Semilattice graded weak Hopf algebra and its related quantum G-double

Abstract

The major purpose of this paper is to construct a weak Hopf algebra with grading from a family of Hopf algebras, and then to gain a related quantum G-double with regular R-matrix. First, over a field k, we introduce a so-called semilattice graded weak Hopf algebra H=⊕α∈YHα. Then the quantum G-double D′(H) of H is obtained in case that H is commutative. Moreover, it is shown that D′(H) is semisimple (respectively, von Neumann regular) if and only if H is a semisimple (respectively, von Neumann regular) Hopf algebra. At last, a nontrivial example of semilattice graded weak Hopf algebras is obtained.