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Recent developments around partial actions

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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.

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Notes

  1. Notice that in [72] the authors defined a partial group action on an algebra assuming that each \(X_g\) is a right ideal generated by an idempotent.

  2. Such a global action with a natural additional assumption is called an enveloping action.

  3. The semigroup \({{\mathcal {S}}}(G)\) was denoted by E(G) in [129,130,131].

  4. The author thanks Fernando Abadie for drawing his attention to this fact.

  5. The authors of [69] use the term idempotent partial action, nevertheless we prefer to employ the latter name in a different sense.

  6. The semigroup \({{\mathcal {S}}}_3(G)\) was denoted by \(E_3(G)\) in [130, 131].

  7. In [137] A is additive and the zero element is denoted by \(\infty .\)

  8. In the case of a partial action on a semilattice, the corresponding semidirect product construction was essentially known much earlier without using the notion of a partial action (see, in particular, [249, 256]).

  9. In fact, it is assumed in [169] that \(\alpha \) is proper, i.e. each \(D_g\) is non-zero, however, later in Ferrero [170] observed that this condition is unnecessary for the proof of this result.

  10. Eric Jespers, Stanley Orlando Juriaans, Boris Novikov.

  11. The preprint [71] was a preliminary version of [72].

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Acknowledgements

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

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  1. Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, SP, 05508-090, Brazil

    M. Dokuchaev

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  1. M. Dokuchaev
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Correspondence to M. Dokuchaev.

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Dedicated to Antonio Paques on the occasion of his 70th birthday.

This work was partially supported by CNPq and Fapesp of Brazil.

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Dokuchaev, M. Recent developments around partial actions. São Paulo J. Math. Sci. 13, 195–247 (2019). https://doi.org/10.1007/s40863-018-0087-y

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