On reflection algebras and twisted Yangians

TL;DR

This paper explores the boundary symmetries of open quantum spin chains with different boundary conditions, identifying their algebraic structures as boundary or twisted Yangians and demonstrating their conserved quantities.

Contribution

It provides a unified quantum-level framework for deriving boundary non-local charges and shows their role as conserved symmetries in open $gl(n)$ spin chains.

Findings

Boundary non-local charges are coproduct realizations of boundary quantum algebras.

The framework applies to both soliton preserving and non-preserving boundary conditions.

Several boundary non-local charges are shown to be conserved quantities.

Abstract

It is known that integrable models associated to rational $R$ matrices give rise to certain non-abelian symmetries known as Yangians. Analogously `boundary' symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay and Short from the field theory point of view, in the context of the principal chiral model on the half line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the $N$ site $gl(n)$ open quantum spin chain. In particular, the open spin chain with two distinct types of boundary conditions known as soliton preserving and soliton non-preserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary non-local charges directly at the quantum level. The non-local charges are simply coproduct realizations of particular boundary quantum algebras called `boundary' or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the symmetry of the open spin chain, namely we show that a number of the boundary non-local charges are in fact conserved quantities