Duals of pointed Hopf algebras

被引:20
|
作者
Beattie, M [1 ]
机构
[1] Mt Allison Univ, Dept Math & Comp Sci, Sackville, NB E4L 1E6, Canada
关键词
D O I
10.1016/S0021-8693(03)00034-6
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
In this paper, we study the duals of some finite-dimensional pointed Hopf algebras working over an algebraically closed field k of characteristic 0. In particular, we study pointed Hopf algebras with coradical k[Gamma] for Gamma a finite abelian group, and with associated graded Hopf algebra of the form B(V) k[Gamma]YD. As a corollary to a B(V) # k[Gamma] where B(V) is the Nichols algebra of V = circle plus(i) V-gi(chii) is an element of (k[Gamma])(k[Gamma])yD. general theorem on duals of coradically graded Hopf algebras, we have that the dual of B(V) # k[Gamma] is B(W) # k[<(Gamma)over cap>] where W = circle plus(i) W-chii(gi) is an element of (k[<(Gamma)over cap>]YD)-Y-k[<(Gamma)over cap>]. This description of the dual is used to explicitly describe the Drinfel'd double of B(V) # k[F]. We also show that the dual of a nontrivial lifting A of B(V) # k[F] which is not itself a Radford biproduct, is never pointed. For V a quantum linear space of dimension 1 or 2, we describe the duals of some liftings of B(V) # k[F]. We conclude with some examples where we determine all the irreducible finite-dimensional representations of a lifting of B(V) # k[Gamma] by computing the matrix coalgebras in the coradical of the dual. (C) 2003 Elsevier Science (USA). All rights reserved.
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收藏
页码:54 / 76
页数:23
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