In particular, the coefficients of this vertex operator endow with an action of the Virasoro algebra with central charge . acts semisimply on with integer eigenvalues that are bounded below. Under the grading provided by the eigenvalues of , the multiplication on is homogeneous in the sense that if and are homogeneous, then is homogeneous of degree . The identity has degree 0, and the conformal element has degree 2. . A homomorphism of vertex algebras is a map of the underlying vector spaces that respects the additional identity, translation, and multiplication structure. Homomorphisms of vertex operator algebras have "weak" and "strong" forms, depending on whether they respect conformal vectors. Commutative vertex algebras A vertex algebra is commutative if all vertex operators commute with each other. This is equivalent to the property that all products lie in , or that . Thus, an alternative definition for a commutative vertex algebra is one in which all vertex operators are regular at . Given a commutative vertex algebra, the constant terms of multiplication endow the vector space with a commutative and associative ring structure, the vacuum vector is a unit and is a derivation. Hence the commutative vertex algebra equips with the structure of a commutative unital algebra with derivation. Conversely, any commutative ring with derivation has a canonical vertex algebra structure, where we set , so that restricts to a map which is the multiplication map with the algebra product. If the derivation vanishes, we may set to obtain a vertex operator algebra concentrated in degree zero. Any finite-dimensional vertex algebra is commutative. Thus even the smallest examples of noncommutative vertex algebras require significant introduction. Basic properties The translation operator in a vertex algebra induces infinitesimal symmetries on the product structure, and satisfies the following properties: , so is determined by . (skew-symmetry) For a vertex operator algebra, the other Virasoro operators satisfy similar properties: (quasi-conformality) for all . (Associativity, or Cousin property): For any , the element given in the definition also expands to in . The associativity property of a vertex algebra follows from the fact that the commutator of and is annihilated by a finite power of , i.e., one can expand it as a finite linear combination of derivatives of the formal delta function in , with coefficients in . Reconstruction: Let be a vertex algebra, and let be a set of vectors, with corresponding fields .