Tetrahedron Equation and Quantum R Matrices for Spin Representations of B^{(1)}_n, D^{(1)}_n and D^{(2)}_{n+1}

TL;DR

This paper explores how solutions to the tetrahedron equation can generate quantum R matrices for specific Lie algebra spin representations, using bosonic and fermionic oscillator solutions with boundary conditions.

Contribution

It introduces a method to derive quantum R matrices for B^{(1)}_n, D^{(1)}_n, and D^{(2)}_{n+1} from tetrahedron equation solutions involving bosons and fermions.

Findings

Generated quantum R matrices match known spin representation solutions.

Applied boundary conditions to oscillator solutions for tetrahedron equation.

Established a link between tetrahedron solutions and Yang-Baxter solutions.

Abstract

It is known that a solution of the tetrahedron equation generates infinitely many solutions of the Yang-Baxter equation via suitable reductions. In this paper this scheme is applied to an oscillator solution of the tetrahedron equation involving bosons and fermions by using special 3d boundary conditions. The resulting solutions of the Yang-Baxter equation are identified with the quantum R matrices for the spin representations of B^{(1)}_n, D^{(1)}_n and D^{(2)}_{n+1}.