WEAK HOPF-ALGEBRAS AND SMASH PRODUCTS
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 associative 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) andwD(H)≤ wD(A)+wD(X).