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 [Dr1]. Finite-dimensional triangular Hopf algebras were studied by several authors (see e.g. [CWZ,EG,G,M]). In [EG] the authors prove that any finite-dimensional semisimple triangular Hopf algebra over an algebraically closed field of characteristic 0 (say C) is obtained from a group algebra after twisting its comultiplication in the sense of Drinfeld [Dr2]. Twists are easy to construct for abelian groups A, they are just 2-cocycles for A with values in C. A general simple construction of triangular semisimple Hopf algebras which are non-trivial, is the following: take a non-abelian groupG, an abelian subgroup of it A, and a twist J ∈ C[A]⊗C[A] which does not commute with g ⊗ g for all g ∈ G, and twist C[G] by J to obtain C[G] . Examples of such Hopf algebras were constructed by the second author in [G]. They are Hopf algebras of dimension pq where p and q are any prime numbers so that q divides p−1. It was also proved in [G] that the dual of the Drinfeld double of these Hopf algebras is triangular. Nevertheless Hopf algebras which are constructed in this way are not minimal, and their minimal Hopf subalgebras are trivial. In fact, as far as we know, in the literature there are no non-trivial semisimple minimal triangular Hopf algebras. A natural question thus arose: Are there finite-dimensional non-trivial minimal semisimple triangular Hopf algebras? In this paper we describe a method for constructing such Hopf algebras. The paper is organized as follows. First, we show how to construct twists for certain solvable non-abelian groups by iterating twists of their abelian subgroups, and thus obtain non-trivial semisimple triangular Hopf algebras. Second, we show that in some cases this construction gives non-trivial semisimple minimal triangular Hopf algebras. Finally, we show how any non-abelian group which admits a