Spring 2023 MATH 261B 001 LEC

Prerequisites Math 214, 261A, Consent of instructor for undergrads.

Description: Lie Groups, Lie algebras, and integrable systems

Completely integrable systems (classical and quantum) emerged in the last 50 years as one of the areas of most powerful applications of Lie groups and Lie algebras. In this course we will use two important classes of integrable systems to illustrate the main tenets of the general theory:

--- the (quantum) Gaudin systems;

--- the (classical) infinite-dimensional soliton hierarchies of KdV type.

Though the former system can be formulated in terms of a finite-dimensional simple Lie algebra, the commuting quantum Hamiltonians of this system arise naturally from the center of the enveloping algebra of an affine Kac-Moody algebra at the critical level. We will discuss the Bethe Ansatz method for constructing eigenvectors and eigenvalues of these Hamiltonians. The connection of these systems to moduli spaces of G-bundles on complex curves and the Langlands correspondence for complex curves will also be discussed (however, no prior knowledge of the Langlands correspondence is required).

The latter systems (and closely related to them affine Toda models) can be described in terms of a double quotient of the Kac-Moody group of affine type (it is related to the so-called infinite Grassmannian). Commuting classical flows arise from the action of a maximal commutative subalgebra of the corresponding affine Kac-Moody Lie algebra.

All necessary notions will be explained, such as affine Kac-Moody algebras and the corresponding Lie groups, their representations, vertex algebras, etc.

Finally, we will talk about the affinization of the Gaudin models in which the role of a simple Lie algebra g is played by the corresponding affine Kac-Moody algebra. It turns out that in a special case, the resulting model may also be viewed as a quantization of the corresponding KdV system. Thus, we can unify the above two classes of integrable systems.

Time permitting, we will also discuss analogues of the Gaudin models (the so-called Heisenberg XXZ spin chains and their generalizations) associated to the quantum affine algebras.

Recommended Reading: to get a general idea of some of the topics that will be discussed, see these surveys (other sources will be introduced as the course progresses):

E. Frenkel, Affine Algebras, Langlands duality and Bethe Ansatz

E. Frenkel, Gaudin Model and Opers

E. Frenkel, Five Lectures on Soliton Equations

V. Drinfeld and V. Sokolov, Lie Algebras and Equations of KdV Type

Course Webpage will be on bCourses.