Projective summands in tensor products of simple modules of finite dimensional Hopf algebras

摘要

Let H be a finite dimensional Hopf algebra over a field k. We show that H contains a unique maximal Hopf ideal J(w)(H) contained in J(H), the Jacobson radical of H. We give various characterizations of J(w)(H), for example J(w)(H) = Ann(H)((H/J(H))(circle timesn)) for all large enough n. The smallest positive integer n with this property is denoted by l(w)(H). We prove that l(w)(H) equals the smallest number n such that (H/J(H))(circle timesn) contains every projective indecomposable H/J(w)(H)-module as a direct summand. This also equals the minimal n such that the tensor product of n suitable simple H-modules contains the projective cover of the trivial H/J(w)(H)-module as a direct summand. We define projective homomorphisms between H-modules, which are used to obtain various reciprocity laws for tensor products of simple H-modules and their projective indecomposable direct summands. We also discuss some consequences of our general results in case H = kG is a group algebra of a finite group G and k is a field of characteristic p.