A new Composition-Diamond lemma for associative conformal algebras

Abstract

Let [Formula: see text] be the free associative conformal algebra generated by a set [Formula: see text] with a bounded locality [Formula: see text]. Let [Formula: see text] be a subset of [Formula: see text]. A Composition-Diamond lemma for associative conformal algebras is first established by Bokut, Fong and Ke in 2004 [L. A. Bokut, Y. Fong and W.-F. Ke, Composition-Diamond Lemma for associative conformal algebras, J. Algebra 272 (2004) 739–774] which claims that if (i) [Formula: see text] is a Gröbner–Shirshov basis in [Formula: see text], then (ii) the set of [Formula: see text]-irreducible words is a linear basis of the quotient conformal algebra [Formula: see text], but not conversely. In this paper, by introducing some new definitions of normal [Formula: see text]-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 [Formula: see text] of [Formula: see text], [Formula: see text] has a unique reduced Gröbner–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.