Using computer algebra system SINGULAR:PLURAL for computations in noncommutative polynomial algebras

Levandovskyy, Viktor; Schönemann, Hans

doi:10.1145/990353.990363

Cited by 29 publications

(61 citation statements)

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“…As far as we know, Singular:Plural is the only modern system allowing user to do such combined computations. Main computational objects in the system are GR-algebras, where we can compute Gröbner bases for left and two-sided ideals and for left modules, modules of syzygies, free resolutions, intersections with subalgebras and many more (see [21] or [22] for description of all these algorithms). We put a lot of efforts in implementing the algorithms as efficient as possible and the latest tests indicate that the performance of Singular:Plural is quite good.…”

Section: Discussionmentioning

confidence: 99%

“…We put a lot of efforts in implementing the algorithms as efficient as possible and the latest tests indicate that the performance of Singular:Plural is quite good. You can find detailed information on Singular:Plural either in [22] or at http://www.singular.uni-kl.de/plural.…”

Section: Discussionmentioning

confidence: 99%

“…We use the computer algebra system Singular:Plural [22], with its commutative backbone Singular and noncommutative extension Plural. We proceed with the described procedure as follows:…”

Section: Remark 52mentioning

confidence: 99%

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PBW bases, non-degeneracy conditions and applications

Levandovskyy¹

2005

Representations of Algebras and Related Topics

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Abstract. We establish an explicit criteria (the vanishing of non-degeneracy conditions) for certain noncommutative algebras to have Poincaré-BirkhoffWitt basis. We study theoretical properties of such G-algebras, concluding they are in some sense "close to commutative". We use the non-degeneracy conditions for practical study of certain deformations of Weyl algebras, quadratic and diffusion algebras.The famous Poincaré-Birkhoff-Witt (or, shortly, PBW) theorem, which appeared at first for universal enveloping algebras of finite dimensional Lie algebras ([7]), plays an important role in the representation theory as well as in the theory of rings and algebras. Analogous theorem for quantum groups was proved by G. Lusztig and constructively by C. M. Ringel ([6]).Many authors have proved the PBW theorem for special classes of noncommutative algebras they are dealing with ([17], [18]). Usually one uses Bergman's Diamond Lemma ([4]), although it needs some preparations to be done before applying it. We have defined a class of algebras where the question "Does this algebra have a PBW basis?" reduces to a direct computation involving only basic polynomial arithmetic.In this article, our approach is constructive and consists of three tasks. Firstly, we want to find the necessary and sufficient conditions for a wide class of algebras to have a PBW basis, secondly, to investigate this class for useful properties, and thirdly, to apply the results to the study of certain special types of algebras.The first part resulted in the non-degeneracy conditions (Theorem 2.3), the second one led us to the G-and GR-algebras (3.4) and their properties (Theorem 4.7, 4.8), and the third one -to the notion of G-quantization and to the description and classification of G-algebras among the quadratic and diffusion algebras.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

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PBW bases, non-degeneracy conditions and applications

Levandovskyy¹

2005

Representations of Algebras and Related Topics

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“…As described in [6], a general multiplication in a non-commutative G-algebra boils down to the multiplication of y m · x n for a couple of variables x, y such that x a y b is a standard word. In general, the polynomial y m · x n involves other variables as well, but the case, when x, y generate a subalgebra of the type A(q, α, β, γ), appears often enough.…”

Section: Application In Computer Algebra Implementationmentioning

confidence: 99%

“…In [6] it has been proposed to address each pair of non-commuting and nonq-commuting variables separately. To each such pair a matrix M is assigned, such that M ij = y i · x j is a polynomial, written in terms of standard monomials.…”

Section: Application In Computer Algebra Implementationmentioning

confidence: 99%

On Two-Generated Non-commutative Algebras Subject to the Affine Relation

Levandovskyy

Koutschan

Motsak

2011

Lecture Notes in Computer Science

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Abstract. We consider algebras over a field K, generated by two variables x and y subject to the single relation yx = qxy + αx + βy + γ for q ∈ K * and α, β, γ ∈ K. We prove, that among such algebras there are precisely five isomorphism classes. The representatives of these classes, which are ubiquitous operator algebras, are called model algebras. We derive explicit multiplication formulas for y m · x n in terms of standard monomials x i y j for many algebras of the considered type. Such formulas are used in e. g. establishing formulas of binomial type and in an implementation of non-commutative multiplication in a computer algebra system. By using the formulas we also study centers and ring-theoretic properties of the non-commutative model algebras.In this paper we study non-commutative algebras in two generators obeying single affine relation. Many operator algebras, coming from different areas of natural sciences, are built from algebras in two generators, see Sect. 2.1 for examples. One of generators, say x, often corresponds to the operator of the multiplication with the function x. Another operator, say y, corresponds to a linear operator, acting on functions in the variable x.In the main Theorem we identify precisely five types of non-isomorphic algebras, which we call model algebras, among them. Despite the fact that many such algebras have been studied in the literature (see e. g. [3, 7, 2, 1], many aspects and properties are too scattered in the existing literature. Another point of this note is to search systematically for closed form of multiplication formulas on monomials. Such closed forms are needed, among other, in computer algebra, where many sophisticated algorithms heavily rely on basic multiplication among monomials. It is not enough to have such formulas just for model algebras, since isomorphisms do not preserve monomials but turn them into polynomials. It turned out, that there are still several cases, where we were not able to derive

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Effective Algebraic Analysis Approach to Linear Systems over Ore Algebras

Cluzeau¹,

Koutschan²,

Quadrat³

et al. 2020

Algebraic and Symbolic Computation Methods in Dynamical Systems

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The purpose of this paper is to present a survey on the effective algebraic analysis approach to linear systems theory with applications to control theory and mathematical physics. In particular, we show how the combination of effective methods of computer algebra − based on Gröbner basis techniques over a class of noncommutative polynomial rings of functional operators called Ore algebras − and constructive aspects of module theory and homological algebra enables the characterization of structural properties of linear functional systems. Algorithms are given and a dedicated implementation, called OREALGEBRAIC-ANALYSIS, based on the Mathematica package HOLONOMICFUNCTIONS, is demonstrated.

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Using computer algebra system SINGULAR:PLURAL for computations in noncommutative polynomial algebras

Abstract: We give a detailed account of algorithms and applications provided with SINGULAR:PLURAL (we call it PLURAL for short). The poster is done in form of (big) one-page introduction to the capabilities of the system.

Cited by 29 publications

References 4 publications

PBW bases, non-degeneracy conditions and applications

PBW bases, non-degeneracy conditions and applications

On Two-Generated Non-commutative Algebras Subject to the Affine Relation

Effective Algebraic Analysis Approach to Linear Systems over Ore Algebras

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