If is spanned by monomials in the positive weight coefficients of the fields (i.e., finite products of operators applied to , where is negative), then we may write the operator product of such a monomial as a normally ordered product of divided power derivatives of fields (here, normal ordering means polar terms on the left are moved to the right). Specifically, More generally, if one is given a vector space with an endomorphism and vector , and one assigns to a set of vectors a set of fields that are mutually local, whose positive weight coefficients generate , and that satisfy the identity and translation conditions, then the previous formula describes a vertex algebra structure. Examples Heisenberg vertex operator algebra A basic example of a noncommutative vertex algebra is the rank 1 free boson, also called the Heisenberg vertex operator algebra. It is "generated" by a single vector b, in the sense that by applying the coefficients of the field b(z) = Y(b,z) to the vector 1, we obtain a spanning set. The underlying vector space is the infinite-variable polynomial ring C[x1,x2,...], where for positive n, the coefficient b–n of Y(b,z) acts as multiplication by xn, and bn acts as n times the partial derivative in xn. The action of b0 is multiplication by zero, producing the "momentum zero" Fock representation V0 of the Heisenberg Lie algebra (generated by bn for integers n, with commutation relations [bn,bm]=n δn,–m), i.e., induced by the trivial representation of the subalgebra spanned by bn, n ≥ 0. The Fock space V0 can be made into a vertex algebra by the following reconstruction: where :..: denotes normal ordering (i.e. moving all derivatives in x to the right). The vertex operators may also be written as a functional of a multivariable function f as: if we understand that each term in the expansion of f is normal ordered. The rank n free boson is given by taking an n-fold tensor product of the rank 1 free boson. For any vector b in n-dimensional space, one has a field b(z) whose coefficients are elements of the rank n Heisenberg algebra, whose commutation relations have an extra inner product term: [bn,cm]=n (b,c) δn,–m. Virasoro vertex operator algebra Virasoro vertex operator algebras are important for two reasons: First, the conformal element in a vertex operator algebra canonically induces a homomorphism from a Virasoro vertex operator algebra, so they play