Foundations and Classification of Invariant Subalgebras of Grassmann Algebra

TL;DR

This paper provides a comprehensive overview of Grassmann algebra, detailing its foundational properties, construction, and the novel classification of its invariant subalgebras, highlighting its significance in mathematics and physics.

Contribution

It introduces a new classification scheme for invariant subalgebras of Grassmann algebra, expanding understanding of its structural properties.

Findings

Presented the defining properties of Grassmann algebra.

Explored the relationship between exterior product and determinant.

Provided a novel classification of invariant subalgebras.

Abstract

This paper is a documentation of author's reseach, focusing on the topic Grassmann Algebra spanning over July, August 2025 under mentorship provided by DRP Turkiye 2025. Grassmann algebra is a fundamental structure in mathematics with wide-ranging applications across multiple areas of mathematics and physics. Most notably, it serves as the foundation for differential geometry, by constituting the natural setting which differential forms reside. This paper begins with presenting the defining properties of Grassmann Algebra, outlining the working principles of the key mechanism of the algebra, wedge product. Following that, we give an exposition of formal construction of Grassmann algebra from free associative algebra with the goal of emphasizing how these properties are imposed in the structure of the algebra. The intrinsic relationship between the exterior product and the determinant is explored in Section 4. Finally, we investigate invariant subalgebras, one of the primary focuses of this paper. Here, we present a novel classification of invariant subalgebras.