Antonio Paques

Partial Groupoid Actions: Globalization, Morita Theory, and Galois Theory

In this article, we introduce the notion of a partial action of a groupoid on a ring as well as we give a criteria for the existence of a globalization of it. We construct a Morita context associated to a globalizable partial groupoid action, and we introduce the notion of a partial Galois extension, which is related to the strictness of this context.

Crossed Products by Twisted Partial Actions: Separability, Semisimplicity, and Frobenius Properties

In this article, we are concerned with the crossed product S = R☆α G by a twisted partial action α of a group G on a ring R. We give necessary and sufficient (resp., sufficient) conditions for S to be a separable (resp., Frobenius, semisimple) extension of R.

PARTIAL ACTIONS OF ORDERED GROUPOIDS ON RINGS

In this paper, we introduce the notion of a partial action of an ordered groupoid on a ring and we construct the corresponding partial skew groupoid ring. We present sufficient conditions under which the partial skew groupoid ring is either associative or unital. Also, we show that there is a one-to-one correspondence between partial actions of an ordered groupoid G on a ring R, in which the domain of each partial bijection is an ideal, and meet-preserving global actions of the Birget–Rhodes expansion GBR of G on R. Using this correspondence, we prove that the partial skew groupoid ring is a homomorphic image of the skew groupoid ring constructed through the Birget–Rhodes expansion.

Actions of Inverse Semigroups on Algebras

In this article we prove that if S is a faithfully projective R-algebra and H is a finite inverse semigroup acting on S as R-linear maps such that the fixed subring S H = R, then any partial isomorphism between ideals of S which are generated by central idempotents can be obtained as restriction of an R-automorphism of S and there exists a finite subgroup of automorphisms G of S with S G = R.

TWISTED PARTIAL ACTIONS OF HOPF ALGEBRAS

In this work, the notion of a twisted partial Hopf action is introduced as a unified approach for twisted partial group actions, partial Hopf actions and twisted actions of Hopf algebras. The conditions on partial cocycles are established in order to construct partial crossed products, which are also related to partially cleft extensions of algebras. Examples are elaborated using algebraic groups.

A Hopf-Galois correspondence for free algebras

Abstract A Galois correspondence is exhibited between right coideals subalgebras of a finite-dimensional pointed Hopf algebra acting homogeneously and faithfully on a free associative algebra and free subalgebras containing the invariants of this action.

When is a Crossed Product by a Twisted Partial Action Azumaya

In this article, we discuss necessary and sufficient conditions for the crossed product S = R★α G by a twisted partial action α of a finite group G on a ring R to be separable over its center.

GALOIS CORRESPONDENCES FOR PARTIAL GALOIS AZUMAYA EXTENSIONS

Let α be a partial action, having globalization, of a finite group G on a unital ring R. Let Rα denote the subring of the α-invariant elements of R and CR(Rα) the centralizer of Rα in R. In this paper we will show that there are one-to-one correspondences among sets of suitable separable subalgebras of R, Rα and CR(Rα). In particular, we extend to the setting of partial group actions similar results due to DeMeyer [Some notes on the general Galois theory of rings, Osaka J. Math.2 (1965) 117–127], and Alfaro and Szeto [On Galois extensions of an Azumaya algebra, Commun. Algebra25 (1997) 1873–1882].

Duality for Groupoid (Co)Actions

In this article we present Cohen–Montgomery–type duality theorems for groupoid (co)actions.

GLOBALIZATION OF TWISTED PARTIAL HOPF ACTIONS

In this work, we review some properties of twisted partial actions of Hopf algebras on unital algebras and give necessary and sufficient conditions for a twisted partial action to have a globalization. We also elaborate a series of examples.