix u := γ α i.e. ] is unitary, α α + γ γ = 1, αα + q 2 γγ = 1, γγ = γ γ, αγ = qγα, αγ = qγ α.

1. The Notion of Quantum Groups (cont.) Facts:

1. The Notion of Quantum Groups (cont.) Facts: (u t ) 1 = α q 1 γ q 2 γ α

1. The Notion of Quantum Groups (cont.) Facts: (u t ) 1 = α q 1 γ q 2 γ α Hopf algebra structure same as C(SU(2)): (u ij ) = 2 u ik u kj, i, j = 1, 2; k=1 ε(u ij ) = δ ij, i, j = 1, 2; S(u) = u 1.

1. The Notion of Quantum Groups (cont.) G q = Deformation of arbitrary simple compact Lie groups: Soibelman et al. based on Drinfeld-Jimbo s U q (g)

1. The Notion of Quantum Groups (cont.) G q = Deformation of arbitrary simple compact Lie groups: Soibelman et al. based on Drinfeld-Jimbo s U q (g) G u q = Twisting of G q

1. The Notion of Quantum Groups (cont.) G q = Deformation of arbitrary simple compact Lie groups: Soibelman et al. based on Drinfeld-Jimbo s U q (g) Gq u = Twisting of G q Woronowicz s theory: Haar measure, Peter-Weyl, Tannka-Krein, etc. Any other classses of examples besides G q, Gq? u

2. Universal CMQGs A u (Q) and B u (Q) For u = (u ij ), ū := (u ij ), u := ū t ; Q is an n n non-singular complex scalar matrix.

2. Universal CMQGs A u (Q) and B u (Q) For u = (u ij ), ū := (u ij ), u := ū t ; Q is an n n non-singular complex scalar matrix. A u (Q) := C {u ij : u = u 1, (u t ) 1 = QūQ 1 }

2. Universal CMQGs A u (Q) and B u (Q) For u = (u ij ), ū := (u ij ), u := ū t ; Q is an n n non-singular complex scalar matrix. A u (Q) := C {u ij : u = u 1, (u t ) 1 = QūQ 1 } B u (Q) := C {u ij : u = u 1, (u t ) 1 = QuQ 1 }

2. A u(q) and B u(q) (cont.) THEOREM 1. (1) A u (Q) and B u (Q) are CMQGs (in fact quantum automorphism groups).

2. A u(q) and B u(q) (cont.) THEOREM 1. (1) A u (Q) and B u (Q) are CMQGs (in fact quantum automorphism groups). (2) Every CMQG is a quantum subgroup of A u (Q) for some Q > 0;

2. A u(q) and B u(q) (cont.) THEOREM 1. (1) A u (Q) and B u (Q) are CMQGs (in fact quantum automorphism groups). (2) Every CMQG is a quantum subgroup of A u (Q) for some Q > 0; Every CMQG with self conjugate fundamental representation is a quantum subgroup of B u (Q) for some Q.

2. A u(q) and B u(q) (cont.) The C -algebras A u (Q) and B u (Q) are non-nuclear (even non-exact) for generic Q s. e.g. C (F n ) is a quotient of A u (Q) for Q > 0.

2. A u(q) and B u(q) (cont.) The C -algebras A u (Q) and B u (Q) are non-nuclear (even non-exact) for generic Q s. e.g. C (F n ) is a quotient of A u (Q) for Q > 0. B u (Q) = C(SU q (2)) for Q = 0 1, q R, q 0. q 0

2. A u(q) and B u(q) (cont.) The C -algebras A u (Q) and B u (Q) are non-nuclear (even non-exact) for generic Q s. e.g. C (F n ) is a quotient of A u (Q) for Q > 0. B u (Q) = C(SU q (2)) for Q = 0 1 q 0, q R, q 0. Banica s computed the fusion rings of A u (Q) (for Q > 0) and B u (Q) (for Q Q) in his deep thesis.

3. A u (Q) for Positive Q Q > 0 is called normalized if Tr(Q) = Tr(Q 1 ).

3. A u (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 q 1 q 2 q n and q 1 q 2 q n respectively.

3. A u (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 q 1 q 2 q n and q 1 q 2 q n respectively. Then, A u (Q) is isomorphic to A u (Q ) iff, (i) n = n, and

4. B u (Q) with Q Q RI n THEOREM 3. Let Q GL(n, C) and Q GL(n, C) be such that Q Q RI n, Q Q RI n, respectively.

4. B u (Q) with Q Q RI n THEOREM 3. Let Q GL(n, C) and Q GL(n, C) be such that Q Q RI n, Q Q RI n, respectively. Then, B u (Q) is isomorphic to B u (Q ) iff, (i) n = n,

4. B u (Q) with Q Q RI n THEOREM 3. Let Q GL(n, C) and Q GL(n, C) be such that Q Q RI n, Q Q RI n, respectively. Then, B u (Q) is isomorphic to B u (Q ) iff, (i) n = n, and (ii) there exist S U(n) and z C such that Q = zs t Q S. Note: The quantum groups B u (Q) are simple in an appropriate sense: cf. S. Wang: Simple compact quantum groups I, JFA, 256 (2009), 3313-3341.

