WEAK HOPF-ALGEBRAS AND SMASH PRODUCTS

Abstract

The definitions of a u-weak Hopf algebra and the quantum dimension detuM of a representation M by u are given. It is shown that a u-weak Hopf algebra H is semisimple if and only if there is a finite-dimensional projective H-module P such that detuP is invertible. Let X be an asso- ciative algebra and A is a weak Hopf algebra. We investigate the global dimension and the weak dimension of the smash product H ◃▹R A and show that lD(H) ≤ rD(A) + lD(X) and wD(H) ≤ wD(A) + wD(X).