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
HTML articles powered by AMS MathViewer

by Richard E. Block PDF

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

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.

References

Richard E. Block, Commutative Hopf algebras, Lie coalgebras, and divided powers, J. Algebra 96 (1985), no. 1, 275–306. MR 808852, DOI 10.1016/0021-8693(85)90050-X
N. Bourbaki, Éléments de mathématique. Fasc. XXVI. Groupes et algèbres de Lie. Chapitre I: Algèbres de Lie, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1285, Hermann, Paris, 1971 (French). Seconde édition. MR 0271276
Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
Harish-Chandra, On representations of Lie algebras, Ann. of Math. (2) 50 (1949), 900–915. MR 30945, DOI 10.2307/1969586
G. Hochschild, Algebraic Lie algebras and representative functions, Illinois J. Math. 3 (1959), 499–523. MR 125910
G. Hochschild, An addition to Ado’s theorem, Proc. Amer. Math. Soc. 17 (1966), 531–533. MR 194482, DOI 10.1090/S0002-9939-1966-0194482-0
N. Jacobson, A note on Lie algebras of characteristic $p$, Amer. J. Math. 74 (1952), 357–359. MR 47026, DOI 10.2307/2372000
Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485