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.

Recent developments around partial actions

Weak Hopf algebras and weak Yang–Baxter operators

In this paper we introduce the notion of weak Hopf quasigroup as a generalization of weak Hopf algebras and Hopf quasigroups. We obtain its main properties and we prove the fundamental theorem of Hopf modules for these algebraic structures.

https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/2016/0020/0004/AJM-2016-0020-0004-a004.pdf

Weak Hopf Quasigroups

If $\mathcal{C}$
 is a strict braided monoidal category in which the idempotent morphisms have a canonical split, we find conditions under which the antipode of a Hopf (co)quasigroup is an isomorphism. We also prove that the object of (co)invariants of a finite Hopf (co)quasigroup in $\mathcal{C}$
 is an invertible object.

The Antipode and the (co)invariants of a Finite Hopf (co)quasigroup

For functors $L:\A\to \B$ and $R:\B\to \A$ between any categories $\A$ and $\B$, a {\em pairing} is defined by maps, natural in $A\in \A$ and $B\in \B$, $$\xymatrix{\Mor_\B (L(A),B) \ar@ [r]^{\alpha} & \Mor_\A (A,R(B))\ar@ [l]^{\beta}}.$$ 
$(L,R)$ is an {\em adjoint pair} provided $\alpha$ (or $\beta$) is a bijection. In this case the composition $RL$ defines a monad on the category $\A$, $LR$ defines a comonad on the category $\B$, and there is a well-known correspondence between monads (or comonads) and adjoint pairs of functors. 
For various applications it was observed that the conditions for a unit of a monad was too restrictive and weakening it still allowed for a useful generalised notion of a monad. This led to the introduction of {\em weak monads} and {\em weak comonads} and the definitions needed were made without referring to this kind of adjunction. The motivation for the present paper is to show that these notions can be naturally derived from pairings of functors $(L,R,\alpha,\beta)$ with $\alpha = \alpha\dcirc \beta\dcirc \alpha$ and $\beta = \beta \dcirc\alpha\dcirc\beta$. Following closely the constructions known for monads (and unital modules) and comonads (and counital comodules), we show that any weak (co)monad on $\A$ gives rise to a regular pairing between $\A$ and the category of {\em compatible (co)modules}.

/pdf/regular-pairings-of-functors-and-weak-co-monads-1a9m6x8y4y.pdf

Regular pairings of functors and weak (co)monads

The authors present the general theory of cleft extensions for a cocommutative weak Hopf algebra H. For a right H-comodule algebra, they obtain a bijective correspondence between the isomorphisms classes of H-cleft extensions A
 H
 ↪ A, where A
 H
 is the subalgebra of coinvariants, and the equivalence classes of crossed systems for H over A
 H
 . Finally, they establish a bijection between the set of equivalence classes of crossed systems with a fixed weak H-module algebra structure and the second cohomology group $H_{\phi _{Z(A_H )} }^2 $
 (H,Z(A
 H
 )), where Z(A
 H
 ) is the center of A
 H
 .

Crossed products over weak Hopf algebras related to cleft extensions and cohomology

In this paper, we present the general theory and universal properties of weak crossed biproducts. We prove that every weak projection of weak bialgebras induces one of these weak crossed structures. Finally, we compute explicitly the weak crossed biproduct associated with a groupoid that admits an exact factorization.

Weak crossed biproducts and weak projections

A Picard-Brauer Five Term Exact Sequence for Braided Categories

In this paper we study the projections of weak braided Hopf algebras using the notion of Yetter-Drinfeld module associated with a weak braided Hopf algebra. As a consequence, we complete the study of the structure of weak Hopf algebras with a projection in a braiding setting obtaining a categorical equivalence between the category of weak Hopf algebra projections associated with a weak Hopf algebra H living in a braided monoidal category and the category of Hopf algebras in the non-strict braided monoidal category of left-left Yetter-Drinfeld modules over H.

http://www.mat.uc.pt/~workCT/abstracts/Soneira_Calvo.pdf

Projections of weak braided Hopf algebras

In this paper, for a cocommutative Hopf algebra H in a symmetric closed category C with basic object K, we get an isomorphism between the group of isomorphism classes of Galois H-objects with a normal basis and the second cohomology group H2(H,K) of H with coefficients in K. Using this result, we obtain a direct sum decomposition for the Brauer group of H-module Azumaya monoids with inner action:
BMinn(C,H) @ B(C) A H2(H,K)
In particular, if C is the symmetric closed category of C-modules with K a field, H2(H,K) is the second cohomology group introduced by Sweedler in [21]. Moreover, if H is a finitely generated projective, commutative and cocommutative Hopf algebra over a commutative ring with unit K, then the above decomposition theorem is the one obtained by Beattie [5] for the Brauer group of H-module algebras.

/pdf/galois-h-objects-with-a-normal-basis-in-closed-categories-a-18qkmhioeq.pdf

José Manuel Fernández Vilaboa

Papers

Crossed products over weak Hopf algebras related to cleft extensions and cohomology

Weak crossed biproducts and weak projections

A Picard-Brauer Five Term Exact Sequence for Braided Categories

Projections of weak braided Hopf algebras

Galois H-objects with a normal basis in closed categories: a cohomological interpretation