The dimension subalgebra problem for enveloping algebras of Lie superalgebras

Abstract:

Let L be an arbitrary Lie superalgebra over a field of characteristic different from 2. Denote by $\omega u(L)$ the ideal generated by L in its universal enveloping algebra $U(L)$. It is shown that $L \cap \omega u{(L)^n} = {\gamma _n}(L)$ for each $n \geq 1$, where ${\gamma _n}(L)$ is the nth term of the lower central series of L. We also prove that $\omega u(L)$ is a residually nilpotent ideal if and only if L is residually nilpotent. Both these results remain true in characteristic 2 provided we take L to be an ordinary Lie algebra.