Quantum groups, Yang–Baxter maps and quasi-determinants

• Main Author: Zengo Tsuboi
• Format: Article
• Language: English
• Published: Elsevier 2018-01-01
• Series: Nuclear Physics B
• Online Access: http://www.sciencedirect.com/science/article/pii/S0550321317303632

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 Uq(gl(n)). Moreover, the map is identified with products of quasi-Plücker 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.