Operator algebraic properties for free wreath products by quantum - PowerPoint PPT Presentation

Fusion rules for free wreath products Operator algebraic properties for free wreath products by quantum permutation groups -Conf´ erence du GDRE “Noncommutative Geometry and Applications”- Fran¸ cois Lemeux (part. joint work with Pierre Tarrago) Universit´ e de Franche-Comt´ e francois.lemeux@univ-fcomte.fr June 2014, 19th - Villa Mondragone

Fusion rules for free wreath products Sommaire Examples of CQG 1 Motivations 2 Fusion rules for quantum reflection groups Operator algebraic properties for CQG Fusion rules for the free wreath products � Γ ≀ ∗ S + 3 N Operator algebraic properties for � Γ ≀ ∗ S + 4 N Work in progress 5

Fusion rules for free wreath products Examples of CQG Contents Examples of CQG 1 Motivations 2 Fusion rules for quantum reflection groups Operator algebraic properties for CQG Fusion rules for the free wreath products � Γ ≀ ∗ S + 3 N Operator algebraic properties for � Γ ≀ ∗ S + 4 N Work in progress 5

Fusion rules for free wreath products Examples of CQG Definition (Woronowicz 80’) G = ( C ( G ) , ∆) GQC : C ( G ) Woronowicz C ∗ -algebra ; C ( G ) unital, ∆ : C ( G ) → C ( G ) ⊗ min C ( G ) s.t. 1 (∆ ⊗ id )∆ = ( id ⊗ ∆)∆ , 2 { ∆( a )( b ⊗ 1) : a , b ∈ C ( G ) } et { ∆( a )(1 ⊗ b ) : a , b ∈ C ( G ) } lin. dense in C ( G ) ⊗ C ( G ) . Peter-Weyl theory : Corep. u ∈ M N ( C ( G )) ≃ M N ( C ) ⊗ C ( G ), ∆( u ij ) = � N k =1 u ik ⊗ u kj . • Hom( u ; v ) = { T ∈ M n v , n u ( C ) : v ( T ⊗ 1) = ( T ⊗ 1) u } , • u ∼ v , ∃ T invertible T ∈ Hom( u ; v ), • u is irreducible if Hom( u ; u ) = C id. Theorem (Woronowicz) Let G = ( C ( G ) , ∆) be a GQC. The corepresentations of C ( G ) decompose as direct sums of irreducibles.

Fusion rules for free wreath products Examples of CQG We consider the unital C ∗ -algebra defined by generators and relations: C ∗ com − � s ij : 1 ≤ i , j ≤ N : ( s ij ) magic unitary � ≃ C ( S N ) s ij �→ ( σ ∈ S N ⊂ M N ( C ) �→ σ ij ) . Magic unitary: ( s ij ) unitary matrix whose entries are projections which sum up to 1 on each row and column. Removing the commutativity: N ) := C ∗ − � v ij : 1 ≤ i , j ≤ N : ( v ij ) magic unitary � , C ( S + one obtains a new C ∗ -algebra for N ≥ 4. We have the coproduct on C ( S + N ): N � ∆ : C ( S + N ) → C ( S + N ) ⊗ C ( S + N ) , ∆( v ij ) = v ik ⊗ v kj . k =1 S + N = ( C ( S + N ) , ∆) is the quantum permutation group (Wang 98).

