The classification of quantum $\mathbb{P}^2$s was completed by M. Artin et al. decades ago, but the classification of quadratic algebras that are viewed as quantum $\mathbb{P}^3$s is still an open problem. Based on work of M. Van den Bergh, it is believed that a ``generic'' quadratic quantum $\mathbb{P}^3$ should have a finite point scheme and a one-dimensional line scheme. Two families of quadratic quantum $\mathbb{P}^3$s with these geometric properties are presented herein, where each family member has a line scheme that is either a union of lines or is a union of a line, a conic and a curve. Moreover, we prove that, under certain conditions, if $A$ is a quadratic quantum $\mathbb{P}^3$ that contains a subalgebra $B$ that is a quadratic quantum $\mathbb{P}^2$, then the point scheme of $B$ embeds in the line scheme of $A$.

Although it has been known for decades that a quadratic AS-regular algebra of global dimension n n need not have any point modules if n ≥ 5 n \geq 5 , an explicit example appears to be absent, prior to the present article, from the published literature. This situation was brought to the attention of the author by the anonymous referee of D. Rogalski’s survey article Artin-Schelter Regular Algebras , who encouraged the author of the present article to remedy the situation. Consequently, an explicit example is presented herein of a quadratic AS-regular algebra of global dimension five that is a graded Clifford algebra having no point modules.

In 1996, J. J. Zhang introduced the concept of twisting a graded algebra by a twisting system, which generalizes the concept of twisting a graded algebra by an automorphism (the latter concept having been introduced in an article by M. Artin, J. Tate and M. Van den Bergh in 1991). Zhang proved that twisting using a twisting system is an equivalence relation and that certain important algebraic properties are invariant under twisting. We call a twisting system nontrivial if it is not given by an automorphism. However, there are very few examples of nontrivial twisting systems in the literature. In 1997, the second author and K. Van Rompay and, in 1999, B. Shelton and the second author were successful in finding one example each of a nontrivial twisting system. Their twisting systems were constructed on certain quadratic algebras A A (on four generators) using two invertible linear maps t t and τ \tau of A 1 A_1 that satisfy t 2 = t^2 = identity and τ 2 ∈ Aut ( A ) \tau ^2 \in \text {Aut}(A) . We extend their work on twisting systems to any finitely generated quadratic algebra B B using analogous maps that satisfy t n = t^n = identity and τ n ∈ Aut ( B ) \tau ^n \in \text {Aut}(B) , for some n ∈ N n \in \mathbb {N} . We illustrate our new method for producing a nontrivial twisting system on a certain quadratic algebra that is a quantum P 3 \mathbb {P}^3 and whose point scheme is isomorphic to a rank-2 quadric in P 3 \mathbb {P}^3 . We prove that our algebra is not determined by the zero locus of its defining relations.

This article is based on a talk given by the author at the 12th International Conference on Clifford Algebras and their Applications in Mathematical Physics. A generalization, introduced by Cassidy and the author, of a classical Clifford algebra is discussed together with connections between that generalization and a generalization of a graded Clifford algebra. A geometric approach to studying the algebras, viewed through the lens of Artin, Tate and Van den Bergh’s noncommutative algebraic geometry, is also presented.

