New Hopf algebras arising from the generalized lifting method Gast - PowerPoint PPT Presentation

New Hopf algebras arising from the generalized lifting method New Hopf algebras arising from the generalized lifting method Gast´ on Andr´ es Garc´ ıa Universidad Nacional de La Plata CMaLP-CONICET Rings, modules, and Hopf algebras Blas Torrecillas’ 60th birthday May 13-17, 2018 Almer´ ıa

New Hopf algebras arising from the generalized lifting method Based on joint work with D. Bagio, J. M. Jury Giraldi and O. Marquez. [GJG] G. A. Garc´ ıa and J. M. Jury Giraldi , On Hopf algebras over quantum subgroups. J. Pure Appl. Algebra , Volume 223 (2019), Issue 2, 738–768. [BGJM] D. Bagio , G. A. Garc´ ıa , J. M. Jury Giraldi and O. Marquez , On Hopf algebras over duals of Radford algebras. In preparation.

New Hopf algebras arising from the generalized lifting method Introduction – Preliminaries The coradical filtration Let k be an algebraically closed field of characteristic zero and let H be a Hopf algebra over k . As a coalgebra, H has a canonical coalgebra filtration, the coradical filtration { H n } n ≥ 0 : ◮ H 0 ⊆ H 1 ⊆ · · · ⊆ H n ⊆ · · · ◮ � n ≥ 0 H n = H , ◮ ∆( H n ) ⊆ � n i =0 H i ⊗ H n − i . H 0 = coradical of H = sum of all simple subcoalgebras. H n = � n +1 H 0 = H n − 1 ∧ H 0 . H n = { h ∈ H : ∆( h ) ∈ H ⊗ H n − 1 + H 0 ⊗ H } . One has that H 0 = Jac( H ∗ ) ⊥ and H n = (Jac( H ∗ ) n +1 ) ⊥ .

New Hopf algebras arising from the generalized lifting method Introduction – Preliminaries The Lifting Method If H 0 is a Hopf subalgebra, then the filtration is a Hopf algebra filtration and � gr H = H n / H n − 1 , with H − 1 = 0 is a Hopf algebra. Take the homogeneous projection π : gr H ։ H 0 . It has a Hopf algebra section (the inclusion) and gr H ≃ R # H 0 Majid-Radford product or bosonization here R = (gr H ) co π a braided graded Hopf algebra in H 0 H 0 YD . H is called a lifting of R over H 0 .

New Hopf algebras arising from the generalized lifting method Introduction – Preliminaries The Lifting Method Let V = P ( R ) = { r ∈ R : ∆( r ) = r ⊗ 1 + 1 ⊗ r } be the space of primitive elements. The subalgebra B ( V ) of R generated by V is called the Nichols algebra of V : ◮ B ( V ) is graded with B ( V )(0) = k and B ( V )(1) = V . ◮ B ( V )(1) = P ( B ( V )). ◮ B ( V ) is generated by V . Rmk: It is possible to define B ( V ) in terms of the braided vector space ( V , c ): B ( V ) = T ( V ) / J , with J the largest two-sided ideal and coideal J ⊆ ⊕ n ≥ 2 V n .

New Hopf algebras arising from the generalized lifting method Introduction – Preliminaries The Lifting Method The Lifting Method for fin-dim. Hopf algebras [Andruskiewitsch-Schneider] Let A be a finite-dimensional cosemisimple Hopf algebra. ( a ) Determine V ∈ A A YD such that B ( V ) is finite-dimensional. ( b ) For such V , compute all L s.t. gr L ≃ B ( V )# A . ( c ) Prove that for all H such that H 0 = A , then gr H ≃ B ( V )# A . (generation in degree one)

New Hopf algebras arising from the generalized lifting method Introduction – Preliminaries Feature results Assume A = k Γ group algebra over a finite group � pointed Hopf algebras ◮ Classification obtained for Γ abelian. ◮ Few examples for Γ non-abelian: e.g. S 3 , S 4 , D 4 t , Z r ⋉ Z s . Conjecture Any finite-dimensional pointed Hopf algebra H s.t. H 0 ≃ k Γ, with Γ finite non-abelian simple group is trivial, i.e. H ≃ k Γ. Verified for A n with n ≥ 5, almost all sporadic groups, Suzuki-Ree groups and infinite families of finite simple groups of Lie type

