Pointed Hopf actions on fields, II

Abstract

Abstract. This is a continuation of the authors' study of finite-dimensional pointed Hopf algebras H which act inner faithfully on commutative domains. As mentioned in Part I of this work, the study boils down to the case where H acts inner faithfully on a field. These Hopf algebras are referred to as Galois-theoretical.In this work, we provide classification results for finite-dimensional pointed Galois-theoretical Hopf algebras H of finite Cartan type. Namely, we determine when such H of type A ×r 1 and some H of rank two possess the Galois-theoretical property. Moreover, we provide necessary and sufficient conditions for Reshetikhin twists of small quantum groups to be Galois-theoretical.

On actions of Drinfel'd doubles on finite dimensional algebras

“…While the Hopf algebra H n (ζ, m, t) generalize the Taft algebras as bosonizations of quantum linear spaces over finite cyclic groups, there are other coradically graded generalizations and directions to consider for further study. For instance, one could start by considering the actions of finite-dimensional pointed Hopf algebras presented in work of Etingof and Walton [11,12].…”

Actions of quantum linear spaces on quantum algebras

“…Secondly, we consider actions on higher-dimensional algebras, specifically quantum affine spaces and quantum matrix algebras. Finally, we study actions of bosonizations of quantum linear spaces (see [3,13]).…”

Actions of Ore extensions and growth of polynomialH-identities

“…a grading, an action of a group G by automorphisms and anti-automorphisms, an action of a Lie algebra by derivations, since in this case it is reasonable to consider, respectively, graded, G-or differential identities [5,6,29]. The case of a Hopf algebra action is of special interest because of the recent developments in the theory of algebras with Hopf algebra actions [12,16].…”