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The conformal Jacobi identity is [[aλ b]µ c] = [aλ [bµ−λ c]] − [bµ−λ [aλ c]]. It holds for a, b, c iff it holds for b, a, c since [[b−λ−T a]µ c] = [eT ∂λ [b−λ a]µ c] = e−µ∂λ [[b−λ a]µ c] = [[bµ−λ a]µ c]. It holds for a, b, c iff it holds for a, c, b since [[aλ b]µ c] = −eT ∂µ [c−µ [aλ b]], [aλ [bµ−λ c]] = −[aλ e(T −λ)∂µ [c−µ b]] = −eT ∂µ [aλ [c−µ b]], and [bµ−λ [aλ c]] = −eT ∂µ [[aλ c]−µ+λ b]. 2 Products and Semidirect Products We discuss products and semidirect products of vertex Lie algebras.

Define a K[T ]-module morphism Witt → glc (R) by Lλ a := (T + Hλ)a for a ∈ E. A direct calculation shows that R thus becomes a Witt-module. In particular, loop and affine vertex Lie algebras ˜g and gˆ with E = g and Clifford vertex Lie algebras C(B) with E = ΠB are Vir-modules. In these three cases we have Witt → Derc (R) as direct calculations show. Thus semidirect products such as Witt ⋉ ˜g and Vir ⋉ C(B) are defined. For example, if g is a one-dimensional abelian Lie algebra then Witt ⋉ ˜g is isomorphic to the vertex Lie algebra {p∂x + q | p, q ∈ K[x±1 ]} of first order differential operators on K[x±1 ] because [−xn+1 ∂x , xm ] = −mxn+m .

Remark. Let R be a conformal vertex Lie algebra with centre C. Then C = ker T and C ⊂ V0 . Proof. 5 we have ker T ⊂ C. We have C ⊂ ker T since T = L(0) . We have C ⊂ V0 since H = L(1) . ✷ 40 2 Vertex Lie Algebras Proposition. Let R be a vertex Lie algebra and S ⊂ R a subspace that generates R. If L(0) |S = T and L(1) |S is diagonalizable then L(0) = T and L(1) is a dilatation operator. Proof. 3 the operator L(0) is a derivation of R. Thus ker(T − L(0) ) is a conformal subalgebra of R. Since S ⊂ ker(T −L(0) ), we get L(0) = T .

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A Course in Vertex Algebra by Rosellen M.


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