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arXiv · 0907.2034

On $δ$-derivations of Lie algebras and superalgebras

Pasha Zusmanovich
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Abstract

We study $δ$-derivations -- a construction simultaneously generalizing derivations and centroid. First, we compute $δ$-derivations of current Lie algebras and of modular Zassenhaus algebra. This enables us to provide examples of Lie algebras having 1/2-derivations which are divisors of zero, thus answering negatively a question of Filippov. Second, we note that $δ$-derivations allow, in some circumstances, to construct examples of non-semigroup gradings of Lie algebras, in addition to the recent ones discovered by Elduque. Third, we note that utilizing the construction of the Grassmann envelope allows to obtain results about $δ$-(super)derivations of Lie superalgebras from the corresponding results about Lie algebras. In this way, we prove that prime Lie superalgebras do not possess nontrivial $δ$-(super)derivations, generalizing the recent result of Kaygorodov.

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Pasha Zusmanovich. 2019-07-08. On $δ$-derivations of Lie algebras and superalgebras. https://doi.org/10.1016/j.jalgebra.2010.09.032%2010.1016%2Fj.jalgebra.2013.08.036

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