Algorithmic Methods in Non-Commutative Algebra

Bueso, José; Gómez-Torrecillas, José; Verschoren, A.

doi:10.1007/978-94-017-0285-0

Cited by 97 publications

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“…. , v m of V compatible with the reduction system (see [7]) defined by the generating relations (2.1) of H λ,κ which is a total order (see [17,Remarks 7.1 and 7.2] for an example). We check overlap ambiguities of the form ghv, gvw, uvw for g, h in G and u, v, w in V .…”

Section: Poincaré-birkhoff-witt Propertymentioning

confidence: 99%

PBW Deformations of Skew Group Algebras in Positive Characteristic

Shepler

Witherspoon

2014

Algebr Represent Theor

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Abstract. We investigate deformations of a skew group algebra that arise from a finite group acting on a polynomial ring. When the characteristic of the underlying field divides the order of the group, a new type of deformation emerges that does not occur in characteristic zero. This analogue of Lusztig's graded affine Hecke algebra for positive characteristic can not be forged from the template of symplectic reflection and related algebras as originally crafted by Drinfeld. By contrast, we show that in characteristic zero, for arbitrary finite groups, a Lusztig-type deformation is always isomorphic to a Drinfeldtype deformation. We fit all these deformations into a general theory, connecting Poincaré-Birkhoff-Witt deformations and Hochschild cohomology when working over fields of arbitrary characteristic. We make this connection by way of a double complex adapted from Guccione, Guccione, and Valqui, formed from the Koszul resolution of a polynomial ring and the bar resolution of a group algebra.

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Section: Poincaré-birkhoff-witt Propertymentioning

confidence: 99%

PBW Deformations of Skew Group Algebras in Positive Characteristic

Shepler

Witherspoon

2014

Algebr Represent Theor

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“…Thus it is a Noetherian domain of Gel'fand-Kirillov dimension 2, which is Cohen-Macaulay and Auslander-regular [1]. However, the global homological dimension is between 1 and 2.…”

Section: Centers and Ring-theoretic Properties Of Model Algebrasmentioning

confidence: 98%

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

Levandovskyy

Koutschan

Motsak

2011

Computer Algebra in Scientific Computing

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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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“…The following lemma allows to transfer the skew polynomial structure across quotients as in the commutative case, compare also [9,Prop. 3.15].…”

Section: Localization and Evaluationmentioning

confidence: 99%

A skew polynomial approach to integro-differential operators

Regensburger

Rosenkranz

Middeke

2009

Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation

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We construct the algebra of integro-differential operators over an ordinary integro-differential algebra directly in terms of normal forms. In the case of polynomial coefficients, we use skew polynomials for defining the integro-differential Weyl algebra as a natural extension of the classical Weyl algebra in one variable. Its normal forms, algebraic properties and its relation to the localization of differential operators are studied. Fixing the integration constant, we regain the integro-differential operators with polynomial coefficients.

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Algorithmic Methods in Non-Commutative Algebra

Cited by 97 publications

References 0 publications

PBW Deformations of Skew Group Algebras in Positive Characteristic

PBW Deformations of Skew Group Algebras in Positive Characteristic

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

A skew polynomial approach to integro-differential operators

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