x, 68 p. : ill. (some col.) / Motivated by the Adams spectral sequence for computing stable homotopy groups, Priddy defined a class of algebras called Koszul algebras with nice homological properties. Many important algebras arising naturally in mathematics are Koszul, and the Koszul property is often tied to important structure in the settings which produced the algebras. However, the strong defining conditions for a Koszul algebra imply that such algebras must be quadratic. A very natural generalization of Koszul algebras called K 2 algebras was recently introduced by Cassidy and Shelton. Unlike other generalizations of the Koszul property, the class of K 2 algebras is closed under many standard operations in ring theory. The class of K 2 algebras includes Artin-Schelter regular algebras of global dimension 4 on three linear generators as well as graded complete intersections. Our work comprises two distinct projects. Each project was motivated by an aspect of the theory of Koszul algebras which we regard as sufficiently powerful or fundamental to warrant an interpretation for K 2 algebras. A very useful theorem due to Backelin and Fröberg states that if A is a Koszul algebra and I is a quadratic ideal of A which is Koszul as a left A -module, then the factor algebra A/I is a Koszul algebra. We prove that if A is Koszul algebra and A I is a K 2 module, then A/I is a K 2 algebra provided A/I acts trivially on Ext A ( A/I,k ). As an application of our theorem, we show that the class of sequentially Cohen-Macaulay Stanley-Reisner rings are K 2 algebras and we give examples that suggest the class of K 2 Stanley-Reisner rings is actually much larger. Another important recent development in ring theory is the use of A ∞ -algebras. One can characterize Koszul algebras as those graded algebras whose Yoneda algebra admits only trivial A ∞ -structure. We show that, in contrast to the situation for Koszul algebras, vanishing of higher A ∞ -structure on the Yoneda algebra of a K 2 algebra need not be determined in any obvious way by the degrees of defining relations. We also demonstrate that obvious patterns of vanishing among higher multiplications cannot detect the K 2 property. This dissertation includes previously unpublished co-authored material. / Committee in charge: Dr. Brad Shelton, Chair; Dr. Victor Ostrik, Member; Dr. Nicholas Proudfoot, Member; Dr. Arkady Vaintrob, Member; Dr. David Boush, Outside Member

xi, 95 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We investigate some homological properties of graded algebras. If A is an R -algebra, then E (A) := Ext A ( R, R ) is an R-algebra under the cup product and is called the Yoneda algebra. (In most cases, we assume R is a field.) A well-known and widely-studied condition on E(A) is the Koszul property. We study a class of deformations of Koszul algebras that arises from the study of equivariant cohomology and algebraic groups and show that under certain circumstances these deformations are Poincaré-Birkhoff-Witt deformations. Some of our results involve the [Special characters omitted] property, recently introduced by Cassidy and Shelton, which is a generalization of the Koszul property. While a Koszul algebra must be quadratic, a [Special characters omitted] algebra may have its ideal of relations generated in different degrees. We study the structure of the Yoneda algebra corresponding to a monomial [Special characters omitted.] algebra and provide an example of a monomial [Special characters omitted] algebra whose Yoneda algebra is not also [Special characters omitted]. This example illustrates the difficulty of finding a [Special characters omitted] analogue of the classical theory of Koszul duality. It is well-known that Poincaré-Birkhoff-Witt algebras are Koszul. We find a [Special characters omitted] analogue of this theory. If V is a finite-dimensional vector space with an ordered basis, and A := [Special characters omitted] (V)/I is a connected-graded algebra, we can place a filtration F on A as well as E (A). We show there is a bigraded algebra embedding Λ: gr F E (A) [Special characters omitted] E (gr F A ). If I has a Gröbner basis meeting certain conditions and gr F A is [Special characters omitted], then Λ can be used to show that A is also [Special characters omitted]. This dissertation contains both previously published and co-authored materials. / Committee in charge: Brad Shelton, Chairperson, Mathematics; Victor Ostrik, Member, Mathematics; Christopher Phillips, Member, Mathematics; Sergey Yuzvinsky, Member, Mathematics; Van Kolpin, Outside Member, Economics