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Vertex (operator) algebras are known as the mathematical counterparts of chiral algebras in two-dimensional quantum conformal field theory and are analogous to associative algebras in certain aspects (At the same time, in many aspects vertex (operator) algebras are also analogous to both Lie algebras and commutative associative algebras). So, it is important to construct various examples of vertex algebras. In [DLM], a new method was created to obtain vertex algebras from another algebraic concept-local vertex Lie algebra.
In the present paper, we analyze the concepts of local vertex Lie algebras and vertex Poisson algebras in detail, and give some more properties and examples. We show that any faithfifi module for certain vertex algebra constructed from the local vertex Lie algebras has a restricted module structure for the local vertex Lie algebra.That is the converse theorem of the Theorem 4.8 in [DLM]. Motivated by Corollary 5.6 in [Pr], We give out the definition of the homomorphism of local vertex Lie algebras. And we show that the homomorphism of local vertex Lie algebras can uniquely induce a vertex algebra homomorphism between certain vertex algebras constructed from the local vertex Lie algebras.