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# representation theory

# An indecomposable module

Suppose and let be a proper ideal containing all monomials of degree . Then is an indecomposable -module.

# A generalization of Schur’s lemma

(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 .

Brief outline:

- 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.

# Some representation theory

Find all representations over of the group of matrices

where and are in .