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 ω u ( L ) \omega u(L) the ideal generated by L in its universal enveloping algebra U ( L ) U(L) . It is shown that L ∩ ω u ( L ) n = γ n ( L ) L \cap \omega u{(L)^n} = {\gamma _n}(L) for each n ≥ 1 n \geq 1 , where γ n ( L ) {\gamma _n}(L) is the nth term of the lower central series of L. We also prove that ω u ( L ) \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.