Globalization of twisted partial actions

Abstract

Abstract. Let A be a unital ring which is a product of possibly infinitely many indecomposable rings. We establish a criteria for the existence of a globalization for a given twisted partial action of a group on A. If the globalization exists, it is unique up to a certain equivalence relation and, moreover, the crossed product corresponding to the twisted partial action is Morita equivalent to that corresponding to its globalization. For arbitrary unital rings the globalization problem is reduced to an extendibility property of the multipliers involved in the twisted partial action.

Dilations of partial representations of Hopf algebras

“…In [14], the globalization of a partial action of a group G over a unital algebra A is achieved if, and only if, the partial domains A g , for each g ∈ G, are unital ideals, that is, they are ideals of A generated by a central idempotent 1 g . In [16], the globalization for twisted partial actions was obtained. In [4], it was proved that every partial action of a group G over an algebra A, such that the ideals A g are idempotent,…”

Twisted partial actions of Hopf algebras

“…for all h, k, l ∈ H. It is clear that the induced map ω (see equality (15)) satisfies condition (16), and we will show that (17) is also satisfied. In what follows, note that (…”

“…The importance of partial actions and partial representations was reinforced by R. Exel in [24] where, among other results, it was proved that given a field K of characteristic 0, a group G and subgroups H, N ⊆ G with N normal in G and H normal in N, there is a twisted partial action θ of G/N on the group algebra K(N/H) such that the Hecke algebra H(G, H) is isomorphic to the crossed product K(N/H) * θ G/N. More recent algebraic results on twisted partial actions and corresponding crossed products were obtained in [5], [15] and [30]. The algebraic concept of twisted partial actions also motivated the study of projective partial group representations, the corresponding partial Schur Multiplier and the relation to partial group actions with K-valued twistings in [17] and [18], contributing towards the elaboration of a background for a general cohomological theory based on partial actions.…”

When is a Crossed Product by a Twisted Partial Action Azumaya?

“…We say that is global if D g = R for all g ∈ G. Given a twisted global action = T g g∈G u g h g h ∈G×G of a group G on a ring T and R = Te, for some nonzero central idempotent e of T , one may obtain a twisted partial action of G on R by restriction of to R. Indeed, it is enough to take D g = R ∩ g R = R g R = Re g e , g = g D g −1 , and w g h = u g h e g e gh e for all g h ∈ G. Under suitable conditions every twisted partial action of a group G on a ring R can be obtained in this way (see [9]). Given a twisted partial action of a group G on a ring R, we recall that the crossed product R G (see [8,Definition 2.2]) is the direct sum g∈G D g g as an abelian group (in which the g 's are symbols), with the usual addition and where the multiplication is defined by the rule r g g r h h = r g g r h 1 g −1 w g h gh By [8, Theorem 2.4] R G is a ring with identity element 1 R 1 .…”