Actions of Compact Quantum Groups V Free and homogeneous actions I - PowerPoint PPT Presentation

Actions of Compact Quantum Groups V Free and homogeneous actions I Kenny De Commer (VUB, Brussels, Belgium)

Free actions Homogeneous actions Outline Free actions Homogeneous actions

Free actions Homogeneous actions Free actions Definition α X � G free if ∀ x ∈ X, G x = { g ∈ G | xg = x } = { e G } .

Free actions Homogeneous actions A C ∗ -algebraic characterisation Lemma α X � G free iff [( C 0 ( X ) ⊗ 1 G ) α ( C 0 ( X ))] = C 0 ( X ) ⊗ C ( G ) . Proof. α free iff Can : X × G �→ X × X, ( x, g ) �→ ( x, xg ) injective iff Can : C 0 ( X ) ⊗ C 0 ( X ) → C 0 ( X ) ⊗ C ( G ) , F = f ⊗ h �→ Can( F ) = ( f ⊗ 1 G ) α ( h ) , ( x, g ) �→ F ( x, xg ) surjective.

Free actions Homogeneous actions Freeness for compact quantum group actions Definition (Ellwood) α Let X � G . Then α free if [( C 0 ( X ) ⊗ 1 G ) α ( C 0 ( X ))] = C 0 ( X ) ⊗ C ( G ) .

Free actions Homogeneous actions C ∗ -correspondences Definition C 0 ( X ) - C 0 ( Y ) -correspondence: ◮ right Hilbert C 0 ( Y ) -module Γ( E ) , ◮ non-degenerate ∗ -representation λ : C 0 ( X ) → L (Γ( E )) .

Free actions Homogeneous actions Interior tensor product Definition (Interior tensor product) Assume ◮ (Γ( E ) , λ ) is C 0 ( X ) - C 0 ( Y ) correspondence. ◮ (Γ( F ) , τ ) is C 0 ( Y ) - C 0 ( Z ) correspondence. Then C 0 ( X ) - C 0 ( Z ) correspondence (Γ( E ) C 0 ( Y ) Γ( F ) , λ ⊗ C 0 ( Y ) id) : ⊗ separation-completion of Γ( E ) ⊗ alg Γ( F ) , � s ⊗ u, t ⊗ v � = � u, τ ( � s, t � ) v � .

Free actions Homogeneous actions The Galois isometry Lemma α Let X � G . Then ∃ isometry, Galois (or canonical) isometry, G α : L 2 C 0 ( Y ) L 2 Y ( X ) → L 2 Y ( X ) ⊗ L 2 ( G ) , Y ( X ) ⊗ a ⊗ b �→ α ( a )( b ⊗ 1 G ) . Proof. � α ( c )( d ⊗ 1) , α ( a )( b ⊗ 1) � ( E Y ⊗ ϕ )(( d ∗ ⊗ 1) α ( c ∗ a )(( b ⊗ 1)) = E Y ( d ∗ E Y ( c ∗ a ) b ) = = � c ⊗ d, a ⊗ b � .

Free actions Homogeneous actions What’s Galois got to do with it? Theorem (Chase-Harrison-Rosenberg) Let ◮ E ⊆ F finite field extension, ◮ G = Aut E ( F ) . Then ◮ H = Map( G, E ) Hopf algebra over E , ◮ Hopf algebraic coaction α : F → F ⊗ E H, α ( f )( g ) = α g ( f ) , and E ⊆ F Galois if and only if the following map is bijective, F ⊗ E F → F ⊗ E H, a ⊗ b �→ α ( a )( b ⊗ 1) .

Free actions Homogeneous actions Unitarity of the Galois map Theorem (DC-Yamashita; Baum-DC-Hajac) α Let X � G . The following are equivalent. 1. The action is free. 2. The Galois map is unitary. ∼ = → K ( L 2 3. C 0 ( X ⋊ G ) Y ( X )) . Remark: Last condition: ‘saturatedness’ (Rieffel)

Free actions Homogeneous actions Example: Action by compact quantum subgroup Example Let H ⊆ G compact quantum subgroup: π : C ( G ) ։ C ( H ) , ( π ⊗ π ) ◦ ∆ G = ∆ H ◦ π. Then free action α = (id ⊗ π ) ◦ ∆ : C ( G ) → C ( G ) ⊗ C ( H ) . Proof. Exercise.

Free actions Homogeneous actions Example: free action on smash product Lemma Let X � � G . Then ( X ⋊ � G ) � G free. Proof. [ α ( C 0 ( X ⋊ � G ))( C 0 ( X ⋊ � α ( O ( X ⋊ � G ))( O ( X ⋊ � G ) ⊗ 1 G )] ⊇ G ) ⊗ 1 G ) ⊇ ( O ( X ) ⊗ 1 G )∆( O ( G ))( O ( G ) O ( X ) ⊗ 1 G ) = O ( X ) O ( G ) O ( X ) ⊗ O ( G ) O ( X ⋊ � = G ) ⊗ O ( G ) . Corollary (Takesaki-Takai duality) G ⋊ G ) ∼ C 0 ( X ⋊ � = B 0 ( L 2 ( G )) ⊗ C 0 ( X ) (since L 2 X ( X ⋊ � G ) = L 2 ( G ) ⊗ C 0 ( X ) ).

Free actions Homogeneous actions Homogeneous actions Definition α X � G homogeneous (or ergodic) if α ( x ) = x ⊗ 1 G ↔ x ∈ C 1 X . ⇒ C ( X ) unital. Lemma α If X � G homogeneous, then α transitive (in the ordinary sense). Proof. C ( X ) G = C ( X/G ) .

