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Recent developments around partial actions. (English) Zbl 1412.16037

Summary: 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 R. Exel [Partial dynamical systems, Fell bundles and applications. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1405.46003)]l, to our previous surveys, as well as to those by other authors.

MSC:

16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
16T05 Hopf algebras and their applications
08A02 Relational systems, laws of composition
16S35 Twisted and skew group rings, crossed products
20C25 Projective representations and multipliers
46L55 Noncommutative dynamical systems
16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras

Citations:

Zbl 1405.46003
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References:

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