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Register a new userOrders of Nikshychs Hopf algebra
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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.
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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.
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.
Let $W$ be a Coxeter group. The goal of the paper is to construct new Hopf algebras that contain Hecke algebras $H_{bf q}(W)$ as (left) coideal subalgebras. Our Hecke-Hopf algebras ${bf H}(W)$ have a number of applications. In particular they provide new solutions of quantum Yang-Baxter equation and lead to a construction of a new family of endo-functors of the category of $H_{bf q}(W)$-modules. Hecke-Hopf algebras for the symmetric group are related to Fomin-Kirillov algebras, for an arbitrary Coxeter group $W$ the Demazure part of ${bf H}(W)$ is being acted upon by generalized braided derivatives which generate the corresponding (generalized) Nichols algebra.
In this paper, we prove that a non-semisimple Hopf algebra H of dimension 4p with p an odd prime over an algebraically closed field of characteristic zero is pointed provided H contains more than two group-like elements. In particular, we prove that non-semisimple Hopf algebras of dimensions 20, 28 and 44 are pointed or their duals are pointed, and this completes the classification of Hopf algebras in these dimensions.
Let H be a non-semisimple Hopf algebra of dimension 2p^2 over an algebraically closed field of characteristic zero, where p is an odd prime. We prove that H or H^* is pointed, which completes the classification for Hopf algebras of these dimensions.
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