Title: Quantum groups, Yang-Baxter maps and quasi-determinants,
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-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.