# Lie Groups, Lie Algebras, and Their Representation

Weisstein, Eric W.

- Sommersemester 2018: Seminar on Lie groups, Lie algebras and their representations!
- Going to Live in Greece (How to).
- Lie Groups, Lie Algebras, and Their Representations | Veeravalli Seshadri Varadarajan | Springer.
- mathematics and statistics online.
- Canon Debate, The.
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- Communities and Technologies 2007: Proceedings of the Third Communities and Technologies Conference, Michigan State University 2007.

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Fundamental examples, Heisenberg algebras. Semi-simple Lie algebras.

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Categories of representations, irreducible representations. Complete reducibility.

Structure of semi-simple Lie algebras. Root systems, Weyl group.

## Lie Groups, Lie Algebras, and Representations

Representations of finite-dimensional Lie algebras. Highest weight modules, Verma modules, category O. Parametrization of simple modules. Jordan-Holder series, multiplicities.

## Lie Groups, Lie Algebras, and Their Representations

Finite-dimensional representations. To present a circle of ideas around Lie groups, Lie algebras and their representations, it is necessary to make a few principal choices. The first question is how to describe a relation between Lie groups and Lie algebras. To make the book accessible to a broader audience, the author does not suppose knowledge of theory of manifolds.

He restricts the attention to matrix groups. Lie algebra of G is then defined using simple properties of the exponential map. As for the correspondence between Lie group homomorphisms and Lie algebra homomorphisms, the author is using the Baker-Campbell-Hausdorff theorem for its description.