Definition and examples Let ( A , m , 1) be an associative algebra - PowerPoint PPT Presentation

Representations of pointed Hopf algebras over S 3 Agust´ ın Garc´ ıa Iglesias Facultad de Matem´ atica, Astronom´ ıa y F´ ısica Universidad Nacional de C´ ordoba Argentina Advanced School and Conference on Homological and Geometrical Methods in Representation Theory January 18th – February 5th, 2010 ICTP, Miramare - Trieste, Italy

� � � � � Definition and examples Let ( A , m , 1) be an associative algebra with unit over a field k . That is, ( ab ) c = a ( bc ) ∀ a , b , c , ∈ A and a 1 = a = 1 a , ∀ a ∈ A . ◮ These axioms can be codified in the commutativity of the following diagrams: m ⊗ id � id ⊗ 1 � 1 ⊗ id A ⊗ A ⊗ A A ⊗ A A ⊗ k A ⊗ A k ⊗ A � � � ���������� � � � m m id ⊗ m � ∼ ∼ � = � = � � m � A A ⊗ A A

� � � � � � ◮ A coassociative counital k - coalgebra ( C , ∆ , ǫ ) is a k -vector space C together with maps ∆ : C → C ⊗ C (the comultiplication) and ǫ : C → k (the counit) such the following diagrams commute: ∆ id ⊗ ǫ ǫ ⊗ id � k ⊗ C C C ⊗ C C ⊗ k C ⊗ C � � � ���������� � � � ∆ id ⊗ ∆ ∆ � ∼ � ∼ = � = � � ∆ ⊗ id � C ⊗ C ⊗ C C ⊗ C C That is, (∆ ⊗ id)∆( c ) = (id ⊗ ∆)∆( c ) , and ( ǫ ⊗ id)∆( c ) = (id ⊗ ǫ )∆( c ) = c , for every c ∈ C

◮ A bialgebra B is an algebra ( B , m , 1) and a coalgebra ( B , ∆ , ǫ ) such that the maps ∆ : B → B ⊗ B , ǫ : B → k , are algebra maps.

◮ A bialgebra B is an algebra ( B , m , 1) and a coalgebra ( B , ∆ , ǫ ) such that the maps ∆ : B → B ⊗ B , ǫ : B → k , are algebra maps. ◮ A Hopf algebra H is a bialgebra ( H , m , ∆) together with a map S ∈ End( H ) (the antipode) such that the following axioms are satisfied: m ( S ⊗ id)∆( h ) = ǫ ( h )1 , m (id ⊗S )∆( h ) = ǫ ( h )1 . for every h ∈ H .

Examples ◮ Let G be a group and k G the group algebra, that is the vector space with basis { e g : g ∈ G } and multiplication rule e g e h = e gh , g , h ∈ G . Then k G is a Hopf algebra with: ∆( e g ) = e g ⊗ e g , ǫ ( e g ) = 1 , S ( e g ) = e g − 1 for every g ∈ G .

Examples ◮ Let G be a group and k G the group algebra, that is the vector space with basis { e g : g ∈ G } and multiplication rule e g e h = e gh , g , h ∈ G . Then k G is a Hopf algebra with: ∆( e g ) = e g ⊗ e g , ǫ ( e g ) = 1 , S ( e g ) = e g − 1 , g ∈ G . ◮ If g is a Lie algebra, then the universal enveloping algebra U ( g ) is a Hopf algebra via ∆( x ) = x ⊗ 1 + 1 ⊗ x , ǫ ( x ) = 0 , S ( x ) = − x , for every x ∈ g .

Some invariants Let H be a Hopf algebra ◮ The coradical H 0 of H is the sum of all simple sub-coalgebras of H . ◮ If 0 � = h ∈ H satisfies ∆( h ) = h ⊗ h , then h is said to be a grouplike element . The set of grouplike elements of H , G ( H ), forms a group under the multiplication in H .

Some invariants Let H be a Hopf algebra ◮ The coradical H 0 of H is the sum of all simple sub-coalgebras of H . ◮ If 0 � = h ∈ H satisfies ∆( h ) = h ⊗ h , then h is said to be a grouplike element . The set of grouplike elements of H , G ( H ), forms a group under the multiplication in H . ◮ Let Γ be a group and assume G ( H ) ∼ = Γ. H is called pointed if H 0 is the group algebra of Γ.

