Semisimple representation

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Covers the structure of finite dimensional algebras and the characterization of semisimple algebras.

Covers the properties of finite dimensional algebras and non-semisimple representations.

Explores Schur's lemma and its applications in representations of an associative algebra over an algebraically closed field.

Discusses linearly reductive groups and their properties, focusing on completely reducible representations and equivalent modules.

Covers the properties of the coordinate ring of a linearly reductive group.

Introduces the regular representation, a key tool for studying group actions on varieties.

Explores the decomposition of the circle of the coordinate ring of a G variety into a direct sum of simple submodules.

Explores the embedding of chi-varieties into vector spaces with linear actions.

Explores simple models, electronic structure evaluation, and machine learning at the atomic scale.

Explores weight space decomposition and proves that a connected linear algebraic group with all semi-simple elements must be a torus.