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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Richard E. Block
Journal: Proc. Amer. Math. Soc. 98 (1986), 406-410
MSC: Primary 17B10; Secondary 17B30
MathSciNet review: 857931
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0857931-0
PII: S 0002-9939(1986)0857931-0
Keywords: Lie algebra, enveloping algebra, nilpotent ideal, nilpotency class, faithful representation
Article copyright: © Copyright 1986 American Mathematical Society