Matrix factorizations and pentagon maps

TL;DR

This paper introduces a class of matrices linked to factorization problems that correspond to various pentagon and Yang-Baxter maps in non-commutative variables, extending known maps and introducing new ones.

Contribution

It establishes a connection between matrix factorizations and pentagon, reverse-pentagon, and Yang-Baxter maps, including new extensions and mappings in non-commutative settings.

Findings

Order N=2 matrices relate to the homogeneous normalization map.

Order N=3 matrices extend the normalization map and produce new maps.

The work links matrix factorizations to integrable map structures.

Abstract

We propose a specific class of matrices which participate in factorization problems that turn to be equivalent to constant and entwining (non-constant) pentagon, reverse-pentagon or Yang-Baxter maps, expressed in non-commutative variables. In detail, we show that factorizations of order $N=2$ matrices of this specific class are equivalent to the homogeneous normalization map. From order $N=3$ matrices, we obtain an extension of the homogeneous normalization map, as well as novel entwining pentagon, reverse-pentagon and Yang-Baxter maps.