A sandwich in thin lie algebras

A thin Lie algebra is a Lie algebra $L$ , graded over the positive integers, with its first homogeneous component $L_1$ of dimension two and generating $L$ , and such that each non-zero ideal of $L$ lies between consecutive terms of its lower central series. All homogeneous components of a thin Lie algebra have dimension one or two, and the two-dimensional components are called diamonds. Suppose the second diamond of $L$ (that is, the next diamond past $L_1$ ) occurs in degree $k$ . We prove that if $k>5$ , then $[Lyy]=0$ for some non-zero element $y$ of $L_1$ . In characteristic different from two this means $y$ is a sandwich element of $L$ . We discuss the relevance of this fact in connection with an important theorem of Premet on sandwich elements in modular Lie algebras.

Restricted Envelopes of Lie Superalgebras

Certain important results concerning p-envelopes of modular Lie algebras are generalized to the super-case. In particular, any p-envelope of the Lie algebra of a Lie superalgebra can be naturally extended to a restricted envelope of the Lie superalgebra. As an application, a theorem on the representations of Lie superalgebras is given, which is a super-version of Iwasawa's theorem in Lie algebra case. As an example, the minimal restricted envelopes are computed for three series of modular Lie superalgebras of Cartan type.