How to superize $\mathfrak{gl}(\infty)$

TL;DR

This paper introduces new types of Lie superalgebras extending $rak{gl}( ext{infinity})$, characterized by supermatrices with non-zero elements confined within curved regions around the diagonal, opening avenues for further mathematical and physical research.

Contribution

The paper presents novel super versions of $rak{gl}( ext{infinity})$ with matrices bounded by curves, expanding the classification of infinite-dimensional Lie superalgebras.

Findings

New super Lie algebras with curved boundary matrices

Formulation of open questions for future research

Potential applications to dynamical systems like KdV

Abstract

Penkov with co-authors stdied several types of Lie algebra $\mathfrak{gl}(\infty)$. Completely different new types of super versions of $\mathfrak{gl}(\infty)$ are introduced in this paper: these are Lie superalgebras of supermatrices infinite in all directions with non-zero elements of each matrix occupying a region around the main diagonal bounded by certain curves, not straight ligns. Several open questions related with further study of the Lie superalgebras introduced and with their possible applications to dynamical systems, such as KdV, are formulated.