Spectral-Parameter Dependent Yang–Baxter Operators and Yang–Baxter Systems From Algebra Structures

Abstract

For any algebra, two families of colored Yang–Baxter operators are constructed, thus producing solutions to the two-parameter quantum Yang–Baxter equation. An open problem about a system of functional equations is stated. The matrix forms of these operators for two and three dimensional algebras are computed. A FRT bialgebra for one of these families is presented. Solutions for the one-parameter quantum Yang–Baxter equation are derived and a Yang–Baxter system constructed.

Key Words:

• FRT construction
• Two-parameter quantum Yang–Baxter equation
• Yang–Baxter system

2000 Mathematics Subject Classification:

• 16W30
• 81R50

ACKNOWLEDGMENTS

We are grateful to Tomasz Brzeziński for several fruitful discussions. Florin Nichita thanks the European Commission for the Marie Curie Fellowship HPMF-CT-2002-01782 at the University of Wales Swansea, and Deepak Parashar thanks the Royal Commission for the Exhibition of 1851 for a Research Fellowship. Deepak Parashar's research has been partly supported by a grant from the European Science Foundation's program on Noncommutative Geometry. Finally, we thank the referee for drawing our attention to references Okado and Yamane (1991) and Perk and Schultz (1983).

Notes

Communicated by K. Misra.