On quiver-theoretic description for quasitriangularity of Hopf algebras

Abstract

This paper is devoted to the study of the quasitriangularity of Hopf algebras via Hopf quiver approaches. We give a combinatorial description of the Hopf quivers whose path coalgebras give rise to coquasitriangular Hopf algebras. With a help of the quiver setting, we study general coquasitriangular pointed Hopf algebras and obtain a complete classification of the finite-dimensional ones over an algebraically closed field of characteristic 0.

On Coquasitriangular Pointed Majid Algebras

“…The following is our main result which enables us to construct coradically graded coquasitriangular pointed Majid algebras exhaustively on Hopf quivers. This is a quasi analogue of Theorem 4.2 in [13] and the proof is given by adjusting the argument there into our situation.…”

“…(1) The coquasitriangular structure R is sort of a "quasi" skew-symmetric bicharacter of the group G. Clearly, if Φ is trivial, then R is a usual skew-symmetric bicharacter. This is the usual Hopf case as given by Theorem 3.3 in [13]. More generally, if Φ is a coboundary, then by a suitable twisting, we can also go back to the Hopf case.…”

“…As direct consequence, the pointed Majid algebras M + (8), M − (8) and M (32) over Z 2 given in [12], and M (n, s, q) with s = 0 over Z n given in [14] are not coquasitriangular since they are nontrivial graded pointed Majid algebras, that is, pointed Majid algebras which are not twisting equivalent to Hopf algebras. Note that finite-dimensional connected graded cotriangular pointed Hopf algebras over Z n is completely classified by Corollary 6.3 of [13]. Therefore, finite-dimensional connected graded cotriangular Majid algebras over Z n are essentially known by the previous proposition.…”

“…Of course, Proposition 4.3 also implies the corresponding consequence on connected braided pointed finite tensor categories whose invertible objects consisting of the cyclic group Z n . In particular, together with [14,13] in a fairly straightforward way, we get a classification result for braided pointed tensor categories of finite type, i.e., in which there are only finitely many indecomposable objects.…”

“…As before we will work on a dual setting, namely the so-called coquasitriangular Majid algebras, since this allows a wider scope and the convenience of exposition. A recent work of the authors [13] shows that the coquasitriangularity of pointed Hopf algebras can be described by combinatorial property of Hopf quivers. Moreover, the quiver setting helps to give a complete classification of finite-dimensional coquasitriangular pointed Hopf algebras over an algebraically closed field of characteristic 0.…”