New Hopf algebras arising from the generalized lifting method The Generalized Lifting Method What if H 0 is not a Hopf subalgebra? [Andruskiewitsch-Cuadra]: replace the coradical filtration by a more general but adequate one � the standard filtration { H [ n ] } n ≥ 0 ◮ the subalgebra H [0] of H generated by H 0 , called the Hopf coradical , ◮ H [ n ] = � n +1 H [0] . It holds: If S is bijective then H [0] is a Hopf subalgebra of H , H n ⊆ H [ n ] and { H [ n ] } n ≥ 0 is a Hopf algebra filtration of H . In particular, � gr H = H [ n ] / H [ n − 1] is a Hopf algebra n ≥ 0 If H 0 is a Hopf subalgebra, then H [0] = H 0 and the coradical filtration coincides with the standard one.

New Hopf algebras arising from the generalized lifting method The Generalized Lifting Method The Generalized Lifting Method for fin-dim. Hopf algebras [Andruskiewitsch-Cuadra] Let A be a finite-dimensional generated by a cosemisimple coalgebra . ( a ) Determine V ∈ A A YD such that B ( V ) is finite-dimensional. ( b ) For such V , compute all L s.t. gr L ≃ B ( V )# A . ( c ) Prove that for all H such that H [0] = A , then gr H ≃ B ( V )# A . (generation in degree one w.r.t. the standard filtration )

New Hopf algebras arising from the generalized lifting method The Generalized Lifting Method Duals of Radford algebras First goal: Construct new Hopf algebras based on this method. First obstruction: find Hopf algebras generated by their coradicals. Source of examples: quotients of quantum function algebras: Let ξ be a primitive 4-th root of 1 and let K be generated by a , b , c , d satisfying ab = ξ ba , ac = ξ ca , 0 = cb = bc , cd = ξ dc , bd = ξ db , 0 = b 2 = c 2 , a 4 = 1 . a 2 c = b , ad = da , ad = 1 , The coalgebra structure and its antipode are determined by ∆( a ) = a ⊗ a + b ⊗ c , ∆( b ) = a ⊗ b + b ⊗ d , ∆( c ) = c ⊗ a + d ⊗ c , ∆( d ) = c ⊗ b + d ⊗ d , ε ( a ) = 1 , ε ( b ) = 0 , ε ( c ) = 0 , ε ( d ) = 1 S ( a ) = d , S ( b ) = ξ b , S ( c ) = − ξ c , S ( d ) = a .

New Hopf algebras arising from the generalized lifting method The Generalized Lifting Method Duals of Radford algebras K is an 8-dimensional Hopf algebra, it is a quotient of O q ( SL 2 ), and K ∗ is a pointed Hopf algebra � basic Hopf algebra. K ∗ = R 2 , 2 was first introduced by Radford. Duals of general Radford algebras R n , m satisfy this property. Prop-Def (Andruskiewitsch-Cuadra-Etingof) Let ξ ∈ G ′ nm . K n , m = R ∗ n , m is generated by U , X and A satisfying U n = 1 , X n = 0 , A m = U , UX = ω XU , UA = AU , AX = ξ XA . As coalgebra U ∈ G ( K n , m ), X ∈ P 1 , U ( K n , m ) and n − 1 � γ n , k X n − k U k A ⊗ X k A ∆( A ) = A ⊗ A + k =1 1 − ξ n where γ n , k = ( k )! ω ( n − k )! ω .

New Hopf algebras arising from the generalized lifting method The Generalized Lifting Method Step ( a ) – finding modules Write K = K n , m . Step ( a ) : To describe V ∈ K K YD , we use the equivalence K K YD ≃ D ( K cop ) M . D ( K cop ) = D is a non-semisimple Hopf algebra of tame representation type � we describe the simple modules, their projective covers and some indecomposable modules. For 0 ≤ i , j ≤ nm − 1, let r ij ∈ N such that 1 ≤ r ij ≤ n and � i + j m + 1 mod n if m | j , r ij = n if m ∤ j .

