Beijing Institute of Mathematical Sciences and Applications

This is a short (18 class hours) course which aims to get researchers acquainted with quantum symmetric pairs, a contemporary topic which has developed in a rapid pace during the last decade.

A Lie algebra $\mathfrak{g}$ together with the fixed-point subalgebra $\mathfrak{g}^\theta$ of an involutive automorphism $\theta$ is called a symmetric pair. For example, let $\mathfrak{g}$ be a matrix Lie algebra and let $\theta$ be "minus the transpose". For a suitable Lie algebra $\mathfrak{g}$ (for example, finite-dimensional simple), its universal enveloping algebra has a deformation $U_q(\mathfrak{g})$ in the class of Hopf algebras (Drinfeld-Jimbo quantum group, quantized enveloping algebra). It has become clear since the 1990s that, if you arrange the involution $\theta$ carefully, then you can define a quantum symmetric pair $(U_q(\mathfrak{g}),U_q'(\mathfrak{g}^\theta))$. The second member is a coideal subalgebra 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.