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Foundations and Classification of Invariant Subalgebras of Grassmann Algebra

Mithat Konuralp Demir

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.

Foundations and Classification of Invariant Subalgebras of Grassmann Algebra

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.
Paper Structure (26 sections, 30 theorems, 83 equations, 3 figures)

This paper contains 26 sections, 30 theorems, 83 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exterior Algebra
  4. Definition of Exterior Algebra
  5. Wedge Product
  6. Interpretation on Euclidean Geometry
  7. Formal Construction of Exterior Algebra
  8. Polynomial Ring
  9. Exterior Algebra
  10. Tensor Product
  11. Step 1
  12. Step 2
  13. Step 3
  14. Transition to Exterior Algebra
  15. Step 4
  16. ...and 11 more sections

Key Result

Proposition 2.11

Let $x = v_1 \wedge \cdots \wedge v_k$, where $v_i \in V$. If any vector appears more than once (up to scalar multiple), then $x = 0$

Figures (3)

  • Figure 1: A direction in $\mathbb{R}^3$
  • Figure 2: A planar direction in $\mathbb{R}^3$
  • Figure 3: A trivector direction in $\mathbb{R}^3$

Theorems & Definitions (114)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 1.9
  • Remark 1.10
  • ...and 104 more