Boundary solutions of the quantum Yang-Baxter equation and solutions in three dimensions

TL;DR

This paper investigates boundary solutions to the quantum Yang-Baxter equation, constructs explicit boundary quantizations for a broad class of solutions, and provides quantizations for all classical r-matrices in sl(3) wedge sl(3).

Contribution

It introduces the concept of boundary solutions to the qYB equation and constructs explicit boundary quantizations for many solutions, including all classical r-matrices in sl(3).

Findings

Constructed explicit boundary quantizations for a large class of solutions.

Listed and provided quantizations for all classical r-matrices in sl(3).

Enhanced understanding of boundary solutions in quantum integrable systems.

Abstract

Boundary solutions to the quantum Yang-Baxter (qYB) equation are defined to be those in the boundary of (but not in) the variety of solutions to the ``modified'' qYB equation, the latter being analogous to the modified classical Yang-Baxter (cYB) equation. We construct, for a large class of solutions $r$ to the modified cYB equation, explicit ``boundary quantizations'', i.e., boundary solutions to the qYB equation of the form $I+tr+ t^2r_{2} + ...$. In the last section we list and give quantizations for all classical r-matrices in $sl(3) \wedge sl(3)$.