LIE ALGEBRAS

MATH 747

Course Description

Lie algebras and matrix groups. Topics: tangent spaces; exponentials; Baker-Campbell-Hausdorff formula; (nilpotent, solvable, semisimple) Lie algebras; Engel's and Lie's theorems; Levi decomposition; Killing form; sl(2)-representations; root systems; Dynkin diagrams; Weyl groups; Cartan and Borel subalgebras; Serre's theorem. Basic graduate algebra courses (e.g.MATH 541and542orMATH 741and742) are strongly recommended.

Prerequisties

Graduate/professional standing or member of the Pre-Masters Mathematics (Visiting International) Program

Satisfies

This course does not satisfy any prerequisites.

Credits

Not Reported

Offered

Not Reported

Grade Point Average

No change from Historical

Completion Rate

100%

No change from Historical

A Rate

100%

No change from Historical

Class Size

84.68% from Historical

Instructors (2025 Fall)

Sorted by ratings from Rate My Professors

Similar Courses

LINEAR ALGEBRA

MATH 341

The theory of linear algebra with an introduction to proofs and proof writing. Topics include vector spaces, linear dependence, span, basis, linear transformations, kernel, image, inner products and inner product spaces, geometry, eigenvalues, eigenvectors, standard matrix factorizations. Other content includes basic set theory, logical operations, quantifiers, direct and indirect arguments, and induction. Differential equations [MATH 320] not covered.

REPRESENTATION THEORY

MATH 843

Introduction to the representation theory of Lie groups and their combinatorics. Universal enveloping algebras, highest weight modules, induction, restriction, weights, characters, multiplicity formulas, tensor products, Shapovalov forms, filtrations, Kazhdan-Zetlin patterns, Littelmann paths.

TOPICS IN ALGEBRA

MATH 847

Topics may include: Lie groups, algebraic groups, Chevalley groups, simple groups and associated geometries, group cohomology, group rings, Hopf algebras, enveloping algebras, quantum groups, infinite-dimensional Lie algebras, Hecke algebras, automorphic forms, Galois representations, zeta and L-functions, abelian varieties.

MODERN ALGEBRA

MATH 542

Field extensions, roots of polynomials, splitting fields, simple extensions, linear transformations, matrices, characteristic roots, canonical forms, determinants.

APPLIED LINEAR ALGEBRA

MATH 443

Review of matrix algebra. Simultaneous linear equations, linear dependence and rank, vector space, eigenvalues and eigenvectors, diagonalization, quadratic forms, inner product spaces, norms, canonical forms. Discussion of numerical aspects and applications in the sciences.