On $\delta$-derivations of Lie algebras and superalgebras

Abstract

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.

v6: fixed minor mathematical typos; inessential changes in the ancillary file

References in corpus (6)

• $\delta$-derivations of simple Jordan algebras and superalgebras
• $\delta$-derivations of classical Lie superalgebras
• Commutative 2-cocycles on Lie algebras
• Low-dimensional cohomology of current Lie algebras and analogs of the Riemann tensor for loop manifolds
• Lie algebras of smooth sections
• Invariants of Lie algebras extended over commutative algebras without unit

Cited by in corpus (23)

• Generalized derivations of (color) n-ary algebras
• Transposed Poisson structures on Block Lie algebras and superalgebras
• Commutative 2-cocycles on Lie algebras
• Transposed Poisson structures on Witt type algebras
• $\delta$-superderivations of semisimple Jordan superalgebras
• The algebraic and geometric classification of transposed Poisson algebras
• Transposed Poisson structures on Galilean and solvable Lie algebras
• Generalized derivations of multiplicative $n$-ary Hom-$\Omega$ color algebras
• Transposed Poisson structures on generalized Witt algebras and Block Lie algebras
• (n+1)-Ary derivations os semisimple Filippov algebras
• Conservative algebras of 2-dimensional algebras
• δ-derivations of n-ary algebras
• Transposed Poisson structures
• A generalization on derivations of Lie algebras
• δ-derivations of semisimple structurable algebras
• Transposed Poisson structures on solvable and perfect Lie algebras
• $\delta$-superderivations of KKM Double
• Transposed Poisson structures on Lie incidence algebras
• Non-semigroup gradings of associative algebras
• Ternary superderivations of Jordan superalgebras
• $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras
• Generalized Kantor Double
• A $\delta$-first Whitehead Lemma