Crossed products by twisted partial actions and graded algebras

Abstract

For a twisted partial action Θ of a group G on an (associative nonnecessarily unital) algebra A over a commutative unital ring k, the crossed product A Θ G is proved to be associative. Given a G-graded k-algebra B = g∈G B g with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B 1 Θ G for some twisted partial action of G on B 1 . The equalityis one of the ingredients of the criteria, and if it holds and, moreover, B has enough local units, then it is shown that B is stably isomorphic to a crossed product by a twisted partial action of G.

Twisted partial actions of Hopf algebras

“…In [14] a twisted partial action of a group G over a unital κ-algebra A was defined as a triple (H, A), given by e(h) = (h · 1 A ), is central with respect to the convolution product. These partial actions are, in some sense, more akin to partial group actions.…”

“…One of the results in [14] gives a criteria for a non-degenerate G-graded algebra B = ⊕ g∈G B g to have the structure of a crossed product A * G by a twisted partial action of G on A = B e . More specifically, if B satisfies…”

“…The new construction permitted to show that any second countable C * -algebraic bundle, which satisfies a certain regularity condition (automatically verified if the unit fiber algebra is stable), is a C * -crossed product of the unit fiber algebra by a continuous partial action of the base group. The algebraic version of the latter fact was established in [14]. 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.…”

“…Example 2.1. This example is inspired by [10,Proposition 4.9] and suits 1 Definition 2.1 of [14]. An idempotent twisted partial action of a group G on a κ-algebra A is a triple…”

“…If (α, ω) is a twisted partial action of κG on a κ-algebra A arisen from a twisted partial action of a group G on A, as defined in Example 2.1, and the conditions (16) and (17) also hold in this case, then A# (α,ω) κG is an slight generalization of the partial crossed product introduced in [14].…”