Schur’s lemma

Dixmier’s lemma

Let 𝐴 be a K-monoid over 𝕂 and 𝑉 be a simple 𝐴-module. If the cardinality |𝕂| >dim𝕂⁡𝑉, then every 𝐴-module endomorphism 𝜗 ∈𝐴𝖬𝗈𝖽(𝑉,𝑉) is an algebraic element over 𝕂.¹ module

Proof

By Schur’s lemma, 𝐴𝖬𝗈𝖽(𝑉,𝑉) is a division algebra over 𝕂. Suppose 𝜗 ∈𝐴𝖬𝗈𝖽(𝑉,𝑉) is transcendental over 𝕂, i.e. 𝑝(𝜗) =0 iff 𝑝 =0 for 𝑝(𝑥) ∈𝕂[𝑥]. The division algebra generated by 𝜗 is then

𝕂(𝜗)={𝑝(𝜗)𝑞(𝜗):𝑝(𝑥),𝑞(𝑥)∈𝕂[𝑥],𝑞≠0}={𝑓(𝜗):𝑓(𝑥)∈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|≤dim𝕂⁡𝕂(𝑥)=dim𝕂⁡𝕂(𝜗)

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

dim𝕂⁡𝕂(𝜗)≤dim𝕂⁡𝑉

and thus

|𝕂|≤dim𝕂⁡𝑉

a contradiction.

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Footnotes

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