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

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.