Beijing Institute of Mathematical Sciences and Applications

This is a 3-credit course which aims to get new researchers acquainted with quantum symmetric pairs, a contemporary topic which has developed in a rapid pace during the last decade. The concepts and main results can be developed in parallel to the more established theory of quantized universal enveloping algebras (Drinfeld-Jimbo quantum groups).

A (classical) symmetric pair consists of a finite-dimensional simple Lie algebra $\mathfrak{g}$ and the fixed-point subalgebra $\mathfrak{g}^\theta$ of an involutive automorphism $\theta$. For example, let $\mathfrak{g}$ be a classical matrix Lie algebra and let $\theta$ be "minus the transpose. More generally, $\mathfrak{g}$ can be a Kac-Moody algebra (e.g. affine Lie algebra).

The universal enveloping algebra $U(\mathfrak{g})$ has a deformation $U_q(\mathfrak{g})$
in the class of Hopf algebras (Drinfeld-Jimbo quantum group). It became clear in the 1990s that, if you arrange $\theta$ carefully, then the Hopf subalgebra $U(\mathfrak{g}^\theta)$ can be quantized as a (one-sided) coideal subalgebra $U_q'(\mathfrak{g}^\theta)$ of $U_q(\mathfrak{g})$, also known as $\imath$-quantum group.

We will discuss the basic constructions and their motivations. We will explore some of the representation theory, some applications in mathematical physics and some variations of the construction.