Recent developments around partial actions

Abstract

We give an overview of publications on partial actions and related concepts, paying main attention to some recent developments on diverse aspects of the theory, such as partial actions of semigroups, of Hopf algebras and groupoids, the globalization problem for partial actions, Morita theory of partial actions, twisted partial actions, partial projective representations and the Schur multiplier, cohomology theories related to partial actions, Galois theoretic results, ring theoretic properties and ideals of partial crossed products. Among the applications we consider in more detail the case of the Carlsen-Matsumoto C *-algebra related to an arbitrary subshift, but also mention many others. The total number of publications directly related to partial actions and partial representations is more than 130, so that it is impossible even to describe briefly the content of all of them within the constraints of the present survey. Thus, the majority of them are only cited with respects to specific topics, trying to give an idea about the involved matter. In order to complete the picture, we refer the reader to a recent book by Ruy Exel, to our previous surveys, as well as to those by other authors.

Partial Galois cohomology and related homomorphisms

“…The latter are becoming an object of intensive research and have their origins in the theory of operator algebras, where they, together with the corresponding crossed products and partial representations, form the essential ingredients of a new and successful method to study C * -algebras generated by partial isometrics, initiated by R. Exel in [25], [26], [27], [28] and [29]. The first algebraic results on these new concepts, established in [29], [18], [44], [45], [34] and [14], and the development of a Galois theory of partial actions in [19], stimulated a growing algebraic activity around partial actions (see the surveys [13] and [31]). In particular, partial Galois theoretic results have been obtained in [3], [6], [7], [32], [38], [42], and applications of partial actions were found to graded algebras in [14] and [17], to tiling semigroups in [35], to Hecke algebras in [30], to automata theory in [24], to restriction semigroups in [10] and [37] and to Leavitt path algebras in [33].…”

Simplicity and Chain Conditions for Ultragraph Leavitt Path Algebras via Partial Skew Group Ring Theory

“…The theory of partial skew group rings has been in constant development recently; see, for example, [5,7], where simplicity criteria are described, [14], where chain conditions are studied, and [4] (and the 283 references therein cited), where most of the recent developments in the theory are compiled. In our case we use partial skew ring theory to characterize artinian ultragraph path algebras and give simplicity criteria for these algebras.…”

Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics

“…The theory of partial skew group rings, which is still quite young, is less developed than its analytical counterpart, but it has evolved quickly in recent years. In particular many important algebras, such as Leavitt path algebras [27] and ultragraph Leavitt path algebras [30], can be realized as partial skew group rings and general results about the theory, as the ones in [3,17,24,25,32], have been applied to study these algebras (see [15] for a comprehensive overview of developments in the theory of partial actions). This recent development of the area indicates that the theory of non-commutative rings may benefit from the theory of partial skew group rings.…”