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Results of Vancliff, Van Rompay and Willaert in 1998 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. In 2010, Cassidy and Vancliff generalized the notion of a GCA to that of a graded skew Clifford algebra (GSCA). The results in this article show that prior results may be extended, with suitable modification, to GSCAs. In particular, using the notion of {\mu}-rank introduced recently by the authors, the point modules over a regular GSCA are determined by (noncommutative) quadrics of {\mu}-rank at most two that belong to the noncommutative quadric system associated to the GSCA.

In 2010, Cassidy and Vancliff extended the notion of a quadratic form on n generators to the noncommutative setting. In this article, we suggest a notion of 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, our methods entail rewriting an arbitrary quadratic form as a sum of products. In so doing, we find analogs for 2 x 2 minors and determinant of a 3 x 3 matrix in this noncommutative setting.

We introduce a generalization, called a skew Clifford algebra, of a Clifford algebra, and relate these new algebras to the notion of graded skew Clifford algebra that was defined in 2010. In particular, we examine homogenizations of skew Clifford algebras, and determine which skew Clifford algebras can be homogenized to create Artin-Schelter regular algebras. Just as (classical) Clifford algebras are the Poincar\' e-Birkhoff-Witt (PBW) deformations of exterior algebras, skew Clifford algebras are the $\mathbb{Z}_2$-graded PBW deformations of quantum exterior algebras. We also determine the possible dimensions of skew Clifford algebras and provide several examples.

In the 1990s, in work of Le Bruyn and Smith and in work of Le Bruyn and Van den Bergh, it was proved that point modules and line modules over the homogenization of the universal enveloping algebra of a finite-dimensional Lie algebra describe useful data associated to the Lie algebra. In particular, in the case of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, there is a correspondence between Verma modules and certain line modules that associates a pair $(\mathfrak{h},\,\phi)$, where $\mathfrak{h}$ is a two-dimensional Lie subalgebra of $\mathfrak{sl}_2(\mathbb{C})$ and $\phi \in \mathfrak{h}^*$ satisfies $\phi([\mathfrak{h}, \, \mathfrak{h}]) = 0$, to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra $\mathfrak{sl}(1|1)$ and for a color Lie algebra associated to the Lie algebra $\mathfrak{sl}_2$.

Let $k$ be an algebraically closed field and ${\sf G}(2,k^4)$ the Grassmannian of 2-planes in $k^4$. We associate to each 6-dimensional subspace $R$ of the space of 4x4 matrices over $k$ a closed subscheme ${\bf X}_R \subseteq {\sf G}(2,k^4)$. We show that each irreducible component of ${\bf X}_R$ has dimension at least one and when ${\rm dim}({\bf X}_R)=1$, then ${\rm deg}({\bf X}_R)=20$ where degree is computed with respect to the ambient ${\mathbb P}^5$ under the Pl\"ucker embedding ${\sf G}(2,k^4) \to {\mathbb P}^5$. We give two examples involving elliptic curves: in one case ${\bf X}_R$ is the secant variety for a quartic elliptic curve, so ${\rm dim}({\bf X}_R)=2$, in the other ${\bf X}_R$ is a curve having 7 irreducible components, three of which are elliptic curves, and four of which are smooth conics., Comment: minor revsion after referee comments

This article is based on a talk given by the author at MSRI in the workshop "Connections for Women" in January 2013, while being a part of the program "Noncommutative Algebraic Geometry and Representation Theory" at MSRI. One purpose of the exposition is to motivate and describe the geometric techniques introduced by M. Artin, J. Tate and M. Van den Bergh in the 1980s at a level accessible to graduate students. Additionally, some advances in the subject since the early 1990s are discussed, including a recent generalization of complete intersection to the noncommutative setting, and the notion of graded skew Clifford algebra and its application to classifying quadratic regular algebras of global dimension at most three. The article concludes by listing some open problems.

A quantum ${\mathbb P}^3$ is a noncommutative analogue of a polynomial ring on four variables, and, herein, it is taken to be a regular algebra of global dimension four. It is well known that if a generic quadratic quantum ${\mathbb P}^3$ exists, then it has a point scheme consisting of exactly twenty distinct points and a one-dimensional line scheme. In this article, we compute the line scheme of a family of algebras whose generic member is a candidate for a generic quadratic quantum ${\mathbb P}^3$. We find that, as a closed subscheme of ${\mathbb P}^5$, the line scheme of the generic member is the union of seven curves; namely, a nonplanar elliptic curve in a ${\mathbb P}^3$, four planar elliptic curves and two nonsingular conics.

We prove that if $A$ is a regular graded skew Clifford algebra and is a twist of a regular graded Clifford algebra $B$ by an automorphism, then the subalgebra of $A$ generated by a certain normalizing sequence of homogeneous degree-two elements is a twist of a polynomial ring by an automorphism, and is a skew polynomial ring. We also present an example that demonstrates that this can fail when $A$ is not a twist of $B$., Comment: 7 Pages

In recent work of T. Cassidy and the author, a notion of complete intersection was defined for (non-commutative) regular skew polynomial rings, defining it using both algebraic and geometric tools, where the commutative definition is a special case. In this article, we extend the definition to a larger class of algebras that contains regular graded skew Clifford algebras, the coordinate ring of quantum matrices and homogenizations of universal enveloping algebras. Regular algebras are often considered to be non-commutative analogues of polynomial rings, so the results herein support that viewpoint., Comment: This paper replaces the preprint "Defining the Notion of Complete Intersection for Regular Graded Skew Clifford Algebras", and also has a paragraph written correctly that is garbled by the publisher in the published version (paragraph after Example 3.8)

We prove that quadratic regular algebras of global dimension three on degree-one generators are related to graded skew Clifford algebras. In particular, we prove that almost all such algebras may be constructed as a twist of either a regular graded skew Clifford algebra or of an Ore extension of a regular graded skew Clifford algebra of global dimension two. In so doing, we classify all quadratic regular algebras of global dimension three that have point scheme either a nodal cubic curve or a cuspidal cubic curve in P^2.