What question did this study set out to answer?

The research focuses on classifying the solutions of the generalized Yang-Baxter matrix equation for both nonsingular and singular matrices.

June 6, 2026Open Access

THE GENERALIZED YANG-BAXTER MATRIX EQUATION A (XA) ᵐ=X (AX) ᵐ OVER M₂ (F)

Key Points

The research focuses on classifying the solutions of the generalized Yang-Baxter matrix equation for both nonsingular and singular matrices.
Expanded the characterization of nonsingular solutions for a general field F.
Examined the set of singular solutions for arbitrary 2x2 matrices.
Provided a comprehensive description of the full solution set.
Enhanced understanding of both nonsingular and singular solutions for the Yang-Baxter equation.
Developed a complete classification of solutions for all 2x2 matrices.
Characterized the solution set for arbitrary positive integers m.

Abstract

Abstract This paper investigates the generalized Yang-Baxter matrix equation A (XA) ᵐ=X (AX) ᵐ A (X A) m = X (A X) m for X M₂ (F) X ∈ M 2 (F) over a general field F F with char (F) 2 char (F) ≠ 2. Building upon our previous research 4 which characterized the set of nonsingular solutions S ₅^m (A) ^* S F m (A) ∗ for a nonsingular A, this study completes the classification by characterizing nonsingular solutions for singular A and describing the set of singular solutions S ₅^m (A) ^ S F m (A) ∘ for arbitrary 2 2 2 × 2 matrices A. Consequently, these results provide a comprehensive description of the full solution set S ₅ᵐ (A) S F m (A) for all 2 2 2 × 2 matrices and positive integers m.

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