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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 Poincaré-Birkhoff-Witt (PBW) deformations of exterior algebras, skew Clifford algebras are the Z 2 -graded PBW deformations of quantum exterior algebras. We also determine the possible dimensions of skew Clifford algebras and provide several examples. [ABSTRACT FROM AUTHOR]

The attempted classification of regular algebras of global dimension four, so-called quantum P 3 s, 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 P 3 . 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 P 5 via the Plücker embedding, it is the union of one nonplanar elliptic curve in a P 3 , 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. [ABSTRACT FROM AUTHOR]

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 5 , 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. [ABSTRACT FROM AUTHOR]

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 subalgebraC, while a GSCA is a finite module over a (typically noncommutative) analogous subalgebraR. We consider the case that a regular GSCA is a twist of a GCA by an automorphism, and we prove, in this case,Ris a skew polynomial ring and a twist ofCby an automorphism. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: 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 . [Copyright &y& Elsevier]

Abstract: We present an algebro-geometric technique for constructing regular Clifford algebras A of global dimension four with associated point scheme consisting of a prespecified finite number of points. In particular, if A has more than one point in its point scheme, then the number of points in the point scheme can be obtained from the number of intersection points of two planar cubic divisors; these cubic divisors correspond to regular Clifford subalgebras of A of global dimension three. If A has exactly a finite number, n, of distinct points in its point scheme, then and all these possibilities occur. We also prove that if a regular Clifford algebra R of global dimension has exactly a finite number, n, of distinct isomorphism classes of point modules, then n is odd if and only if R is an Ore extension of a regular Clifford subalgebra of R of global dimension . [Copyright &y& Elsevier]

Abstract: A result of M. Artin, J. Tate and M. Van den Bergh asserts that a regular algebra of global dimension three is a finite module over its center if and only if the automorphism encoded in the point scheme has finite order. We prove that the analogous result for a regular algebra of global dimension four is false by presenting families of quadratic, noetherian regular algebras A of global dimension four such that (i) A is an infinite module over its center, (ii) A has a finite point scheme, which is the graph of an automorphism of finite order, and (iii) A has a one-parameter family of line modules. Such algebras are candidates for generic regular algebras of global dimension four. The methods used to construct the algebras provide new techniques for creating other potential candidates. [Copyright &y& Elsevier]