Schur’s lemma

Quillen’s lemma

Let be a K-monoid over and be a simple -module. If has a filtration such that and the associated graded algebra is a finitely-generated commutative -algebra, then every -module endomorphism is an algebraic element over .¹ #m/thm/module

Proof

proof

Corollaries

The following algebras fulfil the hypothesis:

1. The Universal enveloping algebra of a finite-dimensional Lie algebra, since by Poincaré-Birkhoff-Witt theorem .

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Footnotes

1. 1969. On the endomorphism ring of a simple module over an enveloping algebra, p. 171 ↩