Actions of Compact Quantum Groups I

Transcription

1 Actions of Compact Quantum Groups I Definition Kenny De Commer (VUB, Brussels, Belgium)

2 Course material Material that will be treated: Actions and coactions of compact quantum groups. Actions on C -algebras and Hilbert modules. Crossed products. Free actions, ergodic actions, and their interrelationship.

3 Outline Lecture I Compact quantum groups Actions of compact quantum groups on compact quantum spaces Actions on non-compact quantum spaces

4 Compact quantum groups Definition (Woronowicz) Compact quantum group (CQG) G: unital C -algebra C(G), unital -homomorphism, comultiplication : C(G) C(G) C(G) s.t. coassociativity: ( id) = (id ), cancellation: [(C(G) 1 G ) (C(G))] = [ (C(G))(1 G C(G))] = C(G) C(G). Here: [S] = closed linear span of S (in some Banach space).

5 Classical CQG Lemma X, Y compact Hausdorff: C(X) C(Y ) = C(X Y ), (a b)(x, y) = a(x)b(y). Example G compact Hausdorff group CQG (C(G), ), : C(G) C(G) C(G), f ( (f) : (g, h) f(gh)). Conversely: CQG G with C(G) commutative G = Spec(C(G)) compact Hausdorff group.

6 C(G)-corepresentations Definition Unitary C(G)-corepresentation: finite dimensional Hilbert space H, U B(H) C(G) s.t. U unitary, (id )(U) = U 12 U 13, where U 12 = U 1 etc. U B(H) C(G) δ : H H C(G), ξ U(ξ 1 G ) s.t....?

7 G-representations Definition G compact quantum group. (Continuous finite dimensional unitary left) G-representation π: finite dimensional Hilbert space H π, linear map δ π : H π H π C(G) s.t. right comodule: (id ) δ π = (δ π id) δ π, isometric: δ π(ξ) δ π(η) = ξ, η 1 C(G), density: [δ π(h)(1 C(G))] = H C(G). Here H π = B(C, Hπ), so (ξ a) (η b) = ξ η a b = ξ, η a b. Density condition automatically satisfied. C(G)-corepresentations G-representations.

8 Classical representations Example Let G compact Hausdorff group. Then G-representations as compact quantum group G-representations as compact group by δ π : H π H π C(G) = C(G, H π ) π : G H π H π, (g, ξ) π(g)ξ = δ π (ξ)(g).

9 The canonical Hopf -algebra Theorem (Woronowicz) Let Then O(G) = {(ξ id)δ π (η) π G-representation, ξ, η H π }. (O(G), ) Hopf -algebra, (O(G),, ɛ, S), O(G) dense in C(G), (O(G), ) unique dense Hopf -algebra, δ π : H H O(G) is O(G)-comodule: (id ) δ π = (δ π id) δ π, (id H ɛ)δ π = id H.

10 Notation (Sweedler-Heynemann notation) h O(G): (h) = h (1) h (2), ( ι) (h) = (2) (h) = h (1) h (2) h (3),... Example Let h O(G). Then Hence (h (1) )(1 S(h (2) )) = h (1) h (2) S(h (3) ) = h (1) ɛ(h (2) )1 = h 1. (Linear span) (O(G))(1 O(G)) = O(G) O(G). alg

11 Universal C -algebra Lemma G CQG. Universal C -envelope C(G u ) of O(G) exists. CQG G u by u : C(G u ) C(G u ) C(G u ). Definition G u universal CQG (associated to G).

12 Right actions of compact quantum groups on C -algebras Definition (Podleś) Right action X G: Compact quantum group G, C -algebra C(X) (with X compact quantum space ), Unital -homomorphism, right coaction α : C(X) C(X) C(G) s.t. coaction property: (α id G ) α = (id X ) α, density (Podleś condition): [α(c(x))(1 X C(G))] = C(X) C(G).

13 Right translations Example Let G compact quantum group. Then G G by : C(G) C(G) C(G).

14 Half-classical case Lemma (All C(G) commutative) G compact Hausdorff group, C -algebra C(X), G α C(X) continuous action: (g, a) α g (a) continuous, each α g -automorphism, α gh = α g α h, αe = id X, for e G identity element. X G, α : C(X) C(X) C(G) = C(G, C(X)), a (α(a) : g α g (a)).

