Solution of the Yang–Baxter System for Quantum Doubles (original) (raw)

Solution of a Yang-Baxter system

Eprint Arxiv Math 9811016, 1998

Yang-Baxter system related to quantum doubles is introduced and large class of both continuous and discrete symmetries of the solution manifold are determined. Strategy for solution of the system based on the symmetries is suggested and accomplished in the dimension two. The complete list of invertible solutions of the system is presented.

Yang-Baxter systems, solutions and applications

1997

Two types of Yang-Baxter systems play roles in the theoretical physics -- constant and colour dependent. The constant systems are used mainly for construction of special Hopf algebra while the colour or spectral dependent for construction of quantum integrable models. Examples of both types together with their particular solutions are presented. The complete solution is known only for the constant system related to the quantized braided groups in the dimension two. The strategy for solution of the system related to quantum doubles is suggested and partial results are presented.

Quantization of certain skew-symmetric solutions of the classical Yang-Baxter equation

An explicit quantization is given of certain skew-symmetric solutions of the classical Yang-Baxter, yielding a family of RRR-matrices which generalize to higher dimensions the Jordanian RRR-matrices. Three different approaches to their construction are given: as twists of degenerations of the Shibukawa-Ueno Yang-Baxter operators on meromorphic functions; as boundary solutions of the quantum Yang-Baxter equation; via a vertex-IRF transformation from solutions to the dynamical Yang-Baxter equation.

Introduction to the Yang-Baxter Equation with Open Problems

The Yang-Baxter equation first appeared in theoretical physics, in a paper by the Nobel laureate C. N. Yang, and in statistical mechanics, in R. J. Baxter's work. Later, it turned out that this equation plays a crucial role in: quantum groups, knot theory, braided categories, analysis of integrable systems, quantum mechanics, non-commutative descent theory, quantum computing, non-commutative geometry, etc. Many scientists have found solutions for the Yang-Baxter equation, obtaining qualitative results (using the axioms of various algebraic structures) or quantitative results (usually using computer calculations). However, the full classification of its solutions remains an open problem. In this paper, we present the (set-theoretical) Yang-Baxter equation, we sketch the proof of a new theorem, we state some problems, and discuss about directions for future research.

The upper triangular solutions to the three-state constant quantum Yang-Baxter equation

Journal of Physics A: Mathematical and General, 1993

In this article we present all nonsingular upper triangular solutions to the constant quantum Yang-Baxter equation R k1k2 j1j2 R l1k3 k1j3 R l2l3 k2k3 = R k2k3 j2j3 R k1l3 j1k3 R l1l2 k1k2 in three state case. The upper triangular ansatz implies 729 equations for 45 variables. Fortunately many of the equations turned out to be simple allowing us to start breaking the problem into smaller ones. In the end we had a total of 552 solutions, but many of them were either inherited from two-state solutions or subcases of others. The final list contains 35 nontrivial solutions, most of them new.

Nonstandard Solutions of the Yang–Baxter Equation

1998

Explicit solutions of the quantum Yang-Baxter equation are given corresponding to the non-unitary solutions of the classical Yang-Baxter equation for sl(5). Mathematics Subject Classifications (1991): Primary 81R50, 17B37; Secondary 16W30.