Completely Reducible Representations

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Kirillov Paradigm for Heisenberg Group

Explores the Kirillov paradigm for the Heisenberg group and unitary representations.

G-modules and G-varieties

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

Group representations constructions

Explores the construction of group representations through various methods and provides an illustrative example using the standard representation of sr2 on c2.

Group Cohomology

Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.

Representation of Groups

Explores the theory of representation of groups and the concept of complete reducibility.

Lie Groups and Lorentz Transformations

Covers Lie groups, Lorentz transformations, boosts, rotations, and complexified Lie algebras.

Tensor Products of Modules

Covers tensor algebras, symmetric and exterior algebras, and tensor products of modules.

Free Resolution of Modules

Covers free resolution of modules and projective modules in module theory.

Rings and Modules: Lecture 1

Covers rings, modules, exact sequences, and finitely generated modules in the context of Noetherian and Artinian properties.

Connected groups of semisimple elements

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