Semisimple Hopf actions on Weyl algebras

We study actions of semisimple Hopf algebras H on Weyl algebras A over a field of characteristic zero. We show that the action of H on A must factor through a group algebra; in other words, if H acts inner faithfully on A, then H is cocommutative. The techniques used include reduction modulo a prime number and the study of semisimple cosemisimple Hopf actions on division algebras.

Orders of Nikshychs Hopf algebra

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.

On the existence of orders in semisimple Hopf algebras

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.

On two finiteness conditions for Hopf algebras with nonzero integral

A Hopf algebra is co-Frobenius when it has a nonzero integral. It is proved that the composition length of the indecomposable injective comodules over a co-Frobenius Hopf algebra is bounded. As a consequence, the coradical filtration of a co-Frobenius Hopf algebra is finite; this confirms a conjecture by Sorin Du{a}scu{a}lescu and the first author. The proof is of categorical nature and the same result is obtained for Frobenius tensor categories of subexponential growth. A family of co-Frobenius Hopf algebras that are not of finite type over their Hopf socles is constructed, answering so in the negative another question by the same authors.

On the structure of (co-Frobenius) Hopf algebras

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).

A sequence to compute the Brauer group of certain quasi-triangular Hopf algebras

A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra $B$ in a braided monoidal category $C$, and under certain assumptions on the braiding (fulfilled if $C$ is symmetric), we construct a sequence for the Brauer group $BM(C;B)$ of $B$-module algebras, generalizing Beatties one. It allows one to prove that $BM(C;B) cong Br(C) times Gal(C;B),$ where $Br(C)$ is the Brauer group of $C$ and $Gal(C;B)$ the group of $B$-Galois objects. We also show that $BM(C;B)$ contains a subgroup isomorphic to $Br(C) times Hc(C;B,I),$ where $Hc(C;B,I)$ is the second Sweedler cohomology group of $B$ with values in the unit object $I$ of $C$. These results are applied to the Brauer group of a quasi-triangular Hopf algebra that is a Radford biproduct $B times H$, where $H$ is a usual Hopf algebra over a field $K$, the Hopf subalgebra generated by the quasi-triangular structure $R$ is contained in $H$ and $B$ is a Hopf algebra in the category ${}_HM$ of left $H$-modules. The Hopf algebras whose Brauer group was recently computed fit this framework. We finally show that $BM(K,H,R) times Hc({}_HM;B,K)$ is a subgroup of the Brauer group $BM(K,B times H,R),$ confirming the suspicion that a certain cohomology group of $B times H$ (second lazy cohomology group was conjectured) embeds into $BM(K,B times H,R).$ New examples of Brauer groups of quasi-triangular Hopf algebras are computed using this sequence.

On the Hopf-Schur group of a field

Let k be any field. We consider the Hopf-Schur group of k, defined as the subgroup of the Brauer group of k consisting of classes that may be represented by homomorphic images of Hopf algebras over k. We show here that twisted group algebras and abelian extensions of k are quotients of cocommutative and commutative Hopf algebras over k, respectively. As a consequence we prove that any tensor product of cyclic algebras over k is a quotient of a Hopf algebra over k, revealing so that the Hopf-Schur group can be much larger than the Schur group of k.