Suppose and let be a proper ideal containing all monomials of degree . Then is an indecomposable -module.
(Dixmier) Let be a -algebra (in fact, any uncountable algebraically closed field will do) and be a simple -module having a countable basis as a -vector space. Then .
- By Schur’s lemma, is a division algebra. Moreover, it has a countable basis.
- Suppose does not act as a scalar. Then is trascendental over .
- This is absurd since it would imply has uncountable dimension.
Find all representations over of the group of matrices
where and are in .