Dixmier's lemma

Let $𝐴$ be a $𝕂$ -monoid over $𝕂$ and $𝑉$ be a simple $𝐴$ -module. If the cardinality $| 𝕂 | > d i m 𝕂 𝑉$ , then every $𝐴$ -module endomorphism $𝜗 \in 𝐴 𝖬 𝗈 𝖽 (𝑉, 𝑉)$ is an algebraic element over $𝕂$ .¹ #m/thm/module

Proof

By Schur's lemma, $𝐴 𝖬 𝗈 𝖽 (𝑉, 𝑉)$ is a division algebra over $𝕂$ . Suppose $𝜗 \in 𝐴 𝖬 𝗈 𝖽 (𝑉, 𝑉)$ is transcendental over $𝕂$ , i.e. $𝑝 (𝜗) = 0$ iff $𝑝 = 0$ for $𝑝 (𝑥) \in 𝕂 [𝑥]$ . The division algebra generated by $𝜗$ is then

𝕂 (𝜗) = {𝑝 (𝜗) 𝑞 (𝜗) : 𝑝 (𝑥), 𝑞 (𝑥) \in 𝕂 [𝑥], 𝑞 \neq 0} = {𝑓 (𝜗) : 𝑓 (𝑥) \in 𝕂 (𝑥)}

where $𝕂 (𝑥)$ is the field of rational functions for $𝕂$ , and we have a straightforward isomorphism of division $𝕂$ -algebras $𝕂 (𝑥) ≅ 𝕂 (𝜗)$ . By Lower bound on the dimension of the field of rational functions we have the inequality

| 𝕂 | \leq d i m 𝕂 𝕂 (𝑥) = d i m 𝕂 𝕂 (𝜗)

Since $𝑉$ is a vector space over $𝕂 (𝜗)$ with scalar multiplication given by the action of $𝜗$ , we have

d i m 𝕂 𝕂 (𝜗) \leq d i m 𝕂 𝑉

and thus

| 𝕂 | \leq d i m 𝕂 𝑉

a contradiction.

#state/tidy | #lang/en | #SemBr

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

Dixmier's lemma

Footnotes