Representations and Lie Theory Seminar

Schedule of talks:

Abstracts

August 30: Quantum Knizhnik-Zamolodchikov (qKZ for short) equations are a family of difference equations whose coefficient matrix is given in terms of an R--matrix with spectral parameter and a regular semisimple operator called a dynamical parameter. They were discovered by I. Frenkel and N. Reshetikhin in 1996, as the equations satisfied by certain quantum correlation functions, and were shown to degenerate to the famous KZ--differential equations of conformal field theory. In this talk, I will introduce (rational and trigonometric) qKZ equations and derive the Yang--Baxter equation (with spectral parameter) as their integrability condition. I will present a conjectural description of the monodromy of these equations, which is a theorem of Etingof--Moura, and Etingof--Varchenko in the trigonometric case.

September 6: The study for finding solutions to the Yang-Baxter equation, which is called R-matrix, has been decades. Yangians are one of the parameter-dependent Hopf algebras for solving non-constant solutions of the Yang-Baxter equation. In this talk, I will explain the properties of R-matrix in 2-tensor space Y_\hbar(\mathfrak{g})\otimes Y_\hbar(\mathfrak{g}) and its action on Yangian representation, mostly focused when \mathfrak{g}=\sl_2. Also, we will discuss the Abelianization method, which is an idea to decompose R-matrix nicely, and how to construct each component.