Recent developments around partial actions

  • M. Dokuchaev


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 action Partial representation Crossed product 

Mathematics Subject Classification

Primary 16W22 16S35 20C99 46L55 Secondary 08A02 13B05 16S10 16S36 16T05 16T15 16W50 20C15 20C25 20F10 20L05 20M18 20M25 20M30 20M50 22A22 46L05 54H15 



The author thanks Fernando Abadie and Mykola Khrypchenko for many useful comments.


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© Instituto de Matemática e Estatística da Universidade de São Paulo 2018

Authors and Affiliations

  1. 1.Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrazil

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