Almost-triangular Hopf Algebras

Abstract

In this paper, we consider a finite dimensional semisimple cosemisimple quasitriangular Hopf algebra \((H,R\,)\) with \(R^{\,21}R\in C(H\otimes H\,)\) (we call this type of Hopf algebras almost-quasitriangular) over an algebraically closed field \(k\). We denote by \(B\) the vector space generated by the left tensorand of \(R^{\,21}R\). Then \(B\) is a sub-Hopf algebra of \(H\). We proved that when \(\dim B\) is odd, \(H\) has a triangular structure and can be obtained from a group algebra by twisting its usual comultiplication [14]; when \(\dim B\) is even, \(H\) is an extension of an abelian group algebra and a triangular Hopf algebra, and may not be triangular. In general, an almost-triangular Hopf algebra can be viewed as a cocycle bicrossproduct.