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.2 The notation U h (gl(n)) is often used in literatures for this form of presentation of the quantum algebra. The deformation parameter q is related to h as q = e h . We assume that q is generic.3 The normalization commonly used in literatures can be obtained by the replacement:, L −(2) 21 = (L −(1) L −(2) ) 21 (L −(1) L −(2) ) 22 (L −(1) L −(2) ) 23 J 21 J 22 J 23 J 31 J 32 J 33 , L −(2) 31 = (L −(1) L −(2) ) 31 (L −(1) L −(2) ) 32

Integrable matrix models in discrete space-time

“…[119] the authors obtained a particular lattice discretization of the N = 2 isotropic Landau-Lifshitz model, its implicit form makes it less appealing for concrete applications. In more recent works [103,104], several formal connections between quantum Yang-Baxter maps (representing the adjoint action of the universal R-matrix of a quantum group) and their classical limits (and discrete-time dynamics) have been uncovered, indicating that classical Yang-Baxter maps in a way naturally descend from the associated quantized algebraic structure. In this respect, our results indicate that, at the set-theoretic level, the emergent classical Yang-Baxter map does not show explicit dependence on the underlying Lie algebra.…”

“…Let us briefly elucidate the origin of the Yang-Baxter map (see [81,[100][101][102], or [103,104] for more recent accounts which discuss its connection to quasitriangular Hopf algebras [105,106]). To this end it is convenient to regard the discrete zero-curvature condition (5) as a re-factorization problem [107] for a pair of Lax matrices…”

Anisotropic Landau-Lifshitz model in discrete space-time

“…• The two-body propagator Φ is a classical Yang-Baxter map [66,[84][85][86], obeying the braided Yang-Baxter relation on the Cartesian product M ×3 1 (suppressing the anisotropy parameter)…”

Brief Review: Simulation of Novel Systems Using Duality Quantum Algorithm

“…Therefore, the YBE has become a significant theoretical tool in physics today. In recent year, it is found gradually that there are natural links between the YBE and one of the hottest frontier research, quantum information and computing [51][52][53][54][55][56][57][58][59][60][61][62][63][64], that the braiding operators in the YBE are universal quantum gates and quantum entanglement has close relationship with the YBE. Having attracted lots of attention, the YBE is being investigated in quantum entanglement, quantum correlation, and topological quantum computing intensively.…”