Residual finiteness of color Lie superalgebras

Abstract:

A (color) Lie superalgebra $L$ over a field $K$ of characteristic $\ne 2, 3$ is called residually finite if any of its nonzero elements remains nonzero in a finite-dimensional homomorphic image of $L$. In what follows we are looking for necessary and sufficient conditions under which all finitely generated Lie superalgebras satisfying a fixed system of identical relations are residually finite. In the case $\operatorname {char}\;K = 0$ we show that a variety $V$ satisfies this property if and only if $V$ does not contain all center-by-metabelian algebras and every finitely generated algebra of $V$ has nilpotent commutator subalgebra.