On Degenerations and Deformations of Sklyanin Algebras

A subfield of noncommutative algebra, entitled noncommutative projective algebraic geometry, was launched in the 1980s through Michael Artin, John Tate, William Schelter, and Michel van den Bergh's classification of noncommutative projective spaces. The toughest challenge was the investigation of three-dimensional Sklyanin algebras, denoted Skly3. The ring-theoretic and homological behavior of Skly3 could not be determined using purely algebraic techniques and the geometry of smooth projective curves was surprisingly used to achieve the desired classification results. In this dissertation, we employ variants of Skly3 to broaden both the class of non-commutative algebras and the type of geometric data featured in the theory of this rapidly expanding field.;Traditional techniques of this area study noncommutative graded algebras with use of projective geometric data of finite type. Even so, our research objectives include: (1) the analysis of degenerate three-dimensional Sklyanin algebras, Sdeg, which are associated to geometric data not of finite type, and (2) the study of the representation theory of ungraded deformations, Sdef, of Skly3.;We have completed problem (1), details of which are provided in Chapter 3 of this dissertation. Namely, we conclude that the geometry of Sdeg yields an analogue to a result of Artin-Tate-van den Bergh, namely that a generalized twisted homogeneous coordinate ring arises as a factor of Sdeg. This has resulted in several open problems regarding noncommutative graded rings and corresponding noncommutative coordinate rings; such further directions are also discussed in Chapter 3.;We only give a partial answer to problem (2), yet this is still sufficient as the work has several consequences not only for the advancement of the representation theory of noncommutative algebras, but also for Berenstein et al.'s study of noncommutative vacua in string theory. In particular, we present results on the dimensions of (not necessarily graded) simple finite-dimensional modules of Skly3 in Chapter 4. We also analyze the representation theory of a central extension D of Skly 3 in Chapter 5 by investigating the existence of fat point modules over D and the structure of the center of D. This is all with a view towards understanding the irreducible finite-dimensional representations of Sdef.