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Register a new userNon-existence of Hopf orders for a twist of the alternating and symmetric groups
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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 Drinfeld twists of group algebras for the following groups: $A_n$, the alternating group on $n$ elements, with $n geq 5$; and $S_{2m}$, the symmetric group on $2m$ elements, with $m geq 4$ even. The twist for $A_n$ arises from a $2$-cocycle on the Klein four-group contained in $A_4$. The twist for $S_{2m}$ arises from a $2$-cocycle on a subgroup generated by certain transpositions which is isomorphic to $mathbb{Z}_2^m$. This provides more examples of complex semisimple Hopf algebras that can not be defined over number rings. As in the previous family known, these Hopf algebras are simple.
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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 Drinfeld 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 Kaplanskys sixth conjecture if and only if it admits a weak order, in the sense of Rumynin and Lorenz, over the integers.
Let $p$ be an odd prime number and $K$ a number field having a primitive $p$-th root of unity $zeta.$ We prove that Nikshychs non-group theoretical Hopf algebra $H_p$, which is defined over $mathbb{Q}(zeta)$, admits a Hopf order over the ring of integers $mathcal{O}_K$ if and only if there is an ideal $I$ of $mathcal{O}_K$ 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 $mathcal{O}_K$ exists, it is unique and we describe it explicitly.
Let $p$ be a prime number and $q=p^m$, with $m geq 1$ if $p eq 2,3$ and $m>1$ otherwise. Let $Omega$ be any non-trivial twist for the complex group algebra of $mathbf{PSL}_2(q)$ arising from a $2$-cocycle on an abelian subgroup of $mathbf{PSL}_2(q)$. We show that the twisted Hopf algebra $(mathbb{C} mathbf{PSL}_2(q))_{Omega}$ does not admit a Hopf order over any number ring. The same conclusion is proved for the Suzuki group $^2!B_2(q)$ and $mathbf{SL}_3(p)$ when the twist stems from an abelian $p$-subgroup. This supplies new families of complex semisimple (and simple) Hopf algebras that do not admit a Hopf order over any number ring. The strategy of the proof is formulated in a general framework that includes the finite simple groups of Lie type. As an application, we combine our results with two theorems of Thompson and Barry and Ward on minimal simple groups to establish that for any finite non-abelian simple group $G$ there is a twist $Omega$ for $mathbb{C} G$, arising from a $2$-cocycle on an abelian subgroup of $G$, such that $(mathbb{C} G)_{Omega}$ does not admit a Hopf order over any number ring. This partially answers in the negative a question posed by Meir and the second author.
In this paper we introduce a trace-like invariant for the irreducible representations of a finite dimensional complex Hopf algebra H. We do so by considering the trace of the map induced by the antipode S on the endomorphisms End(V) of a self-dual module V. We also compute the values of this trace for the representations of two non-semisimple Hopf algebras: u_q(sl_2) and D(H_n(q)), the Drinfeld double of the Taft algebra.
We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf algebra with injective antipode is a deformation of the bosonization of the Hopf coradical by its diagram, a connected graded Hopf algebra in the category of Yetter-Drinfeld modules over the latter. We discuss the steps needed to classify Hopf algebras in suitable classes accordingly. For the class of co-Frobenius Hopf algebras, we prove that a Hopf algebra is co-Frobenius if and only if its Hopf coradical is so and the diagram is finite dimensional. We also prove that the standard filtration of such Hopf algebras is finite. Finally, we show that extensions of co-Frobenius (resp. cosemisimple) Hopf algebras are co-Frobenius (resp. cosemisimple).
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