Superderivations of direct and semidirect sum of Lie superalgebras

Abstract

It is well known that superderivation of a Lie superalgebra is certain generalization of derivation of a Lie algebra. This paper is devoted to investigate the structure and dimension of superderivation algebra Der(G) of G where G is a direct sum of two finite dimensional Lie superalgebras L and K having no non-trivial common direct factor. We also introduce some of its subsuperalgebras. Moreover, we create a condition which shows the isomorphism between superderivation of direct sum and direct sum of superderivations of two Lie superalgebras. Later on, we take G as a semidirect sum of two Lie superalgebras and obtain the structure of $\mathit{Der}(G:K)$ which is a subsuperalgebra of Der(G) that contains those superderivations mapping K to itself. Finally, we give some conditions under which $\mathit{Der}(G:K)$ is also a semidirect sum.

Keywords:

• Cohomology of Lie superalgebra
• direct sum
• semidirect sum
• superderivation

2020 Mathematics Subject Classification:

• 17B05
• 17B40
• 17B56