Lecture Notes
General definitions
- Lecture 01
(08/26/2026) Definitions of a Lie algebra, subalgebra, ideal,
homomorphism of Lie algebras, representations. Examples:
gl(n), Der(algebra), Lie(associative algebra).
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Lecture 02
(08/28/2026) Direct and semidirect products. Operations on
representations: direct sum, tensor product, dual, Hom.
Irreducible and indecomposable representations.
-
Lecture 03
(08/31/2026) Schur's lemma. Examples. Universal enveloping algebra
of a Lie algebra. Definition of a Hopf algebra.
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Lecture 04
(09/02/2026)
Hopf algebras. Examples of a group algebra, tensor algebra,
universal enveloping algebra.
An excellent reference for Hopf algebras by P. Cartier
-
Lecture 05
(09/04/2026)
Filtrations and their associated graded. Canonical filtration
on the enveloping algebra. Poincare-Birkhoff-Witt theorem
(statement only).
A proof of the PBW theorem
-
Lecture 06
(09/09/2026)
Corollaries of the PBW theorem. Free Lie algebras.
Presentation by generators and relations.
Solvable, nilpotent, ... Lie algebras
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Lecture 07
(09/11/2026)
Solvable and nilpotent Lie algebras. Lie's theorem.
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Lecture 08
(09/14/2026)
Corollaries of Lie's theorem. Engel's theorem.
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Lecture 09
(09/16/2026)
Radical ideal. Simple, semisimple and reductive Lie algebras.
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Lecture 10
(09/18/2026)
Invariant forms. Forms associated to finite-dimensional representations.
Killing form. Non-degeneracy implies reductivity.
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Lecture 11
(09/21/2026)
Jordan-Chevalley decomposition theorem. Cartan's criterion
of solvability.
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Lecture 12
(09/23/2026)
Cartan's criterion of semisimplicity. Semisimplicity of the
adjoint representation of semisimple Lie algebras.
-
Lecture 13
(09/25/2026)
Applications of Cartan's criterion: all derivations are inner,
Jordan-Chevalley decomposition for semisimple Lie algebras.
-
Lecture 14
(09/28/2026)
Casimir element and Casimir tensor. Eigenvalues
of the Casimir element.
Complete reducibility for semisimple
Lie algebras.
-
Lecture 15
(09/30/2026)
First Lie algebra cohomology space. Relation with
extensions.
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Lecture 16
(10/02/2026)
Vanishing of the first cohomology for semisimple
Lie algebas (Whitehead's lemma). Weyl's complete
reducibility theorem. Jordan-Chevalley decomposition
for representations.
Representation theory of sl(2).
-
Lecture 17
(10/05/2026)
Classification of finite-dimensional, irreducible
representations of sl(2).
-
Lecture 18
(10/07/2026)
Weights, multiplicities and Weyl symmetry.
Properties of exponential.
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Lecture 19
(10/09/2026)
Weyl group action. Direct proof of Weyl symmetry.
Completion of U(sl(2)).
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Lecture 20
(10/12/2026)
Characters. Clebsch-Gordon rule. Category O and Verma
modules.
Cartan-Killing theorem.
-
Lecture 21
(10/19/2026)
Overview of the structure theorem of E. Cartan and W. Killing.
Example of sl(n+1).
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Lecture 22
(10/21/2026)
Existence of Cartan subalgebras. From (g,h) to root system.
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Lecture 23
(10/23/2026)
Real form of Cartan. Positivity of Killing form.
Root systems. Hyperplane arrangement and wall-chamber
structure. Base of a root system.
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Lecture 24
(10/26/2026)
Rays of cones vs walls of chambers. Cartan matrix.
Weyl group: generation by simple reflections; transitivity
of its action on the set of chambers. Rank 2 case.
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Lecture 25
(10/28/2026)
Cartan matrices (finite type) and their Dynkin diagrams. Classification
of Dynkin diagrams associated to Cartan matrices.
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Lecture 26
(10/30/2026)
Simple reflections. Presentation of the Weyl group. W-action on
the set of chambers is free and transitive.
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Lecture 27
(11/02,04/2026)
Uniqueness of Cartan subalgebras. Cartan subalgebras are centralizers of
generic elements.