Lie algebaic characterization of supercommutative space

TL;DR

This paper extends classical Lie algebraic characterizations to super and graded spaces, linking algebraic structures with geometric and physical theories like quantum gravity and superstring theory.

Contribution

It proves a super version of a classical result, characterizing super and graded spaces via their super Lie algebras of vector fields.

Findings

Super Lie algebraic characterization of super spaces

Explicit description of isomorphisms of super vector fields

Connection between algebraic and geometric structures in supergeometry

Abstract

During the last decades algebraization of space turned out to be a promising tool at the interface between Mathematics and Theoretical Physics. Starting with works by Gel'fand-Kolmogoroff and Gel'fand-Naimark, this branch developed as from the fortieth in two directions: algebraic characterization of usual geometric space on the one hand, and algebraically defined noncommutative space, which is known to be tightly related with e.g. quantum gravity and super string theory, on the other hand. In this note, we combine both aspects, prove a superversion of Shanks and Pursell's classical result stating that any isomorphism of the Lie algebras of compactly supported vector fields is implemented by a diffeomorphism of underlying manifolds. We thus provide a super Lie algebraic characterization of super and graded spaces and describe explicitly isomorphisms of the super Lie algebras of super vector fields.