# A Course in Vertex Algebra by Rosellen M.

By Rosellen M.

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Let R be a conformal vertex Lie algebra of CFT-type and R′ a conformal vertex Lie subalgebra. Then CR (R′ ) is a conformal vertex Lie subalgebra with conformal vector L − L′ and cˆL−L′ = cˆL − cˆL′ . Proof. The Remark shows that Rc := CR (R′ ) = ker L′−1 and that L′ and Lc := L − L′ commute. Thus Lc ∈ Rc and Rc is a graded vertex Lie subalgebra. We have Lc−1 = L−1 = T and Lc0 = L0 = H on Rc . Moreover, Lc = L − L′ ∈ R2 . 9 (ii). The vector Lc is quasi-primary because Lc , L′ commute. We have ✷ cˆLc = cˆL − cˆL′ because L2 L = L′2 L′ + Lc2 Lc .

2 Conformal Operators We discuss the general linear unbounded vertex Lie algebra glc (R). Let g be a differential Lie algebra. The translation covariant g-valued power series form an unbounded vertex Lie algebra g[[µ]]T := g[[µ]] ∩ g[[z ±1 ]]T . 1. Note that g[[µ]]T ∩ g[µ] = 0. The projection a(z) → aµ maps g[[z ±1 ]]T to g[[µ]]T . If R is a K[T ]-module then gl(R) is a differential Lie algebra with T = [T, ]. To give an unbounded λ-product on R is equivalent to giving a K[T ]module morphism R → gl(R)[[µ]]T , a → aµ .

Ii) The integration-by-parts formula implies resz p(z)∂w δ(z, w) = ∂w p(w) = resz (∂z p(z))δ(z, w) = −resz p(z)∂z δ(z, w). This proves the first identity. The second identity follows from the first: δ(z − x, w) = e−x∂z δ(z, w) = ex∂w δ(z, w) = δ(z, w + x). (iii) This follows from the fact that n → −n − 1 is an involution of Z. 6 Conformal Skew-Symmetry We prove that local distributions with values in a skew-symmetric algebra satisfy conformal skew-symmetry. Conformal skew-symmetry for an unbounded conformal algebra is [aλ b] = − ζ ab [b−λ−T a].