Let R R denote a 6-dimensional subspace of the ring M 4 ( k ) M_4(\Bbbk ) of 4 × 4 4 \times 4 matrices over an algebraically closed field k \Bbbk . Fix a vector space isomorphism M 4 ( k ) ≅ k 4 ⊗ k 4 M_4(\Bbbk ) \cong \Bbbk ^4 \otimes \Bbbk ^4 . We associate to R R a closed subscheme X R {\mathbf X}_R of the Grassmannian of 2-dimensional subspaces of k 4 \Bbbk ^4 , where the reduced subscheme of X R {\mathbf X}_R is the set of 2-dimensional subspaces Q ⊆ k 4 Q \subseteq \Bbbk ^4 such that ( Q ⊗ k 4 ) ∩ R ≠ { 0 } (Q \otimes \Bbbk ^4) \cap R \ne \{ 0\} . Our main result is that if X R {\mathbf X}_R has minimal dimension (namely, one), then its degree is 20 when it is viewed as a subscheme of P 5 \mathbb {P}^5 via the Plücker embedding. We present several examples of X R \mathbf X_R that illustrate the wide range of possibilities for it; there are reduced and non-reduced examples. Two examples involve elliptic curves: in one case, X R {\mathbf X}_R is a P 1 \mathbb {P}^1 -bundle over an elliptic curve the second symmetric power of the curve; in the other, it is a curve having seven irreducible components, three of which are quartic elliptic space curves, and four of which are smooth plane conics. These two examples arise naturally from a problem having its roots in quantum statistical mechanics. The scheme X R \mathbf X_R appears in non-commutative algebraic geometry: under appropriate hypotheses, it is isomorphic to the line scheme L \mathcal {L} of a certain graded algebra determined by R R . In that context, it has been an open question for several years to describe such L \mathcal {L} of minimal dimension, i.e., those L \mathcal {L} of dimension one. Our main result implies that if dim ⁡ ( L ) = 1 \dim (\mathcal {L}) = 1 , then, as a subscheme of P 5 \mathbb {P}^5 under the Plücker embedding, deg ⁡ ( L ) = 20 \deg (\mathcal {L}) = 20 .

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 ([5, 6]). In particular, in the case of the Lie algebra sl2(C), there is a correspondence between Verma modules and certain line modules that associates a pair (h, φ), where h is a two-dimensional Lie subalgebra of sl2(C) and φ ∈ h∗ satisfies φ([h, h]) = 0, to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra sl(1|1) and for a color Lie algebra associated to the Lie algebra sl2.

Inspired by the work of Le Bruyn and Smith in (

Inspired by the work of Le Bruyn and Smith in (Proc. Amer. Math. Soc. 118(3): 725–730, 1993) and the work of Shelton and Vancliff in (Comm. Alg. 30(5): 2535-2552, 2002), we analyze certain graded algebras related to the Lie algebra $\mathfrak {sl}(2,\Bbbk)$ using geometric techniques in the spirit of Artin, Tate and Van den Bergh. In particular, we discuss the point schemes and line schemes of certain quadratic quantum $\mathbb {P}^{3}$ s associated to the Lie superalgebra $\mathfrak {sl}(1|1)$ , to a quantized enveloping algebra, $\mathcal {U}_q(\mathfrak {sl}(2,\Bbbk))$ , of $\mathfrak {sl}(2,\Bbbk)$ , and to a color Lie algebra $\mathfrak {sl}_k(2,\Bbbk)$ , respectively. The geometry we consider identifies certain normal elements in the universal enveloping algebra of $\mathfrak {sl}(1|1)$ and in $\mathcal {U}_q(\mathfrak {sl}(2,\Bbbk))$ .

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 s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) , there is a correspondence between Verma modules and certain line modules that associates a pair ( h , ϕ ) (\mathfrak {h},\,\phi ) , where h \mathfrak {h} is a 2 2 -dimensional Lie subalgebra of s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) and ϕ ∈ h ∗ \phi \in \mathfrak {h}^* satisfies ϕ ( [ h , h ] ) = 0 \phi ([\mathfrak {h}, \, \mathfrak {h}]) = 0 , to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra s l ( 1 | 1 ) \mathfrak {sl}(1|1) and for a color Lie algebra associated to the Lie algebra s l 2 \mathfrak {sl}_2 .

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 sl(2)(C), there is a correspondence between Verma modules and certain line modules that associates a pair (h, phi), where h is a 2-dimensional Lie subalgebra of sl(2)(C) and phi is an element of h* satisfies phi([h, h]) = 0, to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra sl(1 vertical bar 1) and for a color Lie algebra associated to the Lie algebra sl(2).

