Differential graded vertex Lie algebras

This is the continuation of the study of differential graded (dg) vertex algebras defined in our previous paper [Caradot et al., "Differential graded vertex operator algebras and their Poisson algebras," J. Math. Phys. 64, 121702 (2023)]. The goal of this paper is to construct a functor from the category of dg vertex Lie algebras to the category of dg vertex algebras which is left adjoint to the forgetful functor. This functor not only provides an abundant number of examples of dg vertex algebras, but it is also an important step in constructing a homotopy theory [see Avramov and Halperin, "Through the looking glass: A dictionary between rational homotopy theory and local algebra," in Algebra, Algebraic Topology and their Interactions, Lecture Notes in Mathematics, edited by J. E. Roos (Springer, Berlin, Heidelberg, 1986), Vol. 1183, pp. 1-27 and D. G. Quillen, Homotopical Algebra, Lecture Notes in Mathematics (Springer, Berlin, Heidelberg, 1967), Vol. 43] in the category of vertex algebras. Vertex Lie algebras were introduced as analogues of vertex algebras, but in which we only consider the singular part of the vertex operator map and the equalities it satisfies. In this paper, we extend the definition of vertex Lie algebras to the dg setting. We construct a pair of adjoint functors between the categories of dg vertex algebras and dg vertex Lie algebras, which leads to the explicit construction of dg vertex (operator) algebras. We will give examples based on the Virasoro algebra, the Neveu-Schwarz algebra, and dg Lie algebras.