Technical ingredients ◮ A rack X = ( X , ⊲ ) is a pair ( X , ⊲ ), where X is a non-empty set and ⊲ : X × X → X is a function, such that φ i = i ⊲ ( · ) : X → X is a bijection ∀ i ∈ X , and i ⊲ ( j ⊲ k ) = ( i ⊲ j ) ⊲ ( i ⊲ k ) , ∀ i , j , k ∈ X . ◮ A 2-cocycle q is a function q : X × X → k ∗ , ( i , j ) �→ q ij such that q i , j ⊲ k q j , k = q i ⊲ j , i ⊲ k q i , k , ∀ i , j , k ∈ X .

◮ Given ( X , q ), let R be the set of equivalence classes in X × X for the relation generated by ( i , j ) ∼ ( i ⊲ j , i ). Let C ∈ R , ( i , j ) ∈ C . Take i 1 = j , i 2 = i , and recursively, i h +2 = i h +1 ⊲ i h . Set n ( C ) = # C and � n ( C ) q i h +1 , i h = ( − 1) n ( C ) � � R ′ = C ∈ R | . h =1

◮ Given ( X , q ), let R be the set of equivalence classes in X × X for the relation generated by ( i , j ) ∼ ( i ⊲ j , i ). Let C ∈ R , ( i , j ) ∈ C . Take i 1 = j , i 2 = i , and recursively, i h +2 = i h +1 ⊲ i h . Set n ( C ) = # C and � n ( C ) q i h +1 , i h = ( − 1) n ( C ) � � R ′ = C ∈ R | . h =1 ◮ Let F be the free associative algebra in the variables { T l } l ∈ X . If C ∈ R ′ , consider the quadratic polynomial n ( C ) � φ C = η h ( C ) T i h +1 T i h ∈ F , h =1 where η 1 ( C ) = 1 and η h ( C ) = ( − 1) h +1 q i 2 i 1 q i 3 i 2 . . . q i h i h − 1 , h ≥ 2.

The algebra H ( Q ) A quadratic lifting datum , or ql-datum, Q consists of ◮ a rack X , ◮ a 2-cocycle q , ◮ a finite group G , ◮ an action · : G × X → X , ◮ a function g : X → G , ◮ a family of 1-cocyles ( χ i ) i ∈ X : G → k ( i. e. χ i ( ht ) = χ i ( t ) χ t · i ( h ), for all i ∈ X , h , t ∈ G ), ◮ a collection ( λ C ) C ∈R ′ ∈ k , ( R ′ ⊂ X × X ) subject to a (non-trivial!) set of compatibilty axioms.

Given a ql-datum Q , we define the algebra H ( Q ) by generators { a i , H t : i ∈ X , t ∈ G } and relations: H e = 1 , H t H s = H ts , t , s ∈ G ; H t a i = χ i ( t ) a t · i H t , t ∈ G , i ∈ X ; φ C ( { a i } i ∈ X ) = λ C (1 − H g i g j ) , C ∈ R ′ , ( i , j ) ∈ C .

Given a ql-datum Q , we define the algebra H ( Q ) by generators { a i , H t : i ∈ X , t ∈ G } and relations: H e = 1 , H t H s = H ts , t , s ∈ G ; H t a i = χ i ( t ) a t · i H t , t ∈ G , i ∈ X ; φ C ( { a i } i ∈ X ) = λ C (1 − H g i g j ) , C ∈ R ′ , ( i , j ) ∈ C . Recall that: n ( C ) � φ C ( { a i } i ∈ X ) = η h ( C ) a i h +1 a i h . h =1

Given a ql-datum Q , we define the algebra H ( Q ) by generators { a i , H t : i ∈ X , t ∈ G } and relations: H e = 1 , H t H s = H ts , t , s ∈ G ; H t a i = χ i ( t ) a t · i H t , t ∈ G , i ∈ X ; φ C ( { a i } i ∈ X ) = λ C (1 − H g i g j ) , C ∈ R ′ , ( i , j ) ∈ C . Recall that: n ( C ) � φ C ( { a i } i ∈ X ) = η h ( C ) a i h +1 a i h . h =1 ◮ H ( Q ) is a pointed Hopf algebra if we define the elements H t to be group-likes and the elements a i to be ( H g i , 1)-primitives. ◮ G ( H ( Q )) is a quotient of the group G . And thus any H ( Q )-module W is G -module W | G , by restriction.