Fusion rules for free wreath products Examples of CQG We denote by NC ( k , l ) the set of non-crossing partitions on k + l points:   · · · ·     p = P non-crossing diagram . P     · · · Theorem (Banica 99) N ( v ⊗ k ; v ⊗ l ) = span { T p : p ∈ NC ( k , l ) } , T p ∈ B ( C N ⊗ k ; C N ⊗ l ) : Hom S + T p ( e i 1 ⊗ · · · ⊗ e i k ) = � j 1 ,..., j l δ p ( i , j ) e j 1 ⊗ · · · ⊗ e j l . Corollaire (Banica 99) The irreducible corepresentations of S + N can be labeled by N with • v (0) = 1 is the trivial representation and v = 1 ⊕ v (1) . • v ( k ) = ( v ( k ) ∗ ) is equivalent to v ( k ) , ∀ k ∈ N . ij • ∀ k , l ∈ N , v ( k ) ⊗ v ( l ) = � 2 min( k , l ) v ( k + l − r ) (Clebsch-Gordan). r =0

Fusion rules for free wreath products Examples of CQG Definition (Bichon 00’) H + N (Γ) := ( C ( H + N (Γ)) , ∆) where C ( H + N (Γ)) is the C ∗ -algebra generated by the elements a ij ( g ) , i , j = 1 , . . . , N s.t. ∀ g , h ∈ Γ , • a ij ( g ) a ik ( h ) = δ j , k a ij ( gh ) , a ji ( g ) a ki ( h ) = δ j , k a ji ( gh ) , • � i a ij ( e ) = 1 = � j a ij ( e ) , • ∆( a ij ( g )) = � N k =1 a ik ( g ) ⊗ a kj ( g ) . Bichon : H + N (Γ) ≃ � Γ ≀ ∗ S + N where N ) := C ∗ (Γ) ∗ N ∗ C ( S + N ) / � g ( i ) v ij − v ij g ( i ) = 0 � C ( � Γ ≀ ∗ S + via a ij ( g ) �→ g ( i ) v ij = v ij g ( i ) . Example • Γ = { e } trivial : S + N . • Γ = Z / s Z : quantum reflection groups H s + N .

Fusion rules for free wreath products Motivations Contents Examples of CQG 1 Motivations 2 Fusion rules for quantum reflection groups Operator algebraic properties for CQG Fusion rules for the free wreath products � Γ ≀ ∗ S + 3 N Operator algebraic properties for � Γ ≀ ∗ S + 4 N Work in progress 5

Fusion rules for free wreath products Motivations Fusion rules for quantum reflection groups • Fusion rules for quantum reflection groups Banica and Vergnioux obtained a combinatorial description of the intertwiner spaces for H s + = H + N ( Z / s Z ) and then deduced the fusion N rules: Theorem (Banica, Vergnioux 08) The irreducible representations of H s + can be labelled by the worlds N ( i 1 , ..., i k ) whose letters are in Z / s Z , with involution ( i 1 , . . . , i k ) = ( − i k , . . . , − i 1 ) and the fusion rules: ( i 1 , . . . , i k ) ⊗ ( j 1 , . . . , j l ) = ( i 1 , . . . , i k − 1 , i k , j 1 , j 2 , . . . , j l ) ⊕ ( i 1 , . . . , i k − 1 , i k + j 1 , j 2 , . . . , j l ) ⊕ δ i k + j 1 , 0[ s ] ( i 1 , . . . , i k − 1 ) ⊗ ( j 2 , . . . , j l )

Fusion rules for free wreath products Motivations Operator algebraic properties for CQG • Operator algebraic properties for CQG Let G = ( C ( G ) , ∆) be a GQC whose Haar state h is a trace. σ w , C r ( G ) = π h ( C ( G )) ≃ C ( G ) / ker ( π h ) . L ∞ ( G ) := C r ( G ) Notations: • Pol ( G ) ⊂ C ( G ) sub- ∗ -algebra (dense) generated by the coefficients of irreducible corepresentations, • C ( G ) 0 = C ∗ − � � i U ii : U ∈ Irr ( G ) � central algebra. Some results: • C r ( U + N ) is simple with unique trace, N ≥ 2 (Banica 99). • C r ( O + N ) is simple with unique trace, L ∞ ( O + N ) is a full II 1 factor, N ≥ 3 (Vaes and Vergnioux 07). • C r ( S + N ) is simple with unique trace, L ∞ ( S + N ) is a full II 1 factor, N ≥ 8 (Brannan 13). • L ∞ ( O + N ) , L ∞ ( U + N ), L ∞ ( S + N ) have the Haagerup property, N ≥ 2 (Brannan 12, 13).

