Quantum groups, Yang-Baxter maps and quasi-determinants

TL;DR

This paper explores the structure of quantum Yang-Baxter maps derived from quasi-triangular Hopf algebras, providing quasi-determinant formulas and their classical limits, advancing understanding of integrable systems and algebraic structures.

Contribution

It introduces a quasi-determinant expression for the quantum Yang-Baxter map associated with $U_q(gl(n))$ and links it to quasi-Plücker coordinates, offering new insights into quantum and classical integrable models.

Findings

Derived a quasi-determinant formula for the quantum Yang-Baxter map.

Connected the map to quasi-Plücker coordinates and L-operators.

Reduced the quantum map to classical ratios of determinants in the quasi-classical limit.

Abstract

For any quasi-triangular Hopf algebra, there exists the universal R-matrix, which satisfies the Yang-Baxter equation. It is known that the adjoint action of the universal R-matrix on the elements of the tensor square of the algebra constitutes a quantum Yang-Baxter map, which satisfies the set-theoretic Yang-Baxter equation. The map has a zero curvature representation among L-operators defined as images of the universal R-matrix. We find that the zero curvature representation can be solved by the Gauss decomposition of a product of L-operators. Thereby obtained a quasi-determinant expression of the quantum Yang-Baxter map associated with the quantum algebra $U_{q}(gl(n))$. Moreover, the map is identified with products of quasi-Pl\"{u}cker coordinates over a matrix composed of the L-operators. We also consider the quasi-classical limit, where the underlying quantum algebra reduces to a Poisson algebra. The quasi-determinant expression of the quantum Yang-Baxter map reduces to ratios of determinants, which give a new expression of a classical Yang-Baxter map.