5. A u (Q) and B u (Q) for Arbitrary Q

5. A u (Q) and B u (Q) for Arbitrary Q Note: A u (Q) = C(T), B u (Q) = C (Z/2Z) for Q GL(1, C).

5. A u (Q) and B u (Q) for Arbitrary Q Note: A u (Q) = C(T), B u (Q) = C (Z/2Z) for Q GL(1, C). THEOREM 4. Let Q GL(n, C). Then there exists positive matrices P j such that

5. A u (Q) and B u (Q) for Arbitrary Q Note: A u (Q) = C(T), B u (Q) = C (Z/2Z) for Q GL(1, C). THEOREM 4. Let Q GL(n, C). Then there exists positive matrices P j such that A u (Q) = A u (P 1 ) A u (P 2 ) A u (P k ).

5. A u(q) and B u(q) for Arbitrary Q THEOREM 5. Let Q GL(n, C). Then there exist positive matrices P i (i k) and matrices Q j (j l) with Q j Qj s are nonzero scalars

5. A u(q) and B u(q) for Arbitrary Q THEOREM 5. Let Q GL(n, C). Then there exist positive matrices P i (i k) and matrices Q j (j l) with Q j Qj s are nonzero scalars and that B u (Q) = A u (P 1 ) A u (P 2 ) A u (P k ) B u (Q 1 ) B u (Q 2 ) B u (Q l ).

5. A u(q) and B u(q) for Arbitrary Q COROLLARY of THEOREM 4. (1). Let Q = diag(e iθ 1P 1, e iθ 2P 2,, e iθ k P k ), with positive matrices P j and distinct angles 0 θ j < 2π, j = 1,, k, k 1

5. A u(q) and B u(q) for Arbitrary Q COROLLARY of THEOREM 4. (1). Let Q = diag(e iθ 1P 1, e iθ 2P 2,, e iθ k P k ), with positive matrices P j and distinct angles 0 θ j < 2π, j = 1,, k, k 1 Then A u (Q) = A u (P 1 ) A u (P 2 ) A u (P k ). (2). Let Q GL(2, C) be a non-normal matrix. Then A u (Q) = C(T).

5. A u(q) and B u(q) for Arbitrary Q COROLLARY of THEOREM 5. (1). Let Q = diag(t 1, T 2,, T k ) be such that T j Tj = λ j I nj, where the λ j s are distinct non-zero real numbers (the n j s need not be different), j = 1,, k, k 1. Then B u (Q) = B u (T 1 ) B u (T 2 ) B u (T k ). [ ] 0 T (2). Let Q = q T 1, where T GL(n, C) and q is a 0 complex but non-real number.

6. Important Related Work A Conceptual Breakthrough:

6. Important Related Work A Conceptual Breakthrough: Debashish Goswami used A u (Q) to prove the existence of quantum isometry groups for very general classes of noncommutative spaces in the sense of Connes.

6. Important Related Work (cont.) Banica s Fusion Theorem 1:

6. Important Related Work (cont.) Banica s Fusion Theorem 1: The irreducible representations π x of the quantum group A u (Q) are parameterized by x N N,

6. Important Related Work (cont.) Banica s Fusion Theorem 1: The irreducible representations π x of the quantum group A u (Q) are parameterized by x N N, where N N is the free monoid with generators α and β and anti-multiplicative involution ᾱ = β, and the classes of u, ū are α, β respectively.

6. Important Related Work (cont.) Banica s Fusion Theorem 2:

6. Important Related Work (cont.) Banica s Fusion Theorem 2: The irreducible representations π k of the quantum group B u (Q) are parameterized by k Z + = {0, 1, 2, },

6. Important Related Work (cont.) Banica s Fusion Theorem 2: The irreducible representations π k of the quantum group B u (Q) are parameterized by k Z + = {0, 1, 2, }, and one has the fusion rules π k π l = π k l π k l +2 π k+l 2 π k+l

7. Open Problems (1) Construct explicit models of irreducible representations π x (x N N) of the quantum groups A u (Q) for positive Q > 0.

7. Open Problems (1) Construct explicit models of irreducible representations π x (x N N) of the quantum groups A u (Q) for positive Q > 0. (2) Construct explicit models of irreducible representations π k (k Z + ) of the quantum groups B u (Q) for Q with Q Q = ±1.

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