Example Let Q λ be the ql-datum: ◮ X = O 3 2 the rack over the conjugacy class of transpositions, ◮ q ≡ − 1, that is q ij = − 1 ∀ i ∈ X , ◮ G = S 3 , ◮ · : G × X → X the conjugation, ◮ g : X ֒ → G the inclusion, ◮ χ i ( t ) = sgn( t ) , ∀ i ∈ X , t ∈ G , ◮ { λ C } C ∈R ′ = { 0 , λ } .

Example Let Q λ be the ql-datum: ◮ X = O 3 2 the rack over the conjugacy class of transpositions, ◮ q ≡ − 1, that is q ij = − 1 ∀ i ∈ X , ◮ G = S 3 , ◮ · : G × X → X the conjugation, ◮ g : X ֒ → G the inclusion, ◮ χ i ( t ) = sgn( t ) , ∀ i ∈ X , t ∈ G , ◮ { λ C } C ∈R ′ = { 0 , λ } . Then A λ = H ( Q λ ) is the algebra presented by generators { a i , H r : i ∈ O 3 2 , r ∈ S 3 } and relations: H e = 1 , H r H s = H rs , r , s ∈ S 3 ; H j a i = − a jij H j , i , j ∈ O 3 2 ; a 2 (12) = 0; a (12) a (23) + a (23) a (13) + a (13) a (12) = λ (1 − H (12) H (23) ) .

◮ The algebras A λ were introduced in (AG). ◮ A λ is a Hopf algebra of dimension 72. If H is a finite-dimensional pointed Hopf algebra with G ( H ) ∼ = S 3 , then either H ∼ = kS 3 , H ∼ = A 0 or H ∼ = A 1 . This is Thm. 4.5 in (AHS) (together with (MS,AG,AZ)). ◮ The algebras H ( Q ) were introduced in (GG). They generalize the algebras A λ and were used to classify pointed Hopf algebras over S 4 . (AG) Andruskiewitsch, N. and Gra˜ na, M., From racks to pointed Hopf algebras , Adv. in Math. 178 (2), 177–243 (2003). (AHS) Andruskiewitsch, N., Heckenberger, I. and Schneider, H.J., The Nichols algebra of a semisimple Yetter-Drinfeld module , arXiv:0803.2430v1. (GG) Garc´ ıa, G. A. and Garc´ ıa Iglesias, A., Pointed Hopf algebras over S 4 . Israel Journal of Math. Accepted. Also available at arXiv:0904.2558v1 [math.QA]

H ( Q ) -modules over G -characters. ◮ Let � G the set of irreducible representations of G . ◮ Let G ab = G / [ G , G ], � G ab = Hom( G , k ∗ ) ⊆ � G . ◮ If χ ∈ � G , and W is a G -module, we denote by W [ χ ] the isotypic component of type χ , and by W χ the corresponding simple G -module.

Isotypical modules Let ρ ∈ � G ab . ◮ There exists ¯ ρ ∈ hom alg ( H ( Q ) , k ) such that ¯ ρ | G = ρ if and only if 0 = λ C (1 − ρ ( g i g j )) if ( i , j ) ∈ C and 2 | n ( C ) , (1) and there exists a family { γ i } i ∈ X of scalars such that γ j = χ j ( t ) γ t · j ∀ t ∈ G , j ∈ X , (2) γ i γ j = λ C (1 − ρ ( g i g j )) if ( i , j ) ∈ C and 2 | n ( C ) + 1 . (3) Assume X is indecomposable and let W be an H ( Q )-module such that W = W [ ρ ] for a unique ρ ∈ � G ab , dim W = n . ◮ W is simple if and only if n = 1. If, in addition, χ i ( g i ) � = 1 , ∀ i ∈ X , then W ∼ = S ⊕ n ρ .