Class Of Noncommutative Algebras Research Articles

Cancellation of Morita and skew types

We study both Morita cancellative and skew cancellative properties of noncommutative algebras as initiated recently in several papers and explore which classes of noncommutative algebras are Morita cancellative (respectively, skew cancellative). Several new results concerning these two types of cancellations, as well as the classical cancellation, are proved.

Noncommutative Auslander theorem

In the 1960s Maurice Auslander proved the following important result. Let $R$ be the commutative polynomial ring $\mathbb {C}[x_1,\dots ,x_n]$, and let $G$ be a finite small subgroup of $\textrm {GL}_n(\mathbb {C})$ acting on $R$ naturally. Let $A$ be the fixed subring $R^G:=\{a\in R|g(a)=a \text { for all } g\in G \}$. Then the endomorphism ring of the right $A$-module $R_A$ is naturally isomorphic to the skew group algebra $R\ast G$. In this paper, a version of the Auslander theorem is proven for the following classes of noncommutative algebras: (a) noetherian PI local (or connected graded) algebras of finite injective dimension, (b) universal enveloping algebras of finite-dimensional Lie algebras, and (c) noetherian graded down-up algebras.

Computation of Non-Commutative Gröbner Bases in Grassmann and Clifford Algebras

Tensor, Clifford and Grassmann algebras belong to a wide class of non-commutative algebras that have a Poincare-Birkhoff-Witt (PBW) “monomial” basis. The necessary and sufficient condition for an algebra to have the PBW basis has been established by T. Mora and then V. Levandovskyy as the so called “non-degeneracy condition”. This has led V. Levandovskyy to a re-discovery of the so called G-algebras (previously introduced by J. Apel) and GR-algebras (Grobner-ready algebras). It was T. Mora who already in the 1990s considered a comprehensive and algorithmic approach to Grobner bases for commutative and non-commutative algebras. It was T. Stokes who eighteen years ago introduced Grobner left bases (GLB) and Grobner left ideal bases (GLIB) for left ideals in Grassmann algebras, with the GLIB bases solving an ideal membership problem. Thus, a natural question is to first seek Grobner bases with respect to a suitable admissible monomial order for ideals in tensor algebras T and then consider quotient algebras T/I. It was shown by Levandovskyy that these quotient algebras possess a PBW basis if and only if the ideal I has a Grobner basis. Of course, these quotient algebras are of great interest because, in particular, Grassmann and Clifford algebras of a quadratic form arise this way. Examples of G-algebras include the quantum plane, universal enveloping algebras of finite dimensional Lie algebras, some Ore extensions, Weyl algebras and their quantizations, etc. Examples of GR-algebras, which are either G algebras or are isomorphic to quotient algebras of a G-algebra modulo a proper two-sided ideal, include Grassmann and Clifford algebras. After recalling basic concepts behind the theory of commutative Grobner bases, a review of the Grobner bases in PBW algebras, G-,and GR-algebras will be given with a special emphasis on computation of such bases in Grassmann and Clifford algebras. GLB and GLIB bases will also be computed.

Linear equations over noncommutative graded rings

We call a graded connected algebra R effectively coherent, if for every linear equation over R with homogeneous coefficients of degrees at most d, the degrees of generators of its module of solutions are bounded by some function D ( d ) . For commutative polynomial rings, this property has been established by Hermann in 1926. We establish the same property for several classes of noncommutative algebras, including the most common class of rings in noncommutative projective geometry, that is, strongly Noetherian rings, which includes Noetherian PI algebras and Sklyanin algebras. We extensively study so-called universally coherent algebras, that is, such that the function D ( d ) is bounded by 2 d for d ≫ 0 . For example, finitely presented monomial algebras belong to this class, as well as many algebras with finite Groebner basis of relations.

HOMOLOGICAL PROPERTIES OF ASSOCIATIVE ALGEBRAS: THE METHOD OF HELICES

Homological properties of associative algebras arising in the theory of helices are studied. A class of noncommutative algebras is introduced in which it is natural (from the viewpoint of the theory of helices) to deform projective spaces and also certain Fano varieties. It is shown that in the case of deformations of the projective plane this approach leads to algebras associated with automorphisms of two-dimensional cubic curves.

Almost commutative algebra and differential calculus on the quantum hyperplane

A notion of almost commutative algebra is given that makes it possible to extend differential geometric ideas associated with commutative algebras in a simple manner to certain classes of noncommutative algebras. As an example differential calculus on the N-dimensional quantum hyperplane is discussed.