Schur’s lemma

Quillen’s lemma

Let 𝐴 be a K-monoid over 𝕂 and 𝑉 be a simple 𝐴-module. If 𝐴 has a filtration {𝐹𝑖𝐴}∞𝑖=1 such that 1 ∈𝐹0𝐴 and the associated graded algebra is a finitely-generated commutative 𝕂-algebra, then every 𝐴-module endomorphism 𝜗 ∈𝐴𝖬𝗈𝖽(𝑉,𝑉) is an algebraic element over 𝕂.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 𝐺∙𝑈(𝔤) =𝑆∙𝔤.

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Footnotes

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