A new Composition-Diamond lemma for associative conformal algebras

Abstract

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.