A Geometric Realization of $sl(6,\mathbb{C})$

Abstract

Given an orientable weakly self-dual manifold $X$ of rank two, we build a geometric realization of the Lie algebra $\mathbf{sl}(6,\mathbb{C})$ as a naturally defined algebra ${\mathcal{L}}_{\mathbb{C}}$ of endomorphisms of the space of differential forms of $X$. We provide an explicit description of Serre generators in terms of natural generators of ${\mathcal{L}}_{\mathbb{C}}$. This construction gives a bundle on $X$ which is related to the search for a natural Gauge theory on $X$. We consider this paper as a first step in the study of a rich and interesting algebraic structure.