On $δ$-derivations of Lie algebras and superalgebras

Details On $δ$-derivations of Lie algebras and superalgebras On $δ$-derivations of Lie algebras and superalgebras

2009-07-12

arxiv.org/abs/0907.2034v3

J. Algebra 324 (2010), 3470-3486

Mathematics

Rings and Algebras

15 pages; v3: English, TeX and bibliographic corrections

Scientific paper

10.1016/j.jalgebra.2010.09.032

We study $\delta$-derivations -- a construction simultaneously generalizing derivations and centroid. First, we compute $\delta$-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 $\delta$-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 $\delta$-(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 $\delta$-(super)derivations, generalizing the recent result of Kaygorodov.

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