Regular and biregular module algebras

Abstract EN

Motivated by the study of von Neumann regular skew group rings as carried out by Alfaro, Ara, and del R´ıo in [1], we investigate regular and biregular Hopf module algebras.

If A is an algebra with an action by an affine Hopf algebra H, then any H-stable left ideal of A is a direct summand if and only if AH is regular and the invariance functor (−)H induces an equivalence between AH-Mod and the Wisbauer category of A as A#H-module.

Analogously we show a similar statement for the biregularity of A relative to H where AH is replaced by R = Z(A) ∩ AH using the module theory of A as an module over Ae H the enveloping Hopf algebroid of A and H.

We show that every two-sided H-stable ideal of A is generated by a central H-invariant idempotent if and only if R is regular and Am is H-simple for all maximal ideals m of R.

Further sufficient conditions are given for A#H and AH to be regular.