Free actions Homogeneous actions Homogeneity and reduced and universal actions Lemma α If X � G homogeneous, then α u and α red homogeneous. Proof. Y = X / G = X u / G u = X red / G red .

Free actions Homogeneous actions Associated von Neumann algebra Definition α Let X � G homogeneous. Then invariant state ϕ X on C ( X ) , ∀ a ∈ C 0 ( X ) , ϕ X ( a )1 X = (id ⊗ ϕ ) α ( a ) . Lemma α Let X � G homogeneous. ◮ ϕ X is invariant: ∀ a ∈ C ( X ) , ( ϕ X ⊗ id G ) α ( a ) = ϕ X ( a )1 G . ◮ With L ∞ ( X ) = C ( X red ) ′′ ⊆ B ( L 2 ( X , ϕ X )) , normal coassociative ∗ -homomorphism α vN : L ∞ ( X ) → L ∞ ( X ) ⊗ L ∞ ( G ) .

Free actions Homogeneous actions Actions of quotient type Definition (Podle´ s) α Let X � G . One calls α of quotient type if ∃ H ⊆ G and θ : C ( X ) ∼ = C ( H \ G ) = { f ∈ C ( G ) | ( π ⊗ id)∆( f ) = 1 H ⊗ f } such that ( θ ⊗ id) ◦ α = ∆ ◦ θ. Remarks: ◮ Quotient type ⇒ Homogeneous. ◮ G, X classical: Homogeneous ⇒ Quotient type. ◮ In general: Homogeneous � Quotient type.

Free actions Homogeneous actions Example: Standard Podle´ s sphere Definition For q ∈ [ − 1 , 1] \ { 0 } : C ∗ -algebra C ( SU q (2)) as � u 11 � � a � − qb ∗ u 12 C ∗ ( a, b | U = = unitary ) u 21 u 22 b a ∗ is CQG for ‘matrix comultiplication’ ∆( u ij ) = u i 1 ⊗ u 1 j + u i 2 ⊗ u 2 j . Lemma S 1 ⊆ SU q (2) by � a � � z � − qb ∗ 0 → . b a ∗ 0 z ¯ Definition s sphere S 2 q = S 1 \ SU q (2) . Standard Podle´

Free actions Homogeneous actions Concrete representation standard Podle´ s sphere Lemma C ( S 2 q ) generated by X = ab, Z = qb ∗ b and Y = b ∗ a ∗ . Lemma C ( S 2 q ) universal C ∗ -algebra generated by X, Y, Z s.t. ◮ X ∗ = Y , ◮ ◮ Z ∗ = Z , ◮ XZ = q 2 ZX , ◮ ◮ Y Z = q − 2 ZY , ◮ Y X = q − 1 Z − q − 2 Z 2 , ◮ Z − q 2 Z 2 . ◮ XY = q Remark: For q = 1 , | X | 2 + ( Z − 1 2 ) 2 = 1 4 .

Free actions Homogeneous actions Embeddable actions Definition (Podle´ s) α � G . One calls α embeddable if ∃ faithful ∗ -homomorphism Let X θ : C ( X ) ֒ → C ( G ) such that ( θ ⊗ id) ◦ α = ∆ ◦ θ. Remark: ◮ Embeddable ⇒ Homogeneous. ◮ Quotient type ⇒ Embeddable. ◮ Embeddable � Quotient type (e.g. non-standard Podle´ s spheres) ◮ Homogeneous � Embeddable.

Free actions Homogeneous actions Non-embeddable actions Example Let π irreducible left G -representation. ξη ∗ → δ π ( ξ ) δ π ( η ) ∗ Ad π : B ( H π ) → B ( H π ) ⊗ C ( G ) , homogeneous, but not embeddable for dim( H π ) ≥ 2

Free actions Homogeneous actions Homogeneous actions with classical point Lemma α X � G homogeneous. TFAE: 1. α red is embeddable. 2. C ( X u ) has a character.

Free actions Homogeneous actions Proof 1 . ⇒ 2 . θ : C ( X red ) → C ( G red ) equivariant, so θ alg : O G ( X ) → O ( G ) ⇒ θ u : C ( X u ) → C ( G u ) . Then ǫ ◦ θ u character. ◮ If χ character, then equivariant ∗ -homomorphism 2 . ⇒ 1 . θ u : C ( X u ) → C ( G u ) , a �→ ( χ ⊗ id) ◦ α u . ◮ Hence equivariant ∗ -homomorphism θ alg : O G ( X ) → O ( G ) . ◮ Then ϕ ( θ alg ( a ) ∗ θ alg ( a )) = χ ( E Y ( a ∗ a )) . But, by homogeneity, E Y values in C 1 X , so E Y ( a ∗ a ) = χ ( E Y ( a ∗ a ))1 X . ◮ Hence θ r : C ( X red ) ֒ → C ( G red ) since E Y faithful on C ( X red ) .

Free actions Homogeneous actions Boca’s theorems Theorem (Boca) If X � α G homogeneous, then all C ( X ) π finite dimensional. In fact, Boca gives concrete estimate in terms of ‘quantum multiplicity’. Combined with Takesaki-Takai duality: Theorem (Boca) α X � G homogeneous ⇒ ∃ set I and Hilbert spaces H i , C 0 ( X ⋊ G ) ∼ = ⊕ i ∈ I B 0 ( H i )