Composition-diamond lemma for associative conformal algebras.
type: Item
instance of: scholarly article
title: Composition-diamond lemma for associative conformal algebras.
zbMATH Open document ID: 1108.17006
DOI: 10.1016/S0021-8693(03)00341-7
published in: Journal of Algebra
publication date: 2004-03-14T00:00:00Z
review text: The formulation of Shirshov's Composition Lemma, Buchberger's Theorem and Composition-Diamond Lemma for commutative, noncommutative and Lie algebras is given as follows: Shirshov's Composition Lemma -- Buchberger's Theorem. Let \(S\) be a Gröbner-Shirshov (Gröb\-ner) basis for Lie or noncommutative (commutative) polynomials. Let \(\text{Id}(S)\) be the ideal generated by \(S\) in the corresponding ``polynomial'' algebra. If \(f\in\text{Id}(S)\), then the leading associative word \(\overline f\) of the ``polynomial'' \(f\) contains as a subword a leading associative word \(\overline s\) for some \(s\in S\). Composition-Diamond Lemma. Let \({\mathcal K}[X]\) be an algebra of commutative, noncommutative, or Lie polynomials, \(S\subseteq{\mathcal K}[X]\) a set of monic polynomials. Then \(S\) is a Gröbner-Shirshov basis if and only if the ``\(S\)-reduced'' basic monomials \([u]\) (i.e. \(u\neq v\overline sw\) for any \(s\in S\)) form a linear base of the algebra \({\mathcal K}[X|S]\) with defining relations \(S\) \(({\mathcal K}[X|S]={\mathcal K}[X]/\text{Id}(S)\)). In the present paper, the authors prove an analogue of the Shirshov's Composition Lemma -- Buchberger's Theorem for associative conformal algebras. They then apply it to get the Composition-Diamond Lemma for associative conformal relations. As some examples, the Composition-Diamond Lemma is applied to the universal enveloping of some Lie conformal algebras (the Virasoro Lie conformal algebra, the loop Lie conformal algebra and the semi-direct product of the two). Some examples of homogeneous conformal algebras are also analysed. The last result of the paper is devoted to prove that, for some homogeneous conformal algebras, the word problem is algorithmically solvable, while it is unsolvable in general.
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 16S10
zbMATH DE Number: 2055652
zbMATH Keywords: associative conformal algebras
zbMATH Keywords: Gröbner-Shirshov bases
zbMATH Keywords: Composition-Diamond Lemma
author: Leonid A. Bokut'
author: Y. Fong
author: Wen-Fong Ke
describes a project that uses: FPLSA
MaRDI profile type: Publication
Wikidata QID: Q124982526
cites work: Lifting standard bases in filtered structures
cites work: Infinite conformal symmetry in two-dimensional quantum field theory
cites work: The diamond lemma for ring theory
cites work: Q5559306
cites work: Q5665187
cites work: Q4119334
cites work: On algebraically closed and simple Lie algebras
cites work: Q4220120
cites work: Gröbner-Shirshov bases and composition lemma for associative conformal algebras: an example.
cites work: Q4471564
cites work: Q4761797
cites work: SERRE RELATIONS AND GRÖBNER-SHIRSHOV BASES FOR SIMPLE LIE ALGEBRAS I
cites work: Gröbner-Shirshov bases for exceptional Lie algebras. I
cites work: Q4495675
cites work: Q4300256
cites work: Gröbner-Shirshov bases for Lie superalgebras and their universal enveloping algebras
cites work: Gröbner-Shirshov bases for quantum enveloping algebras
cites work: Q4495676
cites work: Vertex algebras, Kac-Moody algebras, and the Monster
cites work: Monstrous moonshine and monstrous Lie superalgebras
cites work: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal
cites work: Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems
cites work: Free differential calculus. IV: The quotient groups of the lower central series
cites work: Q5515373
cites work: Q3808246
cites work: Structure theory of finite conformal algebras
cites work: Q4317713
cites work: Q3983629
cites work: Program for constructing a complete system of relations, basis elements, and commutator table for finitely presented Lie algebras and superalgebras
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Q4222006
cites work: Q5581665
cites work: Jacobi Sums, Irreducible Zeta-Polynomials, and Cryptography
cites work: Q4163654
cites work: Q4341774
cites work: Algorithms for construction of standard Gröbner-Shirshov bases of ideals of free algebras over commutative rings
cites work: On theories with a combinatorial definition of 'equivalence'
cites work: Recursive Unsolvability of a problem of Thue
cites work: Q3134850
cites work: On free conformal and vertex algebras
cites work: Universal enveloping conformal algebras.
cites work: Q3248045
cites work: Q5522249
cites work: Q5519136
cites work: Some Algorithmic Problems for Lie Algebras
cites work: Quantum Field Theory in Terms of Vacuum Expectation Values
full work available at URL: s0021-8693(03)00341-7
OpenAlex ID: W2963127038
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