This page will be updated at least once a week. Please check it regularly for announcements.

Announcement: The second homework is posted.

Pictures are taken from Wikipedia.

Classes: MWF 10:40-11:35AM Physics P129 Instructor: Valentina Kiritchenko e-mail: vkiritch at math dot sunysb dot edu Office: Math 3-102 Phone: 632-8884 Office Hours: M 4-5 pm, W 4-6 pm at Math 3-102 and M 5-6 at MLC

Lie groups handout Lie algebras handout Week by week details Last homework

About this Course: We cover the basic theory of Lie groups and algebras together with the representation theory for classical Lie groups (such as the special linear, orthogonal and symplectic groups). Students are expected to be familiar with most of the material of MAT 530, MAT 531 (Geometry/Topology I-II) and MAT 534, MAT 535 (Algebra I-II).

• A.L. Onishchik, E.B. Vinberg; Lie groups and algebraic groups; Springer, 1990,
• W. Fulton and J. Harris; Representation theory: a first course; Springer, 1999.

Recommended reading:

• F. Adams; Lectures on Lie groups; Univ of Chicago Press, 1983.

September 6 Definition of a Lie group, direct products. Real manifolds admitting the structure of a Lie group: of dimension one (R^1, T^1) and compact of dimension two (only T^2). Left and right translations of a Lie group. Topological properties of Lie groups: the tangent bundle of a Lie group is trivial, compact Lie groups have Euler characteristic zero.	September 8 Lie subgroups. When a cyclic subgroup of T^1 is a Lie group. A subgroup of a Lie group is smooth at all points if it is smooth at the identity. Classical groups.
September 11 Proof that classical groups are smooth manifolds. Their dimensions and tangent spaces at the identity. Diffeomorphisms SO_2(R)=S^1, SO_(1,1)(R)=R*, SU_2(C)=S^3. Isomorphisms between low-dimensional classical groups (without proof).	September 13 Homomorphisms and actions. The image of a Lie group under a homomorphism is not always a Lie group (e.g. take irrational winding of a torus). Orbits and stabilizers. A stabilizer is a Lie subgroup.
September 15 Proof that a stabilizer is a Lie subgroup. Classical groups as stabilizers of GL_n-actions. Coset spaces G/H. Smooth structure on coset spaces.	September 18 The intersections and preimages under a homomorphism of Lie subgroups are Lie subgroups. Homogeneous spaces. Computation of the fundamental group of SO_n(R) using the long exact sequence for fiber bundles. Grassmannians.
September 20 The cell decomposition for projective spaces and Grassmannians. Homology groups of RP^n, CP^n and G(2,4). Definition of a Lie algebra. Jacobi identity as a Leibnitz rule. Homework 1: due Friday September 29.	September 22 Examples of Lie algebras. Lie algebras of classical groups. Adjoint action Ad of a Lie group on the tangent space at the identity.

September 25 Three ways to define the Lie bracket on the tangent space T_eG of a Lie group G: by using the commutator in G, by using the adjoint representation and by identifying T_eG with the space of left invariant vector fields on G.	September 27 The differential of a Lie group homomorphism is a homomorphism of respective Lie algebras. The differential ad of the adjoint map Ad is a Lie algebra representation. Exponential map for linear Lie groups. Properties of the matrix exponent.
September 29 Discussion of the first homework. Gram-Shmidt decomposition and polar decomposition of GL_n. Corollary: GL_n(R), SL_n(R), GL_n(C), SL_n(C) are homotopy equivalent to O_n(R), SO_n(R), U_n(C), SU_n(C), respectively.	October 4 Surjectivity of the exponential map for GL_n(C). Exponential map and one-parameter subgroups. One-parameter subgroups as integral curves for left (or right)-invariant vector fields.
October 6 Exponential map for arbitrary Lie groups. Properties of the exponential map. It is a local diffeomorphism at 0, commutes with homomorphisms, in particular, with the adjoint map and if [X,Y]=0, then exp(X)exp(Y)=exp(X+Y).	October 9 Classification of connected Abelian Lie groups. Existence and uniqueness theorem for a homomorphism of Lie groups with a given homomorphism of Lie algebras.
October 11 Correspondence between connected simply connected Lie groups and Lie algebras. Representation theory. Complete reducibility of representations of compact groups.	October 18 Weyl's unitary trick. Real forms and complexifications. Reductive Lie algebras. Compact real forms. Compact real forms of complex classical Lie algebras.

October 20 Complete reducibility of representations of reductive groups. Criterion for a compact Lie algebra. Adjoint invariant definite inner product on compact Lie algebras.	October 23 Automorphisms of Lie groups and algebras. Derivations. Invariant inner products on the Lie algebras of SO_n(R) and U_n(C). Isomophisms between SU_2(C)/{I,-I} and SO_3(R) and between SL_2(R)/{I,-I} and SO_{2,1}(R).
October 25 Classification of irreducible representations of SL_2(C).	October 27 Classification of irreducible representations of SL_2(C) continued. Homework 2: due Friday, November 10.
October 30 Representation of SL_2(C) on the space of polynomials in two variables. Decomposition into irreducible representations.	November 1 Solvable Lie groups and algebras. Lie theorem.
November 3 Killing form. Examples. Cartan criterion.	November 6 Properties and applications of the Killing form. Decomposition of a semisimple Lie algebra into simple Lie algebras. All derivations of a semisimple Lie algebra are inner. Existence of a Lie group with a given semisimple Lie algebra.