arXiv subjects
Publications and source records attributed to Yevgenia Kashina.
For two families of Yetter-Drinfel'd Hopf algebras considered earlier by the authors, we determine which of them are isomorphic. We also determine which of their Radford biproducts are isomorphic.
We describe in which ways the Radford biproducts of certain eight-dimensional Yetter-Drinfel'd Hopf algebras over the elementary abelian group of order 4 can be written as extensions of Hopf algebras.
Let $f(x) = αx + β\mod 1$ for fixed real parameters $α$ and $β$. For any positive integer $n$, define the Sós permutation $π$ to be the lexicographically first permutation such that $0 \leq f(π(0)) \leq f(π(1)) \leq \cdots \leq f(π(n)) < 1$. In this article we give a bijection between Sós permutations and regions in a partition of the parameter space $(α,β)\in [0,1)^2$. This allows us to enumerate these permutations and to obtain the following "three areas" theorem: in any vertical strip $(a/b,c/d)\times [0,1)$, with $(a/b,c/d)$ a Farey interval, there are at most three distinct areas of regions, and one of these areas is the sum of the other two.
By constructing explicit examples, we show that the core of a group-like element in a cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra over the group ring of a finite abelian group is not always completely trivial.
In this paper we classify, up to equivalence, all semisimple nontrivial Hopf algebras of dimension $2^{2n+1}$ for $n\geq 2$ over an algebraically closed field of characteristic $0$ with the group of group-like elements isomorphic to $\mathbb{Z}_{2^{n}}\times \mathbb{Z}_{2^{n}}$. Moreover we classify all such nonisomorphic Hopf algebras of dimension $32$ and show that they are not twist-equivalent to each other. More generally, given an abelian group of order $2^{m-1}$ we give an upper bound for the number of nonisomorphic nontrivial Hopf algebras of dimension $2^{m}$ which have this group as their group of group-like elements.
We classify integral modular categories of dimension pq^4 and p^2q^2 where p and q are distinct primes. We show that such categories are always group-theoretical except for categories of dimension 4q^2. In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara-Yamagami categories and quantum groups. We show that a non-group-theoretical integral modular category of dimension 4q^2 is equivalent to either one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising from a certain quantum group.
We introduce two kinds of gauge invariants for any finite-dimensional Hopf algebra H. When H is semisimple over C, these invariants are respectively, the trace of the map induced by the antipode on the endomorphism ring of a self-dual simple module, and the higher Frobenius-Schur indicators of the regular representation. We further study the values of these higher indicators in the context of complex semisimple quasi-Hopf algebras H. We prove that these indicators are non-negative provided the module category over H is modular, and that for a prime p, the p-th indicator is equal to 1 if, and only if, p is a factor of dim H. As an application, we show the existence of a non-trivial self-dual simple H-module with bounded dimension which is determined by the value of the second indicator.
We study the higher Frobenius-Schur indicators of modules over semisimple Hopf algebras, and relate them to other invariants as the exponent, the order, and the index. We prove various divisibility and integrality results for these invariants. In particular, we prove a version of Cauchy's theorem for semisimple Hopf algebras. Furthermore, we give some examples that illustrate the general theory.
In this paper we show that for an important class of non-trivial Hopf algebras, the Schur indicator is a computable invariant. The Hopf algebras we consider are all abelian extensions; as a special case, they include the Drinfeld double of a group algebra. In addition to finding a general formula for the indicator, we also study when it is always positive. In particular we prove that the indicator is always positive for the Drinfeld double of the symmetric group, generalizing the classical result for the symmetric group itself.
We prove that, over an algebraically closed field of characteristic zero, a semisimple Hopf algebra that has a nontrivial self-dual simple module must have even dimension. This generalizes a classical result of W. Burnside. As an application, we show under the same assumptions that a semisimple Hopf algebra that has a simple module of even dimension must itself have even dimension.
In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple non-commutative Hopf algebras of dimension 16. Moreover, we prove that non-commutative semisimple Hopf algebras of dimension p^n, p is prime, cannot have a cyclic group of grouplikes.