Noncommutative Gröbner bases and filtered-graded transfer.
type: Item
instance of: scholarly article
title: Noncommutative Gröbner bases and filtered-graded transfer.
zbMATH Open document ID: 1050.16022
author: Huishi Li
published in: Lecture Notes in Mathematics
publication date: 2002-11-21T00:00:00Z
full work available at URL: http://hdl.handle.net/11693/50769
review text: The book starts with a short history of noncommutative Gröbner bases. There are mentioned papers by B. Buchberger (1965, 1985) as well as A. Galigo (1985), J. Appel and W. Lassner (1988), A. Kandri-Rody and V. Weispfenning (1990), G. Bergman (1978), T. Mora (1986, 1994), E. L. Green (1993). I would like to add to this list the paper by \textit{A. I. Shirshov} [Sib. Mat. Zh. 3, 292-296 (1962; Zbl 0104.26004), translated in ACM SIGSAM Bull. 33, No. 2, 3-9 (1999)]. In this paper Shirshov introduced a concept that is now known as Gröbner-Shirshov bases for Lie polynomials \(\text{Lie}(X)\) over the field \(k\) under Lie bracket \([xy]=xy-yx\). Note that Shirshov was dealing with Lie polynomials via noncommutative (associative) polynomials \(k\langle X\rangle\), \(\text{Lie}(X)\subset k\langle X\rangle\). To be more precise, to make the composition (\(S\)-polynomial by Buchberger's later terminology) of two monic Lie polynomials \(f,g\) relative to some word (ambiguity with Bergman's later terminology) \(w\), where \(w=\overline fv=u\overline g\) (\(\overline f\) the leading associative monomial of \(f\) as noncommutative polynomial), he made first the associative composition \((f,g)_w=fv-ug\), and then he put the special Lie bracketing \([fv]-[ug]\) in order to get the Lie composition. The same was done with the notion of an elimination of leading monomial of one Lie polynomial in other: he made first the elimination in the associative sense and then put the special Lie bracketing. He proved the Composition lemma: Let \(S\) be a set of Lie polynomials, \(S^*\) be a completion of \(S\) under all possible multiple compositions. If \(f\in\text{Id}(S)\) then \(\overline f\) contains as a subword \(\overline s\) for some \(s\in S^*\). Shirshov had further assumed that \(S\) is stable in the sense that for any \(f,g\in S^*\) the degree of a composition \((f,g)_w\) (after eliminations of leading words) is greater then the degrees of \(f,g\). He did not use this condition in th…
Mathematics Subject Classification ID: 16Z05
Mathematics Subject Classification ID: 16-02
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 16W70
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 16S15
Mathematics Subject Classification ID: 16S36
zbMATH DE Number: 1833671
zbMATH Keywords: noncommutative Gröbner bases
zbMATH Keywords: Gröbner-Shirshov bases
zbMATH Keywords: Lie polynomials
zbMATH Keywords: skew polynomial rings
zbMATH Keywords: quadric solvable polynomial algebras
zbMATH Keywords: Gelfand-Kirillov dimension
zbMATH Keywords: Weyl algebras
zbMATH Keywords: quantum algebras
reviewed by: Leonid A. Bokut'
MaRDI profile type: Publication
OpenAlex ID: W2000191845
DOI: 10.1007/B84211
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description: scientific article; zbMATH DE number 1833671