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The Twisted Partial Group Algebra and (Co)homology of Partial Crossed Products

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Abstract

Given a group G and a partial factor set \(\sigma \) of G,  we introduce the twisted partial group algebra \({\kappa }_{\textrm{par}}^\sigma G,\) which governs the partial projective \(\sigma \)-representations of G into algebras over a field \(\kappa .\) Using the relation between partial projective representations and twisted partial actions we endow \({\kappa }_{\textrm{par}}^\sigma G\) with the structure of a crossed product by a twisted partial action of G on a commutative subalgebra of \({\kappa }_{\textrm{par}}^\sigma G.\) Then, we use twisted partial group algebras to obtain a first quadrant Grothendieck spectral sequence converging to the Hochschild homology of the crossed product \(A*_{\Theta } G,\) involving the Hochschild homology of A and the partial homology of G,  where \({\Theta }\) is a unital twisted partial action of G on a \(\kappa \)-algebra A with a \(\kappa \)-based twist. An analogous third quadrant cohomological spectral sequence is also obtained.

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Notes

  1. Notice that the equality \(D_g=0\) is not prohibited.

  2. Notice that if G has no elements of order 2 then \(\sigma ' = \sigma ''\), and in this case \(\xi (g)=1\) for all \(g\in G\).

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Acknowledgements

We thank the referee for the useful comments. The first named author was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (Fapesp), process no: 2020/16594-0, and by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), process no: 312683/2021-9. The second named author was supported by Fapesp, process no: 2022/12963-7.

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  1. Departamento de Matemática, Universidade de São Paulo, Rua do Matão, 1010, São Paulo, São Paulo, 05508-090, Brazil

    Mikhailo Dokuchaev & Emmanuel Jerez

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  1. Mikhailo Dokuchaev
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  2. Emmanuel Jerez
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Correspondence to Emmanuel Jerez.

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Dokuchaev, M., Jerez, E. The Twisted Partial Group Algebra and (Co)homology of Partial Crossed Products. Bull Braz Math Soc, New Series 55, 33 (2024). https://doi.org/10.1007/s00574-024-00408-5

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