On the existence of orders in semisimple Hopf algebras

Cuadra, Juan; Meir, Ehud

doi:10.1090/tran/6380

Cited by 10 publications

(30 citation statements)

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“…be such that w is invertible and t 2 = w(ζ p − 1). Assume that there is d ∈ Z[ζ n ] such that 1 2 (d + t) ∈ O Q(ζn,t) . Then, H admits a Q(ζ n , t)/Q(ζ n )-form H ′ which in turn admits an order over Z[ζ n ].…”

Section: 4mentioning

confidence: 99%

Orders of Nikshych's Hopf algebra

Cuadra

Meir

2017

J. Noncommut. Geom.

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Let p be an odd prime number and K a number field having a primitive p-th root of unity ζp. We prove that Nikshych's non group-theoretical Hopf algebra Hp, which is defined over Q(ζp), admits a Hopf order over the ring of integers OK if and only if there is an ideal I of OK such that I 2(p−1) = (p). This condition does not hold in a cyclotomic field. Hence this gives an example of a semisimple Hopf algebra over a number field not admitting a Hopf order over any cyclotomic ring of integers. Moreover, we show that, when a Hopf order over OK exists, it is unique and we describe it explicitly.1 Proposition 1.2. Let X be a Hopf order of H. Any character of H belongs to X ⋆ . As a consequence, any character of H * belongs to X.We will also need the following two results by Larson: Proposition 1.3. [6, Proposition 2.2] Let H be a semisimple Hopf algebra over K and X a Hopf order of H. Denote by Λ X and Λ X ⋆ the R-submodule of left integrals of X and X ⋆ respectively. Then ε(Λ X )ε(Λ X ⋆ ) = (dim H) as ideals in R.Proposition 1.4. [6, Corollary 3.2] With hypotheses as before, assume that X and Y are Hopf orders of H such that X ⊆ Y . If ε(Λ X ) = ε(Λ Y ), then X = Y .

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Section: 4mentioning

confidence: 99%

Orders of Nikshych's Hopf algebra

Cuadra

Meir

2017

J. Noncommut. Geom.

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“…The following special case of Corollary 6 is due to Cuadra and Meir [1,Theorem 3.4]. Note that (14) gives a formula for the Casimir element to be tested for integrality.…”

Section: Frobenius Divisibility For Hopf Algebrasmentioning

confidence: 99%

“…It is our aim in this article to present a unified approach to a number of these generalizations using some general observations on symmetric algebras. Some of this material can be found in the earlier article [9] in the slightly more general setting of Frobenius algebras, and the authors have also greatly benefitted from a reading of the preprint [1] by Cuadra and Meir. 0.2.…”

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confidence: 99%

“…We start with a formulation of Corollary 6, due to Cuadra and Meir [1], that is specific to H; the result (Theorem 8) states that FD for H is equivalent to integrality (over Z) of the Casimir element c λ of H that is associated to the unique integral of λ ∈ H * satisfying λ, 1 = 1. Further results derived in Section 2 from the material on symmetric algebras are: a theorem of Zhu [14] on irreducible representations of H whose character is central in H * , the so-called class equation [6], [15], and a theorem of Schneider [13] on factorizable Hopf algebras.…”

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confidence: 99%

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Frobenius divisibility for Hopf algebras

Jacoby¹,

Lorenz²

2017

Groups, Rings, Group Rings, and Hopf Algebras

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ABSTRACT. We present a unified ring theoretic approach, based on properties of the Casimir element of a symmetric algebra, to a variety of known divisibility results for the degrees of irreducible representations of semisimple Hopf algebras in characteristic 0. All these results are motivated by a classical theorem of Frobenius on the degrees of irreducible complex representations of finite groups.

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“…As in our previous paper , the main idea of the proof is to construct elements

ϕ \in X^{★}

and

x \in X

, by a subtle manipulation of characters and cocharacters, such that

ϕ (x) = s / 2

, with

s

an odd number. For the characters, we directly use the character table of

A_{5}

and

S_{4}

, respectively.…”

Section: Introductionmentioning

confidence: 99%

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

Cuadra

Meir

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...

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On the existence of orders in semisimple Hopf algebras

Cited by 10 publications

References 16 publications

Orders of Nikshych's Hopf algebra

Orders of Nikshych's Hopf algebra

Frobenius divisibility for Hopf algebras

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

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