Generalizations of the Exterior Algebra

Author

Steven Robert Lippold, Bowling Green State University

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:V₂^⊗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:V₃^⊗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^S3V₂ [n] as vector spaces, and a map det^S3:V₂^⊗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≤3d(v_{i,j, k})) = 0 if there exists 1 ≤ x < y < z < t ≤ 3d 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