## Mathematics Ph.D. Dissertations

# Generalizations of the Exterior Algebra

## Date of Award

2023

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy (Ph.D.)

## Department

Mathematics

## First Advisor

Mihai Staic (Committee Chair)

## Second Advisor

Xiangdong Xie (Committee Member)

## Third Advisor

Ben Ward (Committee Member)

## Fourth Advisor

Marlise Lonn (Committee Member)

## Abstract

Staic first introduced the exterior GSC-operad Λ^{S2}_{Vd }as a generalization of the exterior algebra in Staic (2020), which he used to give a linear map *det*^{S2}:V_{2}^{⊗6} → *k* with the property that *det*^{S2}(⊗1≤*i*<*j*≤4(*v** _{i,j}*) = 0 if there exists 1 ≤

*x*<

*y*<

*z*≤ 4 such that v

_{x,y}= v

_{x,z}= v

_{y,z}. In this dissertation, we start by further exploring Λ

^{S2}

_{V.}We get results such as connections to graph theory, dimensions of Λ

^{S2}

_{V}[

*n*] as vector spaces, and a map

*det*

^{S2}:V

_{3}

^{⊗15}→

*k*which generalizes the determinant map. Then, we give a further generalization of the exterior algebra first presented in Staic and Lippold (2022), denoted Λ

^{S3}

_{Vd}. We discuss connections to hypergraphs, the dimensions of Λ

^{S3}V

_{2}[

*n*] as vector spaces, and a map

*det*

^{S3}:V

_{2}

^{⊗20}→

*k*which generalizes the determinant and the map

*det*

^{S2}. Following this, we define maps

*det*

^{S3}: V

_{d}

^{⊗(1/2)d(3d-1)(3d-2)}→

*k*for all

*d*≥ 2 with the property that

*det*

^{S3}(⊗1≤

*i*<

*j*<

*k*≤3

*d*(v

_{i,j, k})) = 0 if there exists 1 ≤

*x*<

*y*<

*z*<

*t*≤ 3

*d*such that v

_{x,y,z}= v

_{ x,y,t}= v

_{x,z,t}=v

_{y,z,t}. Further, we show that

*det*

^{S3}is a

*SL*(

_{d}*k*) invariant nontrivial linear map. Lastly, we further extend these constructions to discuss the graded vector space Λ

^{Sr}

_{V}.

## Recommended Citation

Lippold, Steven Robert, "Generalizations of the Exterior Algebra" (2023). *Mathematics Ph.D. Dissertations*. 99.

https://scholarworks.bgsu.edu/math_diss/99