New Hopf algebras arising from the generalized lifting method The Generalized Lifting Method Step ( a ) – simple modules Definition Let 0 ≤ i , j < nm and write r = r i , j . Let V i , j be the C -vector space with basis B = { v 0 , · · · , v r − 1 } and D -action given by A · v k = ξ i − k v k g · v k = ξ j − km v k ∀ 0 ≤ k ≤ r − 1 , � v k +1 if 0 ≤ k < r − 1 , x · v k = (1 − ξ jn ) v 0 if k = r − 1 , � 0 if k = 0 , X · v k = c k v k − 1 if 0 < k ≤ r − 1 , where c k = ( k ) ω ω − k ( ξ j ω − k +1+ i − 1) , ∀ 1 ≤ k ≤ r − 1 . (1)

New Hopf algebras arising from the generalized lifting method The Generalized Lifting Method Step ( a ) – simple modules / finite-dimensional Nichols algebras Theorem (Bagio, G, Jury Giraldi, Marquez) { V i , j } 1 ≤ i , j < nm is a set of pairwise non-isomorphic simple D-modules. The case n = 2 = m Theorem (G-Jury Giraldi) Let M ∈ K K YD be a finite-dimensional non-simple indecomposable module. Then B ( M ) is infinite-dimensional. Theorem (G-Jury Giraldi, Xiong, Andruskiewitsch-Angiono) Let B ( V ) be a finite-dimensional Nichols algebra over an object V in K K YD . Then V is semisimple and isomorphic either to k χ j = V j , 2 , V 1 , j , V 2 , j , � ℓ =1 or 3 k χ ℓ , V 1 , j ⊕ k χ , V 2 , j ⊕ k χ 3 , V 1 , 1 ⊕ V 1 , 3 , V 2 , 1 ⊕ V 2 , 3 with j = 1 , 3 .

New Hopf algebras arising from the generalized lifting method The Generalized Lifting Method Step ( a ) – Nichols algebras ◮ B ( � n i =1 k χ ℓ i ) = � n i =1 k χ ℓ i , dim B ( � n i =1 k χ ℓ i ) = 2 n . ◮ B ( V 1 , j ) = k � x , y : x 2 + 2 ξ y 2 = 0 , xy + yx = 0 , x 4 = 0 � , dim B ( V 1 , j ) = 8. The braiding is not diagonal � new example! ◮ B ( V 1 , j ⊕ k χ ) = k � x , y , z � / J , with J generated by: x 2 + 2 ξ y 2 = 0 , x 4 = 0 , z 2 = 0 , xy + yx = 0 , zx 2 + (1 − ξ j ) xzx − ξ j x 2 z = 0 , ξ j xyz − ξ j xzy + yzx + zxy = 0 , 1 2 ξ (1 + ξ − j )( xz ) 2 ( yz ) 2 + ( yz ) 4 + ( zy ) 4 = 0 . dim B ( V 1 , j ⊕ k χ ) = 128.

New Hopf algebras arising from the generalized lifting method The Generalized Lifting Method Steps ( b ) and ( c ) – Liftings & generation in degree one Theorem (G-Jury Giraldi, Xiong, Andruskiewitsch-Angiono) Let H be a finite-dimensional Hopf algebra over K . Then H is isomorphic either to ( i ) ( � n i =1 k χ ℓ i )# K with ℓ i = 1 , 3 ; ( ii ) B ( V 2 , j )# K for j = 1 , 3 ; ( iii ) B ( V 2 , j ⊕ k χ 3 )# K ; ( iv ) B ( V 2 , 1 ⊕ V 2 , 3 )# K ( v ) A 1 , j ( µ ) for j = 1 , 3 and some µ ∈ k ; ( vi ) A 1 , j , 1 ( µ, ν ) for j = 1 , 3 and some µ, ν ∈ k . ( vii ) A 1 , 1 , 1 , 3 ( µ, ν ) for j = 1 , 3 and some µ, ν ∈ k .