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The results on Lie homomorphisms of associative algebras are extended to certain associative superalgebras. It is shown that under appropriate conditions a Lie superautomorphism of $A=A_0oplus A_1$ is a sum of a superautomorphism or the negative of a superantiautomorphism and a central map. In particular we consider the situation when $A$ is a central simple algebra and its $mathbb Z_2$-grading is induced by an idempotent. [ABSTRACT FROM AUTHOR]

In this paper we describe all gradings by a finite abelian group $G$ on the following Lie algebras over an algebraically closed field $F$ of characteristic $pneq 2$: $mathfrak {sl}_n(F)$ ($n$ not divisible by $p$), $mathfrak {so}_n(F)$ ($ngeq 5$, $nneq 8$) and $mathfrak {sp}_n(F)$ ($ngeq 6$, $n$ even). [ABSTRACT FROM AUTHOR]

In this paper we find necessary and sufficient conditions on a finite-dimensional Lie superalgebra under which any associative PI-envelope of $L$ is finite-dimensional. We also extend M. Scheunert's result which enables one to pass from color Lie superalgebras to the ordinary ones, to the case of gradings by an arbitrary abelian group. [ABSTRACT FROM AUTHOR]