Composition-diamond lemma for associative conformal algebras.
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.
scientific article; zbMATH DE number 2055652
scientific article