Nilpotent Matrix Representation of Exterior Derivative Operator in Noncommutative Geometry

TL;DR

This paper presents a nilpotent matrix representation of the exterior derivative operator in noncommutative geometry, simplifying the construction of extended gauge theories by translating algebraic relations into matrix form.

Contribution

It introduces a novel matrix representation that streamlines the algebraic complexity in noncommutative geometry and gauge theory construction.

Findings

Matrix representation is nilpotent and preserves algebraic relations.

Simplifies the construction of extended gauge theories.

Reduces complexity compared to traditional algebraic methods.

Abstract

In this letter, we show a nilpotent matrix representation of the exterior derivative operator in noncommutative geometry (NCG), by translating the noncommutative relations of the algebraic formalization into the original one. As a result, we can construct the extended gauge theory without complicated algebraic rules nor excessively large differential algebra and non-monomorphic representation.