Algebra. Part 2

Annotation

The second part of the year-long Algebra course focuses on representations of finite-dimensional and infinite-dimensional Lie algebras.

Representations of Lie algebras provide an infinitesimal approach to the study of symmetries of various mathematical and physical systems. The classes of Lie algebras whose representations are of main importance in mathematics and theoretical physics include semisimple Lie algebras and their generalizations such as Kac–Moody algebras, loop and current algebras, Virasoro algebras etc.

The aim of the course is to introduce these main classes of Lie algebras arising in mathematics and theoretical physics with a focus on their representation theory. The core of the course is the theory of semisimple finite-dimensional Lie algebras and their representations. The constructions and results of this theory motivate infinite-dimensional generalizations such as Kac–Moody algebras, etc.

Representation theory of Lie algebras developed in this course is applied in many fields of mathematics and also will be used in quantum field theory courses as well as in various special courses.

Course plan

1. Reminder on linear representations of Lie groups and Lie algebras: basic definitions and constructions (differential of a Lie group representation, adjoint representation, dual representation, direct sum, tensor product), relation between properties of a Lie group representation and of its differential.
2. Universal enveloping algebra. The Poincaré–Birkhoff–Witt theorem. Representations of a Lie algebra and of its universal enveloping algebra.
3. Semisimple Lie algebras, Killing form, Casimir element. The Weyl theorem on complete reducibility of finite-dimensional representations of semisimple Lie algebras. Automorphisms and derivations of semisimple Lie algebras.
4. Representation theory of the Lie algebra sl2 and the Lie groups SL2, SU2 and SO3. Classification of irreducible finite-dimensional representations. Clebsch–Gordan formula for tensor product decompositions.
5. Cartan and Borel subalgebras of a semisimple complex Lie algebra, root decomposition. Root systems of classical Lie algebras, classification of semisimple Lie algebras via root systems.
6. Irreducible finite-dimensional representations of semisimple Lie algebras: the highest weight theory.
7. Representations of GLn and SLn in tensor spaces: Schur functors, Weyl modules, Young diagrams.
8. Generalized Cartan matrices. Kac–Moody algebras, triangular and root decompositions.
9. Real and imaginary roots. The Weyl group of a Kac–Moody algebra.
10. Representations of Kac–Moody algebras. The category O, highest weight modules and Verma modules. Integrable representations, their characters, complete reducibility.
11. Affine Kac–Moody algebras and loop algebras. Integrable representations and Macdonald identities.
12. Heisenberg algebras, oscillator algebras, and Virasoro algebras. Representations of these Lie algebras, vacuum vector and central charge.

Literature

1. Fulton, W., and Harris, J. Representation theory. A first course. Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991.
2. Kac, V. Infinite-dimensional Lie algebras. Third edition. Cambridge University Press, Cambridge, 1990
3. Onishchik, A. L., and Vinberg, E. B. Lie groups and algebraic groups. Springer Series in Soviet Mathematics, 93. Springer-Verlag, Berlin-Heidelberg, 1990.