arXiv subjects

Shlomo Gelaki

Publications and source records attributed to Shlomo Gelaki.

59 records · Page 4Linked to original sources

arXiv1998

On Finite-Dimensional Semisimple and Cosemisimple Hopf Algebras in Positive Characteristic

Recently, important progress has been made in the study of finite-dimensional semisimple Hopf algebras over a field of characteristic zero. Yet, very little is known over a field of positive characteristic. In this paper we prove some results on finite-dimensional semisimple and cosemisimple Hopf algebras A over a field of positive characteristic, notably Kaplansky's 5th conjecture on the order of the antipode of A. These results have already been proved over a field of characteristic zero, so in a sense we demonstrate that it is sufficient to consider semisimple Hopf algebras over such a field (they are also cosemisimple), and then to use our Lifting Theorem 2.1 to prove it for semisimple and cosemisimple Hopf algebras over a field of positive characteristic. In our proof of Lifting Theorem 2.1 we use standard arguments of deformation theory from positive to zero characteristic. The key ingredient of the proof is the theorem that the bialgebra cohomology groups of A vanish.

math.QA↗

arXiv1998

On pointed Hopf algebras and Kaplansky's 10th conjecture

In this paper we construct and study two new families of finite dimensional pointed Hopf algebras which generalize Radford's families. We show that over any infinite field which contains a primitive nth root of unity, one of the families contains infinitely many non-isomorphic Hopf algebras of any dimension of the form Nn^2, where 2<n<N are integers so that n divides N. We thus answer in the negative Kaplansky's 10th conjecture from 1975 on the finite number of types of Hopf algebras of a given dimension.

math.QA↗

arXiv1998

A method of construction of finite-dimensional triangular semisimple Hopf algebras

The goal of this paper is to give a new method of constructing finite-dimensional semisimple triangular Hopf algebras, including minimal ones which are non-trivial (i.e. not group algebras). The paper shows that such Hopf algebras are quite abundant. It also discovers an unexpected connection of such Hopf algebras with bijective 1-cocycles on finite groups and set-theoretical solutions of the quantum Yang-Baxter equation defined by Drinfeld.

math.QA↗

arXiv1998

On Semisimple Hopf algebras of dimension pq

In this paper we consider some properties of semisimple Hopf algebras of dimension pq where p and q are distinct primes. These properties are useful for classification of such Hopf algebras. In particular, we show that for such a Hopf algebra H, if H and H^* are of Frobenius type then H or H^* is a group algebra.

math.QA↗

arXiv1997

Some Properties of Finite-Dimensional Semisimple Hopf Algebras

Kaplansky conjectured that if H is a finite-dimensional semisimple Hopf algebra over an algebraically closed field k of characteristic 0, then H is of Frobenius type (i.e. if V is an irreducible representation of H then dimV divides dimH). It was proved by Montgomery and Witherspoon that the conjecture is true for H of dimension p^n, p prime, and by Nichols and Richmond that if H has a 2-dimensional representation then dimH is even. In this paper we first prove that if V is an irreducible representation of D(H), the Drinfeld double of any finite-dimensional semisimple Hopf algebra H over k, then dimV divides dimH (not just dimD(H)=(dimH)^2). In doing this we use the theory of modular tensor categories (in particular Verlinde formula). We then use this statement to prove that Kaplansky's conjecture is true for finite-dimensional semisimple quasitriangular Hopf algebras over k. As a result we prove easily the result of Zhu that Kaplansky's conjecture on prime dimensional Hopf algebras over k is true, by passing to their Drinfeld doubles. Second, we use a theorem of Deligne on characterization of tannakian categories to prove that triangular semisimple Hopf algebras over k are equivalent to group algebras as quasi-Hopf algebras.

q-alg↗