Soliton cellular automata for the affine general linear Lie superalgebra

TL;DR

This paper extends the box-ball system, a cellular automaton modeling water waves, to affine general linear Lie superalgebras using crystal bases, revealing solitonic behavior in this new setting.

Contribution

It introduces a generalization of the Hikami--Inoue box-ball system to affine Lie superalgebras via Kirillov--Reshetikhin crystals, demonstrating solitonic phenomena.

Findings

Solitonic behavior observed in the generalized system

Extension of BBS to affine Lie superalgebras

Use of crystal bases for the generalization

Abstract

The box-ball system (BBS) is a cellular automaton that is an ultradiscrete analogue of the Korteweg--de Vries equation, a non-linear PDE used to model water waves. In 2001, Hikami and Inoue generalised the BBS to the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. We further generalise the Hikami--Inoue BBS to column tableaux using the Kirillov--Reshetikhin crystals for $\hat{\mathfrak{gl}}{(m|n)}$ devised by Kwon and Okado (arXiv:1804.05456), where we find similar solitonic behaviour under certain conditions.