Orders of Nikshych's Hopf algebra

Advances in Mathematics

2017

Self Cite

Abstract. We study Hopf algebras via tools from geometric invariant theory. We show that all the invariants we get can be constructed using the integrals of the Hopf algebra and its dual together with the multiplication and the comultiplication, and that these invariants determine the isomorphism class of the Hopf algebra. We then define certain canonical subspaces Inv i,j of tensor powers of H and H * , and use the invariant theory to prove that these subspaces satisfy a certain non-degeneracy condition. Using this non-degeneracy condition together with results on symmetric monoidal categories, we prove that the spaces Inv i,j can also be described as (A , where A is the group of Hopf automorphisms of H. As a result we prove that the number of possible Hopf orders of any semisimple Hopf algebra over a given number ring is finite. we give some examples of these invariants arising from the theory of Frobenius-Schur Indicators, and from Reshetikhin-Turaev invariants of three manifolds. We give a complete description of the invariants for a group algebra, proving that they all encode the number of homomorphisms from some finitely presented group to the group. We also show that if all the invariants are algebraic integers, then the Hopf algebra satisfies Kaplansky's sixth conjecture: the dimensions of the irreducible representations of H divide the dimension of H.

Section: Finiteness Of the Number Of Ordersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

See 1 more Smart Citation

Semisimple Hopf algebras via geometric invariant theory

Advances in Mathematics

2017

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“…When n = 1, we call them character supported elements. For example, if g and h are characters of H * , and ψ is a character of H, the element ψ(g (1) g (4) )g (3) hS(g (2) ) is character supported. Definition 1.5.…”

Section: Preliminariesmentioning

confidence: 99%

On the existence of orders in semisimple Hopf algebras

Cuadra

Trans. Amer. Math. Soc.

2015

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We show that there is a family of complex semisimple Hopf algebras that do not admit a Hopf order over any number ring. They are Drinfel'd twists of certain group algebras. The twist contains a scalar fraction which makes impossible the definability of such Hopf algebras over number rings. We also prove that a complex semisimple Hopf algebra satisfies Kaplansky's sixth conjecture if and only if it admits a weak order, in the sense of Rumynin and Lorenz, over the integers.Given a twist J for H a new Hopf algebra H J , called Drinfel'd twist of H, can be constructed as follows: H J = H as an algebra, the counit is that of H, and the new comultiplication and antipode are:Here U J := J (1) S(J (2) ) and, if we write J −1 = J −(1) ⊗ J −(2) , its inverse is U −1 J = S(J −(1) )J −(2) . We next describe the particular example of twist for the group algebra that we will use. This construction was used several times in the past, see for example, [6], [8] or [18] and the references therein. Let M be a finite abelian group and M its group of characters. Consider the group algebra KM and its dual Hopf algebra (KM ) * . We identify (KM ) * and K M as Hopf algebras. Assume that char K ∤ |M |. There is an isomorphism of Hopf algebras (KM ) * ≃ KM induced by an isomorphism of groups M ≃ M . Let {ϑ m } m∈M ⊂ (KM ) * be the dual basis of {m} m∈M . If ω : M × M → K × is a normalized 2-cocycle, then J = m,m ′ ∈M ω(m, m ′ )ϑ m ⊗ ϑ m ′ ∈ (KM ) * ⊗ (KM ) *

“…In another direction, we ask if a complex semisimple Hopf algebra that is lower‐semisolvable, as defined by Montgomery and Witherspoon in , always admits a Hopf order over a number ring. Some other questions arising from our previous results on orders in semisimple Hopf algebras can be found in [, p. 2548; , p. 954].…”

Section: Introductionmentioning

confidence: 99%

Non‐existence of Hopf orders for a twist of the alternating and symmetric groups

Cuadra

2019

J. London Math. Soc.

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We prove the non-existence of Hopf orders over number rings for two families of complex semisimple Hopf algebras. They are constructed as Drinfel'd twists of group algebras for the following groups: An, the alternating group on n elements, with n 5, and S2m, the symmetric group on 2m elements, with m 4 even. The twist for An arises from a 2-cocycle on the Klein four-group contained in A4. The twist for S2m arises from a 2-cocycle on a subgroup generated by certain transpositions, which is isomorphic to Z m 2 . This provides more examples of complex semisimple Hopf algebras that cannot be defined over number rings. As in the previous family known, these Hopf algebras are simple.Similarly to the previous family, they are constructed as Drinfel'd twists of certain group algebras. The twist is in turn constructed using the following method due to Movshev [16].Let M be an abelian subgroup of G. Let K be a number field. Suppose that K is large enough so that the group algebra KM splits. Denote by M the character group of M . For φ ∈ M , let e φ be the associated idempotent in KM . If ω : M × M → K × is a normalized 2-cocycle, thenis a twist for KG. The twisting procedure alters the coalgebra structure and the antipode of KG but leaves unchanged the algebra structure. So, as algebras, these Hopf algebras are group algebras.Here we deal with the following two families of groups and twists:(1) The alternating group A n on n elements, with n 5. Consider the double transpositions d 1 = (12)(34) and d 2 = (13)(24). The subgroup M is generated by them. The character group M is generated by ϕ i , for i = 1, 2, given by ϕ i (d j ) = (−1) δij . The 2-cocycle ω is the bicharacter:This family was introduced by Nikshych in [17] and provided the first non-trivial examples of simple and semisimple Hopf algebras.(2) The symmetric group S 2m on 2m elements, with m 4 even.This family was introduced by Bichon in [2] and further discussed by Galindo and Natale in [10]. These examples are also simple.Our main results, Theorems 3.3 and 4.5, are abridged in the following statement:Theorem. Let K be a number field with ring of integers O K . Let R ⊂ K be a Dedekind domain such that O K ⊆ R. Let G be one of the above groups and J the twist arising from the corresponding cocycle. If the twisted Hopf algebra (KG) J admits a Hopf order over R, then 1 2 ∈ R.As a consequence, (KG) J does not admit a Hopf order over any number ring.This theorem implies that the complexified Hopf algebra (CG) J does not admit a Hopf order over any number ring.The strategy of the proof can be outlined as follows. Hopf orders are inherited to Hopf subalgebras and quotient Hopf algebras (Proposition 1.1). For our purposes, this fact allows us to focus immediately on A 5 in the case of A n . The case of S 2m needs several reductions to subgroups and quotient groups to focus ultimately on S 8 . Assume now that H is a Hopf algebra over K and X is a Hopf order of H over R. Consider the dual Hopf algebra H * . The dual order X consists of those ϕ ∈ H * such that ϕ(X) ⊆ R. Th...