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

  • Satyendra Kumar MishraEmail author
  • Sergei Silvestrov
Conference paper
  • 41 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (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.

Keywords

Hom-Gerstenhaber algebras Hom-Lie algebroids Hom-Lie algebras 

MSC 2010 Classification

17B75 17B99 17B35 17B37 17D99 16W10 16W55 

Notes

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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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Theoretical Statistics and Mathematics UnitIndian Statistical Institute, Bangalore CentreBangaloreIndia
  2. 2.Division of Applied MathematicsSchool of Education, Culture and Communication, Mälardalen UniversityVästeråsSweden

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