Reflection matrices, coideal subalgebras and generalized Satake diagrams of affine type

TL;DR

This paper generalizes quantum symmetric pairs using Satake diagrams for affine Kac-Moody algebras, computes reflection matrices as solutions to the quantum reflection equation, and explores their algebraic and representation-theoretic properties.

Contribution

It introduces generalized Satake diagrams for affine algebras, constructs new reflection matrices, and analyzes their properties and symmetries, extending the theory of quantum symmetric pairs.

Findings

Computed explicit intertwiners for types A, B, C, D affine algebras.

Found reflection matrices with at most two nonzero entries per row/column.

Identified symmetries and parameter reductions of the reflection matrices.

Abstract

We present a generalization of the theory of quantum symmetric pairs as developed by Kolb and Letzter. We introduce a class of generalized Satake diagrams that give rise to (not necessarily involutive) automorphisms of the second kind of symmetrizable Kac-Moody algebras $\mathfrak{g}$. These lead to right coideal subalgebras $B_{\mathbf{c},\mathbf{s}}$ of quantized enveloping algebras $U_q(\mathfrak{g})$. In the case that $\mathfrak{g}$ is a twisted or untwisted affine Lie algebra of classical type Jimbo found intertwiners (equivariant maps) of the vector representation of $U_q(\mathfrak{g})$ yielding trigonometric solutions to the parameter-dependent quantum Yang-Baxter equation. In the present paper we compute intertwiners of the vector representation restricted to the subalgebras $B_{\mathbf{c},\mathbf{s}}$ when $\mathfrak{g}$ is of type ${\rm A}^{(1)}_n$, ${\rm B}^{(1)}_n$, ${\rm C}^{(1)}_n$ and ${\rm D}^{(1)}_n$. These intertwiners are matrix solutions to the parameter-dependent quantum reflection equation known as trigonometric reflection matrices. They are symmetric up to conjugation by a diagonal matrix and in many cases satisfy a certain sparseness condition: there are at most two nonzero entries in each row and column. Conjecturally, this classifies all such solutions in vector spaces carrying this representation. A group of Hopf algebra automorphisms of $U_q(\mathfrak{g})$ acts on these reflection matrices, allowing us to show that each reflection matrix found is equivalent to one with at most two additional free parameters. Additional characteristics of the reflection matrices such as eigendecompositions and affinization relations are also obtained. The eigendecompositions suggest that for all these matrices there should be a natural interpretation in terms of representations of Hecke-type algebras.