Inner actions of weak Hopf algebras

摘要

Let R be an associative ring and e, f idempotent elements of R. In this paper we introduce the notion of (e, f)-invertibility for an element of R and use it to define inner actions of weak Hopf algebras. Given a weak Hopf algebra H and an algebra A, we present sufficient conditions for A to admit an inner action of H. We also prove that if A is a left H-module algebra then H acts innerly on the smash product A#H if and only if H is a quantum commutative weak Hopf algebra.