On (Α, Β, Γ)-Derivations OF LIE SUPERALGEBRAS

Abstract

This paper is primarily concerned with (α, β, γ)-derivations of finite-dimensional Lie superalgebras over the field of complex numbers. Some properties of (α, β, γ)-derivations of the Lie superalgebras are obtained. In particular, two examples for (α, β, γ)-derivations of low-dimensional non-simple Lie superalgebras are presented and the super-spaces of (α, β, γ)-derivations for simple Lie superalgebras are determined. Using certain complex parameters we generalize the concept of cohomology cocycles of Lie superalgebras. A special case for the generalization of 1-cocycles with respect to the adjoint representation is exactly (α, β, γ)-derivations. Furthermore, two-dimensional twisted cocycles of the adjoint representation are investigated in detail.

Generalized conformal derivations of Lie conformal algebras

“…It is known that d i = α i Id g , where α i ∈ C (see Theorem 13 in [17]). Therefore, D λ (a) = f (λ )a for any f (λ ) ∈ C[λ ].…”

“…Then, we consider the case when α = δ , β = γ = 1. Switching the place of a and b in (4.6), we get It is known from Theorem 13 in [17] that any (δ , 1, 1)-derivation of the Lie algebra g is zero when δ = 1 and δ = 2, and any (2, 1, 1)-derivation of g is equal to αId g for some complex number α. Therefore, by (4.14), we conclude that when δ = 1 and δ = 2, D λ = 0 and when δ = 2, can be easily obtained from (iii) and (iv).…”

“…In [14,17], (α, β , γ)-derivations of Lie algebras and Lie superalgebras were studied. Moreover, they also obtained some properties of (α, β , γ)-derivations of Lie (super)algebras.…”

Conformal $(\si,\t)$-Derivations on Lie conformal superalgebras

“…Lately, the generalized derivation theory of Lie conformal (super)algebras was developed in [4,9,10]. [8,11] studied the (α, β, γ)-derivations of Lie (super)algebras. [1] studied a kind of new generalized derivations of Lie algebras, that is the (σ, τ)-derivation theory of Lie algebras.…”

О $(n+1)$-арных дифференцированиях полупростых алгебр Филиппова

“…[11]). Понятие -дифференцирования допускает естественное обобщение до ( , , )-дифференци-рования, которое изучалось в работах [15], [16].…”