# A geometric realization of sl(6,C)

## TL;DR

This paper constructs a geometric realization of the Lie algebra sl(6,C) using differential forms on a specific type of manifold, linking algebraic structures to geometric and gauge theory concepts.

## Contribution

It provides an explicit geometric construction of sl(6,C) on a weakly self-dual manifold, including Serre generators, connecting algebraic and geometric frameworks.

## Key findings

- Realization of sl(6,C) as endomorphisms of differential forms
- Explicit description of Serre generators in geometric terms
- Introduction of a bundle related to gauge theory on the manifold

## Abstract

Given an orientable weakly self-dual manifold X of rank two, we build a geometric realization of the Lie algebra sl(6,C) as a naturally defined algebra L of endomorphisms of the space of differential forms of X. We provide an explicit description of Serre generators in terms of natural generators of L. 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.