Braid group actions and quasi K-matrices for quantum symmetric pairs

Abstract

Quantum groups arose in the early 80’s in the investigation of integrable systems in mathematical physics. Quantum groups are a family of non-commutative, non-cocommutative Hopf algebras which arise through deformation quantisation of universal enveloping algebras of Lie algebras or of coordinate rings of affine algebraic groups. In this thesis, we focus on quantum groups coming from universal enveloping algebras, known as ‘quantised enveloping algebras’. One of the fundamental properties of quantised enveloping algebras is that they give rise to a universal R-matrix which provides solutions of the quantum Yang-Baxter equation for each representation. The universal R-matrix allows applications of quantum groups in the construction of invariants of knots and links. The main component of the universal R-matrix is a quasi R-matrix, which has applications in other areas of representation theory, for instance in Lusztig’s and Kashiwara’s theory of canonical bases. Also essential to the theory of quantised enveloping algebras is the existence of a braid group action by algebra automorphisms, due to Lusztig. This braid group action allows the definition of root vectors and PBW bases. Parallel to quantised enveloping algebras is the notion of quantum symmetric pair coideal subalgebras, developed by G. Letzter in a series of papers from 1999 to 2004. These are quantum group analogues of Lie subalgebras which are fixed under an involution. Over the past five years it has become increasingly clear that many of the results for quantised enveloping algebras have analogues in the quantum symmetric pair setting. An important example of this is the construction of a universal K-matrix for quantum symmetric pairs by Balagovi´c and Kolb following earlier work by Bao and Wang. The universal K-matrix provides solutions to the reflection equation, which is an analogue of the quantum YangBaxter equation. The main ingredient of the universal K-matrix is a quasi K-matrix which is an analogue of the quasi R-matrix. The quasi K-matrix recently played a crucial role in the theory of canonical bases for quantum symmetric pairs, developed by Bao and Wang. Until recently, only a recursive formula for the quasi K-matrix was known. The first main result of this thesis is to give an explicit formula for the quasi K-matrix in many cases. This formula closely resembles the known formula for the quasi R-matrix, which admits a factorisation as a product of rank one quasi R-matrices. In particular, the quasi K-matrix has a factorisation into a product of quasi K-matrices for Satake diagrams of rank one. This factorisation depends on the restricted Weyl group of the symmetric Lie algebra similarly to how the quasi R-matrix depends on the Weyl group of the Lie algebra. The key idea is to calculate the quasi K-matrix explicitly in rank one and in rank two