Let R denote a 6-dimensional subspace of the ring M4(k) of 4 × 4 matrices over an algebraically closed field k. Fix a vector space isomorphism M4(k) ∼= k 4 ⊗ k. We associate to R a closed subscheme XR of the Grassmannian of 2-dimensional subspaces of k , where the reduced subscheme of XR is the set of 2-dimensional subspaces Q ⊆ k 4 such that (Q ⊗ k) ∩ R 6= {0}. Our main result is that if XR has minimal dimension (namely, one), then its degree is 20 when it is viewed as a subscheme of P via the Plücker embedding. We present several examples of XR that illustrate the wide range of possibilities for it; there are reduced and non-reduced examples. Two examples involve elliptic curves: in one case, XR is a P -bundle over an elliptic curve, the second symmetric power of the curve; in the other, it is a curve having seven irreducible components, three of which are quartic elliptic space curves, and four of which are smooth plane conics. These two examples arise naturally from a problem having its roots in quantum statistical mechanics. The scheme XR appears in non-commutative algebraic geometry: under appropriate hypotheses, it is isomorphic to the line scheme L of a certain graded algebra determined by R. In that context, it has been an open question for several years to describe such L of minimal dimension, i.e., those L of dimension one. Our main result implies that if dim(L) = 1, then, as a subscheme of P under the Plücker embedding, deg(L) = 20.

The attempted classification of regular algebras of global dimension four, so-called quantum P3s, has been a driving force for modern research in noncommutative algebra. Inspired by the work of Artin, Tate, and Van den Bergh, geometric methods via schemes of d-linear modules have been developed by various researchers to further their classification. In this work, we compute the line scheme of a certain family of algebras whose defining relations involve a scalar α such that almost all of the algebras in the family are considered candidates for a generic quadratic quantum P3. We find that, for almost all α, the algebras have a one-dimensional line scheme consisting of eight curves. Viewing the line scheme as a closed subscheme of P5 via the Plücker embedding, it is the union of one nonplanar elliptic curve in a P3, one nonplanar rational curve with a unique singular point, two planar elliptic curves, and two subschemes, each consisting of the union of a nonsingular conic and a line.

A generalization of the notion of symmetric matrix was introduced by Cassidy and Vancliff in 2010 and used by them in a construction that produces quadratic regular algebras of finite global dimension that are generalizations of graded Clifford algebras. In this article, we further their ideas by introducing a generalization of the matrix transpose map and use it to generalize the notion of skew-symmetric matrix. With these definitions, an analogue of the result that every n × n matrix is a sum of a symmetric matrix and a skew-symmetric matrix holds. We also prove an analogue of the result that the transpose map is an antiautomorphism of the algebra of n × n matrices, and show that the antiautomorphism property of our generalized transpose map is related to the notion of twisting the polynomial ring on n variables by an automorphism.

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 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 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 P-3. We find that, as a closed subscheme of P-3, the line scheme of the generic member is the union of seven curves; namely, a nonplanar elliptic curve in a P-3, four planar elliptic curves and two nonsingular conics. (C) 2015 Elsevier Inc. All rights reserved.

In 2010, a quantized analog of a graded Clifford algebra (GCA), called a graded skew Clifford algebra (GSCA), was proposed by Cassidy and Vancliff, and many properties of GCAs were found to have counterparts for GSCAs. In particular, a GCA is a finite module over a certain commutative subalgebra C, while a GSCA is a finite module over a (typically noncommutative) analogous subalgebra R. We consider the case that a regular GSCA is a twist of a GCA by an automorphism, and we prove, in this case, R is a skew polynomial ring and a twist of C by an automorphism.

In recent work of T. Cassidy and the author, a notion of complete intersection was defined for (noncommutative) 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 noncommutative analogues of polynomial rings, so the results herein support that viewpoint.

Results of Vancliff, Van Rompay and Willaert in 1998 [8] 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, in [4], 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 the results of [8] may be extended, with suitable modification, to GSCAs. In particular, using the notion of μ-rank introduced recently by the authors in [9], 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.

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.