Generalized pentagram maps via Q-nets and refactorization mappings

TL;DR

This paper generalizes pentagram maps using Q-nets and refactorization in Poisson-Lie groups, deriving Lax forms and Poisson brackets, and connects these to discrete integrable systems like the Toda and BKP equations.

Contribution

It introduces a new family of pentagram map generalizations via Q-nets and refactorization, extending previous methods to include Lax forms and Poisson structures.

Findings

Derived Lax form with spectral parameter

Established invariant Poisson brackets

Connected discrete pentagram maps to Toda and BKP equations

Abstract

We introduce a family of generalizations of the pentagram maps related to $Q$-nets. A specific example is considered, and we find the map can be treated as a refactorization mapping in the Poisson-Lie group of pseudo-difference operators. This method was firstly proposed by Izosimov, and we generalize it to fit our needs. Using this description, we obtain the corresponding Lax form with a spectral parameter and invariant Poisson brackets. Finally, we consider the reduction to $B$-nets and the discrete BKP equation, offering a geometric explanation for the discrete-time Toda equation of BKP type proposed by Hirota.