By Rosellen M.
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Additional resources for A Course in Vertex Algebra
Define [a, b] := [a0 b]. Setting λ = µ = 0 in the conformal Jacobi identity, we get: Remark. If an unbounded conformal algebra satisfies the conformal Jacobi identity then [ , ] satisfies the Leibniz identity. The indices λ, µ, and µ−λ in the conformal Jacobi identity are determined by the fact that all three terms of the identity transform in the same way 22 2 Vertex Lie Algebras with respect to T : replacing a by T a, b by T b, or c by T c is equivalent to multiplication with −λ, λ − µ, and T + µ, resp.
The bracket of g(R) is determined by the λ-bracket of R via the weak commutator formula. The algebra g(R) is the Borcherds Lie algebra of R. Let R be a vertex Lie algebra. There exists a Lie algebra g(R) and a morphism Y : R → g(R)[[z ±1 ]] of unbounded conformal algebras such that Y (R) is local and the following universal property holds: for any Lie algebra g and any morphism ψ : R → g[[z ±1 ]] such that ψ(R) is local, there exists a unique algebra morphism φ : g(R) → g such that ψ = φ ◦ Y . The pair (g(R), Y ) is unique up to a unique isomorphism.
For a vector space E, define a subspace of E[[z ±1 , w±1 ]]: E[(z/w)±1 ][[w±1 ]] := an−m,m xm ∈ E[x±1 ] . a(z, w) ∀ n ∈ Z m∈Z If a(z, w) ∈ E[(z/w)±1 ][[w±1 ]] then a(w, w) := well-defined. 3 Vertex Lie Algebras of Distributions 23 a(z + w) := ew∂z a(z) ∈ E[[z ±1 ]][[w]]. Note that a(z + w) = a(w + z) iff a(z) ∈ E[[z]]. Proposition. (i) Let a(z, w) ∈ E[(z/w)±1 ][[w±1 ]]. Then δ(z, w)a(z, w) is well-defined and δ(z, w)a(z, w) = δ(z, w)a(w, w). In particular, resz δ(z, w)a(z, w) = a(w, w). (ii) ∂w δ(z, w) = −∂z δ(z, w) and δ(z − x, w) = δ(z, w + x).
A Course in Vertex Algebra by Rosellen M.