Partial actions and Galois theory

Abstract

In this article, among other results, we develop a Galois theory of commutative rings under partial actions of finite groups, extending the well-known results by In the celebrated paper by Chase, Harrison and Rosenberg [3] the authors developed a Galois theory for commutative ring extensions S ⊃ R, under the assumptions that S is separable over R, finitely generated and projective as an R-module, and the elements of the Galois group G are pairwise strongly distinct R-automorphisms of S. In particular, Theorem 1.3 of that paper gives several equivalent conditions for the definition of a Galois extension and Theorem 2.3 states a one-to-one correspondence between the subgroups of G and the R-subalgebras of S which are separable and G-strong.On the other hand, partial actions of groups have been introduced in the theory of operator algebras giving powerful tools of their study (see, in particular, [6,7,10,16] and [18]). A related concept, that of a partial representation of a group on a Hilbert space, has been defined independently by Exel [7], and Quigg and Raeburn [18]. Several relevant classes of C * -algebras were deeply investigated in [8-10] from the point of view of partial actions and partial representations of groups, including the Cuntz-Krieger algebras introduced in [4].Given a partial action of a group on an object it is natural to ask whether it is a restriction of a global action defined on a bigger object. Such global action is called a globalization or an enveloping action, provided that certain minimality condition is satisfied which guarantees its uniqueness. Globalizations of partial actions where first considered by F. Abadie in his PhD Thesis of 1999 (see also [1]) and independently by Kellendonk and Lawson in [15]. $ This paper was partially supported by CNPq, CAPES, FAPESP and FAPERGS (Brazil). (M. Dokuchaev), mferrero@mat.ufrgs.br (M. Ferrero), paques@ime.unicamp.br (A. Paques).

Twisted partial actions of Hopf algebras

“…The fact that (24) and (27) define a twisted partial action of H ′ 1 = (κS 1 ) ⊗n ⋊ κG on A ′ , which satisfies (16) and (17), will follow from the next easy: Proposition 3.2. Let G be a finite group and L be a cocommutative Hopf algebra over a field κ, such that L is a left κG-module algebra.…”

“…Then using (26) and (25) it is readily seen that the properties (3), (4), (16) and (17) are resumed respectively to the following equalities:…”

“…In particular, the Galois theory of partial group actions developed in [16] inspired further Galois theoretic results in [9], as well as the introduction and study of partial Hopf actions and coactions in [10]. The latter paper became in turn the starting point for further investigation of partial Hopf (co)actions in [2], [3] and [4].…”

“…Given a twisted partial action (α, ω) of a Hopf κ-algebra H on a κ-algebra A, the κ-algebra A# (α,ω) H is called a crossed product by a twisted partial action (shortly, a partial crossed product) if the additional conditions (16) and (17) hold.…”

“…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].…”