A natural Lie superalgebra bundle on rank three WSD manifolds

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.