The first Weyl algebra, A, is naturally Z-graded by letting deg x = 1 and deg y = -1. Sue Sierra studied gr-A, the category of graded right A-modules, computing its Picard group and classifying all rings graded equivalent to A. Paul Smith showed that ,in fact, gr-A is equivalent to the category of quasicoherent sheaves on a certain quotient stack.

In this dissertation, we generalize results of Sierra and Smith by studying the graded module category of certain generalized Weyl algebras. We show that for a generalized Weyl algebra A(f) with base ring k[z] defined by a quadratic polynomial f, the Picard group of gr-A(f) is isomorphic to the Picard group of gr-A. For each A(f), we also construct a commutative ring whose graded module category is equivalent to the quotient category qgr-A(f), the category gr-A(f) modulo its full subcategory of finite-dimensional modules.

In order to study AS-regular algebras of dimension 5, we consider dimension 5 graded iterated Ore extensions generated in degree one. We present an interesting example of an Ore extension with two generators and a degree type first discussed by Floystad and Vatne. We classify the possible degrees of relations and structure of the free resolution for extensions with 3 and 4 generators. We show that every known type of algebra of dimension 5 can be realized by an Ore extension and we consider which of these types cannot be realized by an enveloping algebra. We then investigate the possible bigrading of Ore extensions with degree types that cannot be realized by an enveloping algebra and show that there is no AS-regular algebra with minimal relations of degrees 2, 2, and 3.

Metric approximable groups have been studied since the introduction of sofic groups byGromov [ 13]. Since then, further classes of metric approximable groups have been studied, such as hyperlinear [24], linearly sofic [4 ], and weakly sofic groups [11]. Due to their connection with many open problems such as Gottschalk’s Surjunctivity Conjecture [ 12], Connes’ Embedding Problem [7], and Kaplansky’s Direct Finiteness Conjecture [ 18], metric approximable groups have generated much interest. Recently, metric approximability has been extended from groups to associative algebras through linearly sofic associative algebras [ 4]. These algebras were shown to have many similar properties as linearly sofic groups, such as equivalent characterization through metric x ultraproducts and almost representations. Additionally, non-linearly sofic algebras proved easier to find as compared to the case of linearly sofic groups. However, certain properties that hold for groups have not been shown to hold for associative algebras, such as preservation of linear soficity through certain extensions. In this dissertation, we continue the work in [4 ] by extending the definition of linear soficity further to Lie algebras. Lie algebras are a natural object to study in this area, as they have many similarities to both groups and associative algebras. In §3, we define linear soficity for Lie algebras using metric ultraproducts, as well as give an equivalent characterization through the use of almost representations. We also give some examples of linearly sofic Lie algebras. In §4, we show the connection between linear soficity in Lie and associative algebras by showing that, over fields of characteristic 0, a Lie algebra is linearly sofic if and only if its universal enveloping algebra is. In §5, we look at extensions of linearly sofic Lie algebras. We show that any extension of a linearly sofic Lie algebra by a Lie algebra with an amenable universal enveloping algebra is linearly sofic, as in the case of groups. In §6, we use wreath products to show that a countable metric approximable group is embeddable in a finitely generated metric approximable group of the same kind. In addition, we use a similar argument using wreath products to show that a countable dimensional linearly sofic Lie algebra is embeddable in a finitely generated linearly sofic Lie algebra.

Studying invariant theory of commutative polynomial rings has motivated many developments in commutative algebra and algebraic geometry. For a finite group acting on a polynomial ring, the remarkable Chevalley-Shephard-Todd Theorem proves that the fixed subring is isomorphic to a polynomial ring if and only if the group is generated by pseudo-reflections. Another interesting question is to find properties of a fixed ring for a group action satisfying certain attributes. In recent years, progress was made in work of Jing, J{\o}rgensen, Kirkman, Kuzmanovich, Walton, Zhang, and others to extend the theory to regular algebras which are a noncommutative generalization of polynomial rings.

Naturally, the question arises if the theory generalizes further to non-connected noncommutative algebras. Our objects of study will be preprojective algebras which are certain factor algebras of path algebras corresponding to extended Dynkin diagrams of type $A$, $D$ or $E$. This dissertation answers the question what conditions need to be satisfied by the fixed ring in order to make a rich theory possible. Moreover, we give a sufficient condition on the finite group acting on a preprojective algebra to guarantee that the fixed ring has finite injective dimension and satisfies a generalized Gorenstein condition. Part of this result is the construction of a homological determinant of a non-connected algebra which turns out to be particularly nice for the examined preprojective algebras.