All Non-Commuting Solutions of the Yang-Baxter-like Matrix Equation Which Coefficient Matrix is Similar to diag (λ,J2(λ))

WANG Yun-jie

doi:10.13371/j.cnki.chin.q.j.m.2026.01.008

Chinese Quarterly Journal of Mathematics >

2026 , Vol. 41 >Issue 1: 92 - 110

DOI: https://doi.org/10.13371/j.cnki.chin.q.j.m.2026.01.008

All Non-Commuting Solutions of the Yang-Baxter-like Matrix Equation Which Coefficient Matrix is Similar to diag (λ,J₂(λ))

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Kewen Institute, Jiangsu Normal University, Xuzhou 221116, China

WANG Yun-jie (1978-), male, native of Fuyang, Anhui, associate professor of Kewen Institute, Jiangsu Normal University, engages in large-scale scientific and engineering computation.

Received date: 2025-09-15

Online published: 2026-03-30

Supported by

Supported by National Natural Science Foundation of China (Grant No. 62173161).

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Abstract

Let A be a 3×3 singular or diagonalizable matrix, all solutions to the Yang-Baxter-like matrix equation have been determined. However, finding all solutions for full rank, non-diagonalizable matrices remains challenging. By utilizing classification techniques, we establish all solutions of the Yang-Baxter-like matrix equation in this paper when the coefficient matrix A is similar to non-diagonalizable matrix diag(λ,J₂(λ)) with λ= 0. More specifically, we divide the non-diagonal elements of the solution into 10 different cases. By discussing each situation, we establish all solutions of the YangBaxter-like matrix equation. The results of this work enrich the existing ones.

Key words： Yang–Baxter–like matrix equation; Yang–Baxter equation; Commuting solutions; Non–commuting solutions

Cite this article

WANG Yun-jie . All Non-Commuting Solutions of the Yang-Baxter-like Matrix Equation Which Coefficient Matrix is Similar to diag (λ,J₂(λ))[J]. Chinese Quarterly Journal of Mathematics, 2026 , 41(1) : 92 -110 . DOI: 10.13371/j.cnki.chin.q.j.m.2026.01.008

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