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\(\frac{1}{2}\)-derivations of Lie algebras and transposed Poisson algebras

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

A relation between \(\frac{1}{2}\)-derivations of Lie algebras and transposed Poisson algebras has been established. Some non-trivial transposed Poisson algebras with a certain Lie algebra (Witt algebra, the algebra \({\mathcal {W}}(a,-1)\), the thin Lie algebra and a solvable Lie algebra with abelian nilpotent radical) have been done. In particular, we have developed an example of the transposed Poisson algebra with associative and Lie parts isomorphic to the Laurent polynomials and the Witt algebra. On the other side, it has been proved that there are no non-trivial transposed Poisson algebras with a Lie algebra part isomorphic to a semisimple finite-dimensional algebra, a simple finite-dimensional superalgebra, the Virasoro algebra, \(N=1\) and \(N=2\) superconformal algebras, or a semisimple finite-dimensional n-Lie algebra.

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Authors and Affiliations

  1. Federal University of Technology, Guarapuava, Brazil

    Bruno Leonardo Macedo Ferreira

  2. CMCC, Universidade Federal do ABC, Santo André, Brazil

    Ivan Kaygorodov

  3. Saint Petersburg University, Saint Petersburg, Russia

    Viktor Lopatkin

Authors
  1. Bruno Leonardo Macedo Ferreira
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  2. Ivan Kaygorodov
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  3. Viktor Lopatkin
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Correspondence to Ivan Kaygorodov.

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The work is supported by the Russian Science Foundation under Grant 19-71-10016.

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Ferreira, B.L.M., Kaygorodov, I. & Lopatkin, V. \(\frac{1}{2}\)-derivations of Lie algebras and transposed Poisson algebras. RACSAM 115, 142 (2021). https://doi.org/10.1007/s13398-021-01088-2

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