Algebra. Part 1

Annotation

The first part of the year-long Algebra course focuses on Lie groups and Lie algebras.

The concept of a Lie group incorporates two fundamental ideas of mathematics and natural science: continuity and symmetry. Lie groups provide symmetries of physical systems, both classical and quantum. The theory of Lie groups builds on a synthesis of global and infinitesimal approaches, where the latter one reduces many questions to studying linear structures such as Lie algebras.

The course introduces students to basic concepts of Lie theory including: Lie groups and Lie algebras, their homomorphisms, representations and actions on manifolds, homogeneous spaces, exponential map, connectedness and fundamental group, main classes of Lie groups and Lie algebras, etc. Basic results are proved, and many examples and constructions are considered with an emphasis put on those used in mathematical physics. In particular, Clifford algebras and spinor representations are discussed from the viewpoint of Lie theory.

Fundamental concepts and results of Lie theory developed in this course will be used in such courses as “Representations of finite- and infinite-dimensional Lie algebras”, physical courses on quantum field theory and general relativity, and various special courses.

Course plan

1. Lie groups: definition, examples, constructions. Lie subgroups of Lie groups.
2. Connected components of a Lie group, the identity component, the group of components. Generation of a connected Lie group by an arbitrarily small neighborhood of unity.
3. Invariant vector fields and invariant differential equations on Lie groups. The Lie algebra of a Lie group.
4. Exponential map and one-parameter subgroups. Relation between Lie subgroups and Lie subalgebras. Recovering a connected Lie subgroup from its Lie algebra. Intersection of Lie subgroups.
5. Lie group actions on manifolds. Velocity fields. Orbits and stabilizers, their properties. Transitive actions and quotient spaces.
6. Lie group homomorphisms and their differentials, interaction with exponential maps. Recovering a homomorphism of a connected Lie group from its differential. Kernels, images, and inverse images of Lie subgroups under homomorphisms of Lie groups. Relation between normal Lie subgroups of a Lie group and ideals of its Lie algebra. Quotient Lie groups.
7. Linear representations of Lie groups and Lie algebras. Isotropy representation. Adjoint representation. Basic representation-theoretic constructions: dual representation, direct sum, tensor product. Relation between properties of a Lie group representation and of its differential: invariant subspaces, irreducibility, complete reducibility.
8. Simply connected cover and fundamental group of a Lie group. Connected components and fundamental groups of classical linear Lie groups. An example of a non-linear Lie group. Isomorphisms between classical Lie algebras of small dimension.
9. Integration of homomorphisms of Lie algebras. Equivalence of categories of simply connected Lie groups and their Lie algebras.
10. Clifford algebra, spinor group, and spinors.
11. Commutator subgroup of a Lie group. Solvable Lie groups and Lie algebras. Theorems of Lie and Engel.
12. Semisimple Lie groups and Lie algebras: definition, basic properties, examples.

Literature

1. Helgason, S. Differential geometry, Lie groups, and symmetric spaces. Corrected reprint of the 1978 original. Graduate Studies in Mathematics, 34. American Mathematical Society, Providence, RI, 2001.
2. Onishchik, A. L., and Vinberg, E. B. Lie groups and algebraic groups. Springer Series in Soviet Mathematics, 93. Springer-Verlag, Berlin-Heidelberg, 1990.
3. Serre, J.-P. Lie algebras and Lie groups. 1964 lectures given at Harvard University. Corrected fifth printing of the second (1992) edition. Lecture Notes in Mathematics, 1500. Springer-Verlag, Berlin, 2006.
4. Warner, F. W. Foundations of differentiable manifolds and Lie groups. Corrected reprint of the 1971 edition. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983.