Fusion rules for free wreath products Γ ≀ ∗ S + Fusion rules for the free wreath products � N Contents Examples of CQG 1 Motivations 2 Fusion rules for quantum reflection groups Operator algebraic properties for CQG Fusion rules for the free wreath products � Γ ≀ ∗ S + 3 N Operator algebraic properties for � Γ ≀ ∗ S + 4 N Work in progress 5

Fusion rules for free wreath products Γ ≀ ∗ S + Fusion rules for the free wreath products � N Intertwiner spaces in H + N (Γ): Strategy: Find a CQG G = ( C ( G ) , ∆), • s.t. we have a surjective morphisme π : C ( G ) ։ C ( H + N (Γ)), → If Γ = � S � , | S | = p , G = ∗ p i =1 ( H ∞ + ) N • s.t. the intertwiner spaces in G have a combinatorial description, • s.t. the kernel of π admits a combinatorial description. ⇒ The intertwiners in H + N (Γ) are given by the intertwiners in G = ∗ p i =1 ( H ∞ + ) and by the relations in the kernel. N � Combinatorial description of the intertwiner spaces for tensor products of the corepresentations a ( g ) := ( a ij ( g )) 1 ≤ i , j ≤ N , g ∈ Γ.

Fusion rules for free wreath products Γ ≀ ∗ S + Fusion rules for the free wreath products � N Theorem (L.) Let Γ be discrete N ≥ 4 . Hom H + N (Γ) ( a ( g 1 ) ⊗ · · · ⊗ a ( g k ); a ( h 1 ) ⊗ · · · ⊗ a ( h l )) = span { T p : p ∈ NC Γ ( g 1 , . . . , g k ; h 1 , . . . , h l ) } NC Γ ( g 1 , . . . , g k ; h 1 , . . . , h l ) : NC part. s.t. in each bloc � g i = � h j . Theorem (L.) The irreducible corepresentations of H + N (Γ) can be indexed by the words ( g 1 , . . . , g k ) , g i ∈ Γ , with involution ( g 1 , . . . , g k ) = ( g − 1 k , . . . , g − 1 1 ) and fusion rules: ( g 1 , . . . , g k ) ⊗ ( h 1 , . . . , h l ) = ( g 1 , . . . , g k − 1 , g k , h 1 , h 2 , . . . , h l ) ⊕ ( g 1 , . . . , g k − 1 , g k h 1 , h 2 , . . . , h l ) ⊕ δ g k h 1 , e ( g 1 , . . . , g k − 1 ) ⊗ ( h 2 , . . . , h l ) .

Fusion rules for free wreath products Γ ≀ ∗ S + Operator algebraic properties for � N Contents Examples of CQG 1 Motivations 2 Fusion rules for quantum reflection groups Operator algebraic properties for CQG Fusion rules for the free wreath products � Γ ≀ ∗ S + 3 N Operator algebraic properties for � Γ ≀ ∗ S + 4 N Work in progress 5

Fusion rules for free wreath products Γ ≀ ∗ S + Operator algebraic properties for � N Irreducible + fusion rules for H + N (Γ): allow to prove several interesting properties for the associated operator algebras. Theorem (L.) The von Neumann algebras L ∞ ( H + N (Γ)) have the Haagerup property for all N ≥ 4 and all finite groups Γ . Strategy: • Construct convolution operators on L ∞ ( H + N (Γ)) from states on the central algebra C ( H + N (Γ)) 0 (Brannan 12), • Understand π : C ( H + N (Γ)) 0 → C ( S + N ) 0 ≃ C ([0 , N ]), • Consider the states on C ( H + N (Γ)) 0 , given by ev x ◦ π + estimates on Tchebytchev polynomials.