Point Modules Over Regular Graded Skew Clifford Algebras

Abstract

In this thesis, I consider point modules over regular graded skew Clifford algebras.First, I define a notion of rank (called μ-rank) on noncommutative quadratic forms. To every (commutative) quadratic form is associated a symmetric matrix, and one has the standard notions of rank and determinant function defined on the matrix, and, thus, on the quadratic form. In 2010, in [15], the notion of quadratic form was extended to the noncommutative setting and a one-to-one correspondence was established between these quadratic forms and certain matrices. Using this generalization, I define a notion of rank (called μ-rank) for such noncommutative quadratic forms, where n = 2 or 3. Since writing an arbitrary quadratic form as a sum of squares fails in this context, my methods entail rewriting an arbitrary quadratic form as a sum of products. In so doing, I find analogs for 2 x 2 minors and determinant of a 3 x 3 matrix in this noncommutative setting. Second, I use the μ-rank of a noncommutative quadratic form to determine the point modules over regular graded skew Clifford algebras. Results of Vancliff, Van Rompay and Willaert in 1998 [VVW] prove that point modules over a regular graded Clifford algebra (GCA) are determined by (commutative) quadrics of rank at most two that belong to the quadric system associated to the GCA. The results in this thesis show that the results of [VVW] may be extended, with suitable modification, to GSCAs. In particular, using the notion of μ-rank, the point modules over a regular GSCA are determined by (noncommutative) quadrics of μ-rank at most two that belong to the noncommutative quadric system associated to the GSCA.