ON THE REPRESENTATIONS OF WEAK CROSSED PRODUCTS

Abstract

Let H be a finite-dimensional weak Hopf algebra in the sense of [G. Böhm and K. Szlachányi, A coassociative C*-quantum group with nonintegral dimensions, Lett. Math. Phys.35 (1996) 437–456] which is semisimple as well as its dual H*, A be a finite-dimensional algebra measured by H and A#σH be a weak crossed product. Then we first prove that the representation dimension of A#σH equals to that of A. Moreover, we will show that over an algebraically closed field k, the representation type between A and its weak crossed product algebra A#σH is coincident, which extends the result given by Liu [On the structure of tame graded basic Hopf algebras, J. Algebra 299 (2006) 841–853]. Finally, we show that A#σH is a CM-finite n-Gorenstein algebra if and only if so is A and the global Gorenstein projective dimension of A#σH is equal to that of A.