On nilpotency in the Ado-Harish-Chandra theorem on Lie algebra representations

Abstract

Let L L be a finite-dimensional Lie algebra, over an arbitrary field, and regard L L as embedded in its enveloping algebra UL. Theorem. If K K is an ideal of L L and K K is nilpotent of class q q , then for any r r there exists a finite-dimensional representation ρ \rho of L L which vanishes on all products (in UL) of ≥ q r + 1 \geq qr + 1 elements of K K and is faithful on the subspace of UL spanned by all products of ≤ r \leq r elements of L L . This result sharpens (with respect to nilpotency) the Ado-Iwasawa theorem on the existence of faithful representations and the Harish-Chandra theorem on the existence of representations separating points of UL.