Classifying Quadratic Quantum P^2s by using Graded Skew Clifford Algebras

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

Conjugacy of Cartan subalgebras of complex finite dimensional Leibniz algebras

In the present work the properties of Cartan subalgebras and their connection with regular elements in finite dimensional Lie algebras are extended to the case of Leibniz algebras. It is shown that Cartan subalgebras and regular elements of a Leibniz algebra correspond to Cartan subalgebras and regular elements of a Lie algebra by a natural homomorphism. Conjugacy of Cartan subalgebras of Leibniz algebras is shown.Comment: 11 page

On the Notion of Complete Intersection outside the Setting of Skew Polynomial Rings

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