A new Composition-Diamond lemma for associative conformal algebras

• Published in: arXiv.org
• Main Authors: Ni, Lili, Chen, Yuqun
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 16.01.2016
• Subjects: Algebra; Composition; Diamonds; Mathematics - Rings and Algebras
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1602.03554

Let \(C(B,N)\) be the free associative conformal algebra generated by a set \(B\) with a bounded locality \(N\). Let \(S\) be a subset of \(C(B,N)\). A Composition-Diamond lemma for associative conformal algebras is firstly established by Bokut, Fong, and Ke in 2004 \cite{BFK04} which claims that if (i) \(S\) is a Gr\"obner-Shirshov basis in \(C(B,N)\), then (ii) the set of \(S\)-irreducible words is a linear basis of the quotient conformal algebra \(C(B,N|S)\), but not conversely. In this paper, by introducing some new definitions of normal \(S\)-words, compositions and compositions to be trivial, we give a new Composition-Diamond lemma for associative conformal algebras which makes the conditions (i) and (ii) equivalent. We show that for each ideal \(I\) of \(C(B,N)\), \(I\) has a unique reduced Gr\"obner-Shirshov basis. As applications, we show that Loop Virasoro Lie conformal algebra and Loop Heisenberg-Virasoro Lie conformal algebra are embeddable into their universal enveloping associative conformal algebras.