A natural Lie superalgebra bundle on rank three WSD manifolds

TL;DR

This paper characterizes a natural Lie superalgebra bundle on rank three WSD manifolds, revealing its structure as a product of classical and super Lie algebras with explicit real forms, advancing understanding of geometric symmetries.

Contribution

It explicitly determines the structure of the Lie superalgebra generated by natural operators on rank three WSD manifolds, including a description of its real form and geometric invariance properties.

Findings

The superalgebra is a product of sl(4,a9) and special linear superalgebras.

An explicit real form of the superalgebra is constructed.

The superalgebra preserves the Poincare9 (odd Hermitean) inner product.

Abstract

We determine the structure of the $*$-Lie superalgebra generated by a set of carefully chosen natural operators of an orientable WSD manifold of rank three. This Lie superalgebra is formed by global sections of a natural Lie superalgebra bundle, and turns out to be a product of $\mathbf{sl}(4,\C)$ with the full special linear superalgebras of some graded vector spaces isotypical with respect to a natural action of $\mathbf{so}(3,\R)$. We provide an explicit description of one of the real forms of this superalgebra, which is geometrically natural being made of $\mathbf{so}(3,\R)$-invariant operators which preserve the Poincar\'e (odd Hermitean) inner product on the bundle of forms.