Schur’s lemma

Dixmier’s lemma

Let be a K-monoid over and be a simple -module. If the cardinality , then every -module endomorphism is an algebraic element over .¹ module

Proof

By Schur’s lemma, is a division algebra over . Suppose is transcendental over , i.e. iff for . The division algebra generated by is then

where is the field of rational functions for , and we have a straightforward isomorphism of division -algebras . By Lower bound on the dimension of the field of rational functions we have the inequality

Since is a vector space over with scalar multiplication given by the action of , we have

and thus

a contradiction.

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Footnotes

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