Vertex Coalgebras, Coassociator, and Cocommutator Formulas

Based on the definition of vertex coalgebra introduced by Hubbard, 2009, we prove that this notion can be reformulated using coskew symmetry, coassociator and cocommutator formulas without restrictions on the grading. We also prove that a vertex coalgebra can be defined in terms of dual versions of the axioms of Lie conformal algebra and differential algebra. 1. Introduction The notion of vertex coalgebra was introduced by Hubbard in [1] as a generalization of the notion of vertex operator coalgebra previously studied in [2–4]. Many of the properties described in these works are in some sense dual to the properties satisfied by vertex algebras. Since the original definition of vertex algebra was introduced by Borcherds [5] in the 1980s, several reformulations have been studied. A vertex algebra can be defined using Lie algebra-type axioms or it can be seen as a generalization of a commutative and associative algebra with unit, focusing on the commutator or associator formulas. These formulations are introduced and thoroughly studied in [6–8]. A vertex algebra can also be defined as the deformation of a Poisson vertex algebra, namely, a Lie conformal algebra and a left symmetric differential algebra with unit satisfying certain compatibilities, as described in [9]. These different approaches engender several equivalent definitions of vertex algebra based on different axioms. Our goal is to prove that these approaches can be, in some sense, dualized to obtain equivalent definitions of vertex coalgebra. The study of these approaches is not automatic in the case of vertex coalgebras as there might be axioms in the definition of vertex algebra that do not make sense in their dual version. For instance, axioms such as “weak commutativity” and “weak associativity” do not make sense unless we require the use of grading on a vertex coalgebra (see [1]). In [10], the authors show that with a coefficient approach, the Jacobi identity can be proven to follow from the commutator and associator formulas. Based on that idea, we first obtain a reformulation of the original definition of vertex coalgebra analogous to the original definition of vertex algebra introduced by Borcherds [5]. Then, we prove that the original definition of vertex coalgebra can be reformulated in three equivalent definitions: the first based on the coassociator and coskew symmetry formulas, the second based on the cocommutator formula, and the third based on dual versions of the axioms of Lie conformal algebra and differential algebra, following the ideas developed in [9]. In the next section we