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

Block, Richard E.

doi:10.1090/S0002-9939-1986-0857931-0

On nilpotency in the Ado-Harish-Chandra theorem on Lie algebra representations
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by Richard E. Block

Proc. Amer. Math. Soc. 98 (1986), 406-410

DOI: https://doi.org/10.1090/S0002-9939-1986-0857931-0

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Abstract:

Let $L$ be a finite-dimensional Lie algebra, over an arbitrary field, and regard $L$ as embedded in its enveloping algebra UL. Theorem. If $K$ is an ideal of $L$ and $K$ is nilpotent of class $q$, then for any $r$ there exists a finite-dimensional representation $\rho$ of $L$ which vanishes on all products (in UL) of $\geq qr + 1$ elements of $K$ and is faithful on the subspace of UL spanned by all products of $\leq r$ elements of $L$. 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.

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