Solutions of the Yang–Baxter–like matrix equation with 3×3 diagonalizable coefficient matrix

Abstract

The Yang–Baxter–like matrix equation plays an important role in quantum group theory, knot theory and braid groups, which has received great attention from physicists and mathematicians. A difficult and important open problem is to find all the solutions of the Yang–Baxter–like matrix equation. As a matter of fact, it is difficult to find all of the solutions even when the coefficient matrix is a $3\times 3$ matrix. To the best of our knowledge, when the coefficient matrix is a $3\times 3$ diagonalizable complex matrix with three distinct nonzero eigenvalues, finding all the solutions of the Yang–Baxter–like matrix equation is still an open problem. In order to fill-in this gap, we first present a sufficient and necessary condition for the commuting solutions of the Yang–Baxter–like matrix equation, and give all the commuting solutions of the matrix equation. Second, with the help of a simplified matrix equation, we derive all the non–commuting solutions of the Yang–Baxter–like matrix equation by discussing whether the off–diagonal elements of the solutions are zero or not.

Keywords:

• Yang–Baxter–like matrix equation
• Yang–Baxter equation
• commuting solutions
• non–commuting solutions
• diagonalizable matrix

AMS subject classifications:

• 65F45
• 65F15
• 65F10

Acknowledgments

We would like to express our sincere thanks to the referees for their insightful comments and suggestions that greatly improved the representation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant number 12271518], the Fujian Natural Science Foundation [grant number 2023J01354], the Key Research and Development Project of Xuzhou Natural Science Foundation [grant number KC22288], and the Open Project of Key Laboratory of Data Science and Intelligence Education of the Ministry of Education [grant number DSIE202203].