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International Conference on Stochastic Processes and Algebraic Structures

SPAS 2017: Algebraic Structures and Applications pp 285–315Cite as

A Review on Hom-Gerstenhaber Algebras and Hom-Lie Algebroids

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 317))

Abstract

The aim of the present  article is to review the current progress on Hom-Gerstenhaber algebras and Hom-Lie algebroids. There are two different definitions of Hom-Lie algebroids. The modification was made in the original definition to consider some essential results on Hom-Lie algebroids and discuss some new examples. However, it turns out that for such results such a modification is not required. There are several attempts on defining representations and cohomology of Hom-Lie algebroids. As expected from the case of Hom-Lie algebras, there is no unique cohomology for Hom-Lie algebroids. We discuss about representations and cohomology of Hom-Lie algebroids that yields a differential calculus and dual description for Hom-Lie algebroids. Later on, we summarise the relationship between Hom-Lie algebroids and Hom-Gerstenhaber algebras.

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Acknowledgements

Satyendra Kumar Mishra is grateful to the research environment in Mathematics and Applied Mathematics (MAM), Division of Applied Mathematics at the School of Education, Culture and Communication at Mälardalen University, Västerås, Sweden for providing support and excellent research environment during his visit to Mälardalen University when part of the work on this paper has been performed.

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Authors and Affiliations

  1. Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore Centre, 8th Mile Mysore Road, Bangalore, 560059, India

    Satyendra Kumar Mishra

  2. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Box 883, 72123, Västerås, Sweden

    Sergei Silvestrov

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  1. Satyendra Kumar Mishra
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  2. Sergei Silvestrov
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Correspondence to Satyendra Kumar Mishra .

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Editors and Affiliations

  1. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Sergei Silvestrov

  2. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Anatoliy Malyarenko

  3. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Milica Rančić

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Mishra, S.K., Silvestrov, S. (2020). A Review on Hom-Gerstenhaber Algebras and Hom-Lie Algebroids. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds) Algebraic Structures and Applications. SPAS 2017. Springer Proceedings in Mathematics & Statistics, vol 317. Springer, Cham. https://doi.org/10.1007/978-3-030-41850-2_11

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