Tetrahedron equation and quantum R matrices for infinite dimensional modules of U_q(A^{(1)}_1) and U_q(A^{(2)}_2)

TL;DR

This paper constructs quantum R matrices for infinite-dimensional modules of quantum affine algebras U_q(A^{(1)}_1) and U_q(A^{(2)}_2) by reducing solutions of the tetrahedron equation derived from q-oscillators.

Contribution

It introduces a novel reduction procedure to derive solutions to the Yang-Baxter equation from tetrahedron equation solutions, linking them to quantum R matrices for specific infinite-dimensional modules.

Findings

Constructed solutions to the Yang-Baxter equation from tetrahedron equation solutions.

Identified these solutions with quantum R matrices for modules of U_q(A^{(1)}_1) and U_q(A^{(2)}_2).

Connected the modules to affinization of Verma modules of subalgebras.

Abstract

From the q-oscillator solution to the tetrahedron equation associated with a quantized coordinate ring, we construct solutions to the Yang-Baxter equation by applying a reduction procedure formulated earlier by S. Sergeev and the first author. The results are identified with the quantum R matrices for the infinite dimensional modules of U_q(A^{(1)}_1) and U_q(A^{(2)}_2) corresponding to an affinization of Verma modules of their subalgebras isomorphic to U_q(sl_2) and U_{q^4}(sl_2).