#### Date of Award

5-2022

#### Document Type

Thesis

#### Degree Name

Master of Arts in Mathematics

#### Department

Mathematics

#### First Reader/Committee Chair

Dunn, Corey

#### Abstract

We decompose the space of algebraic curvature tensors (ACTs) on a finite dimensional, real inner product space under the action of the orthogonal group into three inequivalent and irreducible subspaces: the real numbers, the space of trace-free symmetric bilinear forms, and the space of Weyl tensors. First, we decompose the space of ACTs using two short exact sequences and a key result, Lemma 3.5, which allows us to express one vector space as the direct sum of the others. This gives us a decomposition of the space of ACTs as the direct sum of three subspaces, which at this point may or may not be inequivalent or irreducible. We then count the number of nonzero, independent ways of contracting an ACT down to a real number and determine there are exactly three ways of doing so. We conclude with a verification that the subspaces in our decomposition are in fact inequivalent and irreducible by applying another key result, Lemma 3.7, a representation theoretic tool used to sense irreducibility. Since the number of terms in our decomposition (three) is equal to the number of nonzero, independent ways to contract an ACT down to the real numbers (three), we conclude that our decomposition is the orthogonal decomposition of the space of ACTs where the subrepresentations are inequivalent and irreducible.

#### Recommended Citation

Risinger, Katelyn Sage, "THE DECOMPOSITION OF THE SPACE OF ALGEBRAIC CURVATURE TENSORS" (2022). *Electronic Theses, Projects, and Dissertations*. 1486.

https://scholarworks.lib.csusb.edu/etd/1486