The non-commutative algebraic geometry of some quadratic algebras

The purpose of this dissertation is two-fold. Firstly, we apply the techniques of non-commutative algebraic geometry developed by Artin, Tate and Van den Bergh to certain quantum groups; and, secondly, we study the relationship between non-commutative algebraic geometry and symplectic geometry for non-commutative algebras which are graded deformations of commutative algebras.; There are four chapters to this dissertation. In chapter 1 we define a class of algebras depending on some geometric data. These algebras satisfy the "good" homological properties shared by polynomial rings. In contrast, in chapter 2 we study an algebra that does not satisfy these "good" homological properties. However for both kinds of algebras, those in chapter 1 and those in chapter 2, we show that many of their algebraic properties are determined by the corresponding geometric data. The algebras in chapter 1 are quadratic algebras on 4 generators depending on a nonsingular quadric Q in {dollar}IPsp3{dollar}, a line L in {dollar}IPsp3{dollar} and an automorphism {dollar}sigma{dollar} of Q {dollar}cup{dollar} L. We show that the point modules over an algebra in this family are parametrized by Q {dollar}cup{dollar} L and that the line modules are parametrized by the lines in {dollar}IPsp3{dollar} that either lie on Q or meet L. This family contains {dollar}{lcub}cal O{rcub}sb{lcub}q{rcub}(Msb2{dollar}), the coordinate ring of quantum 2 {dollar}times{dollar} 2 matrices; the quantum determinant is the unique (up to a scalar multiple) homogeneous element of degree 2 in {dollar}{lcub}cal O{rcub}sb{lcub}q{rcub}(Msb2{dollar}) that vanishes on the graph of {dollar}sigmamid sb{lcub}Q{rcub}{dollar} but not on the graph of {dollar}sigmamidsb{lcub}L{rcub}{dollar}. The algebra in chapter 2 is a quadratic algebra on 4 generators with six defining relations. We show that the point modules over this algebra are parametrized by a twisted cubic curve C in {dollar}IPsp3{dollar}, and that C determines a certain nilpotent ideal of the algebra.; Chapter 3 focuses on {dollar}{lcub}cal O{rcub}sb{lcub}q{rcub}(Msb{lcub}n{rcub}{dollar}), viewed as a graded flat deformation of a polynomial ring, whereas chapter 4 considers more general graded flat deformations A(q) of a polynomial ring. The purpose of these two chapters is to compare the information obtained from the two kinds of geometric data--algebro-geometric versus symplectic. We show that if the variety parametrizing the point modules (respectively one-dimensional) modules over A(q) is independent of q for all but finitely many q, then it is contained in the variety consisting of the projective (respectively affine) zero-dimensional symplectic leaves. In the case of {dollar}{lcub}cal O{rcub}sb{lcub}q{rcub}(Msb{lcub}n{rcub}){dollar}, if {dollar}qsp2 not={dollar} 1, these varieties coincide.