Total quantum integrals of weak Hopf group coalgebras

Abstract

In this paper, we introduce the concept of total quantum integrals in the case of weak crossed Hopf $\pi$-coalgebras and prove the affineness criterion
for weak quantum Yetter-Drinfel'd $\pi$-modules, which
is a generalization of the total quantum integral introduced by Chen and Wang(Filomat  26: 101-118, (2012)).