Let $k$ be a field and $B$ a finite-dimensional, associative, unital $k$-algebra. For each $1 \le d \le \dim_kB$, let $\operatorname{AlgGr}_d(B)$ denote the projective variety of $d$-dimensional subalgebras of $B$, and let $\operatorname{Aut}_k(B)$ denote the automorphism group of $B$. In this thesis, we are primarily concerned with understanding the relationship between $\operatorname{AlgGr}_d(B)$, the representation theory of $B$, and the representation theory of $\operatorname{Aut}_k(B)$. We begin by proving fundamental structure theorems for the maximal subalgebras of $B$. We show that maximal subalgebras of $B$ come in two flavors, which we call split type and separable type. As a consequence, we provide complete classifications for maximal subalgebras of semisimple algebras and basic algebras. We also demonstrate that the maximality of $A$ in $B$ is related to the representation theory of $B$, through the separability of functors closely associated with the extension $A \subset B$.
The rest of this document showcases applications of these results. For $k = \bar{k}$, we compute the maximal dimension of a proper subalgebra of $B$. We discuss the problem of computing the minimal number of generators for $B$ (as an algebra), and provide upper and lower bounds for basic algebras. We then study $\operatorname{AlgGr}_d(B)$ in detail, again when $B$ is basic. When $d = \dim_kB-1$, we find a projective embedding of $\operatorname{AlgGr}_d(B)$, and explicitly describe its associated homogeneous vanishing ideal. In turn, we provide a simple description of its irreducible components. We find equivalent conditions for this variety to be a finite union of $\operatorname{Aut}_k(B)$-orbits, and describe several classes of algebras which satisfy these conditions. Furthermore, we provide an algebraic description for the orbits of connected maximal subalgebras of type-$\mathbb{A}$ path algebras. Finally, we study the fixed-point variety $\operatorname{AlgGr}_d(B)^{\operatorname{Aut}_k(B)}$ (for general $d$), which connects naturally to the representation theory of $\operatorname{Aut}_k(B)$. We investigate the case where $B$ is a truncated path algebra over $\mathbb{C}$ in detail.
| Identifer | oai:union.ndltd.org:uiowa.edu/oai:ir.uiowa.edu:etd-8391 |
| Date | 01 May 2019 |
| Creators | Sistko, Alexander Harris |
| Contributors | Iovanov, Miodrag C. |
| Publisher | University of Iowa |
| Source Sets | University of Iowa |
| Language | English |
| Detected Language | English |
| Type | dissertation |
| Format | application/pdf |
| Source | Theses and Dissertations |
| Rights | Copyright © 2019 Alexander Harris Sistko |
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