Related papers: Burchnall-Chaundy theory for Ore extensions
We build resolutions for general twisted tensor products of algebras. These bimodule and module resolutions unify many constructions in the literature and are suitable for computing Hochschild (co)homology and more generally Ext and Tor for…
The numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of complexes (algebras) are introduced. Cohomological range leads to the concepts of derived bounded algebras and…
We survey ultradifferentiable extension theorems, i.e., quantitative versions of Whitney's classical extension theorem, with special emphasis on the existence of continuous linear extension operators. The focus is on Denjoy-Carleman classes…
First and second fundamental theorems are given for polynomial invariants of a class of pseudo-reflection groups (including the Weyl groups of type $B_n$), under the assumption that the order of the group is invertible in the base field.…
Generalizations of the (rank 1) Bannai-Ito algebra are obtained from a refinement of the grade involution of the Lie super algebra $\mathfrak{osp}(1,2)$. A hyperoctahedral extension is derived by using a realization of $\mathfrak{osp}(1,2)$…
A double Ore extension is a natural generalization of the Ore extension. We prove that a connected graded double Ore extension of an Artin-Schelter regular algebra is Artin-Schelter regular. Some other basic properties such as the…
We obtain similar types of conclusions as that of Br\"{u}ck [1] for two differential polynomials which in turn radically improve and generalize several existing results. Moreover, a number of examples have been exhibited to justify the…
An expansion formula of a new type with the rest term of Cauchy type is derived in the operator formulation of generalized umbral calculus
We develop deformation theory of algebras over quadratic operads where the parameter space is a commutative local algebra. We also give a construction of a distinguised deformation of an algebra over a quadratic operad with a complete local…
In this paper we investigate extended modules for a special class of Ore extensions. We will assume that $R$ is a ring and $A$ will denote the Ore extension $A:=R[x_1,\dots,x_n;\sigma]$ for which $\sigma$ is an automorphism of $R$,…
In this article, we study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings $R[x;\delta]$, under the hypothesis that $R$ is $s$-unital and $\ker(\delta)$…
We investigate the behavior of extension monads, introduced in the 1990s by the second author, in terms of structure results for infinitely many finitary operations and common constructions in varieties or categories of algebras.…
Using algebraic transformations and equivalent reformulations we derive a number of new results from some earlier ones (by the author) in more accepted terms closely related to well-known conjectures of Bondy and Jung including a number of…
All the basic cohomology groups and reduced cohomology groups of the extended Schr\"odinger-Virasoro conformal algebra with trivial coefficients are completely determined. In particular, we introduce the notion of the relative cohomology of…
In this paper, which is a follow-up of our first paper "Normal forms for ordinary differential operators, I", we extend the theory of normal forms for non-commuting operators, and obtain as an application a commutativity criterion for…
We consider the deconstruction/reconstruction of extensions in varieties of algebras which are modules expanded by multilinear operators. The parametrization of extensions determined by abelian ideals with unary actions agrees with the…
In this paper we prove the universal property of skew $PBW$ extensions generalizing this way the well known universal property of skew polynomial rings. For this, we will show first a result about the existence of this class of…
Recurrent relations for branching coefficients are based on a special type of singular element decomposition. We show that this decomposition can be used to construct the parabolic Verma modules and finally to obtain the generalized…
We study the operator product expansions in the chiral algebra $\mathcal{W}_{\infty}$, first using the associativity conditions in the basis of primary generating fields and second using a different basis coming from the free field…
This paper introduces and systematically studies Weyl-type, Witt-type, and non-associative algebras defined over expolynomial rings -- commutative rings generated by exponential functions $e^{\alpha x}$, exponentials of exponentials $e^{\pm…