15 Proof, Part I Forgetting group structure: Using partitions of unity on G: C(X) C(G) = C(G, C(X)) by a f (g f(g)a). C(X) C(G) C(G) = C(G G, C(X)), etc. continuous G α C(X) by unital -endomorphisms α : C(X) C(G, C(X)) unital -homomorphism. ((id )α)(a)(g, h) = ((α id)α)(a)(g, h) α gh (a) = α g(α h (a)). Conclusion: one-to-one correspondence between α with coaction property, and actions of a group on a C -algebra by endomorphisms. To do: Density α e = id C(X) for e unit G.

16 Proof, Part II -homomorphism α : C(X) C(G) C(X) C(G), Density α surjective. On level of C(G, C(X)) = C(X) C(G): a f α(a)(1 f). F C(G, C(X)), α(f )(g) = α g (F (g)). Assume α e = id C(X). Then α has inverse β, β(f )(g) = α g 1(F (g)). Hence range α dense. If α e id C(X) α e non-trivial idempotent -endomorphism. Put C(X e ) = α e (C(X)) C(X). g G: α g (C(X)) = α e (α g (C(X))) C(X e ). If a / C(X e ), then g a not in range α.

17 Classical Example (All C(G) and C(X) commutative) G compact Hausdorff group, X compact Hausdorff space, X G continuous G C(X), α g (f)(x) = f(x g). Example Consider sphere S N 1 = {z = (z 1,..., z N ) R N i z 2 i = 1}. Then S N 1 O(N) by (z, g) zg.

18 Example: Half-classical I Example Cuntz algebras, O n = C (V 1,..., V n V i V j = δ ij, i V i V i = 1). Then U(n) O n by α u (V i ) = j u ji V j. In particular, S 1 O n by α z (V i ) = zv i.

19 Example: Half-classical II Example (Banica) Free spheres, C(S N 1 + ) =< V 1,..., V N V i = V i, i V 2 i = 1}. Then O(N) C(S N 1 + ) by α g (V i ) = j g ji V j.

20 Left actions of compact quantum groups on C -algebras Definition (Podleś) Left action G H: Compact quantum group G, C -algebra C(X), Unital -homomorphism, left coaction α : C(X) C(G) C(X) s.t. coaction property: density: (id G α) α = ( id X ) α, [(C(G) 1 X )α(c(x))] = C(X) C(G).

21 From left to right Definition Let G CQG. Then G op CQG by C(G op ) = C(G), G op = op G = ς, where Lemma ς : C(G) C(G) C(G) C(G), g h h g. G α X X αop G op.

22 Non-unital C -algebras Definition (Multiplier C -algebras) C 0 (X) non-unital C -algebra ( locally compact quantum space ). Multiplier C -algebra M(C 0 (X)) = C b (X): Concrete: For C 0 (X) B(H) with [C 0 (X) H] = H: C b (X) = {T B(H) a C 0 (X), T a, at C 0 (X)}. Abstract: C b (X) collection maps T : C 0 (X) C 0 (X) s.t. T, a, b C 0 (X), a (T b) = (T b) a. If T C b (X): T (ab) = T (a)b, and C 0 (X) C b (X). Example If X locally compact Hausdorff space, M(C 0 (X)) = C b (X).

23 Morphisms between locally compact quantum spaces Definition -homomorphism π : C 0 (Y) M(C 0 (X)) non-degenerate: [π(c 0 (Y))C 0 (X)] = C 0 (X). Example Let X, Y locally compact Hausdorff spaces. Non-degenerate maps C 0 (Y ) C b (X) continuous maps X Y. Non-degenerate maps C 0 (Y ) C 0 (X) continuous proper maps X Y. Degenerate map C 0 (Y ) C b (X): points of X to infinity. Lemma π : C 0 (Y) C b (X) non-degenerate!π : C b (Y) C b (X).

24 Actions on locally compact quantum spaces Definition Right action X G: Compact quantum group G, C -algebra C 0 (X), non-degenerate -homomorphism, right coaction s.t. coaction property: density: α : C 0 (X) C b (X G) (α id G ) α = (id X ) α, [α(c 0 (X))(1 X C(G))] = C 0 (X) C(G). In particular... α : C 0 (X) C 0 (X) C(G) (proper action).