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6 Supplements 43 The conformal vertex Lie algebra 2-Vir has an involution given by G± → G and J → −J. This follows from G− λ G+ = G+ −λ−T G− = 2L + (−T − 2λ)J + (2ˆ c/3)λ(2) . This involution is the mirror involution. 11 shows that L + T J/2 is a conformal vector of 2-Vir. The topological Virasoro vertex Lie algebra tVir is the vertex Lie algebra 2-Vir endowed with the conformal vector L + T J/2. 11 yields the following description of the topological Virasoro algebra. We have tVir = K[T ](KL ⊕ KQ ⊕ KG ⊕ KJ) ⊕ Kdˆ where L is the conformal vector and cˆL = 0, J is a U (1)-vector with kˆJ = dˆ and ˆ (2) , Q, G are odd, primary, J-primary vectors of weight Lλ J = (T + λ)J − dλ ˆ (2) , Qλ Q = Gλ G = 0.

The functor R → g(R) from vertex Lie algebras to local Lie algebras is left adjoint to g → R(g). The morphism Y : R → R(g(R)) is the unit of adjunction. Proof. Let R be a vertex Lie algebra and g a local Lie algebra. By definition, a local Lie algebra morphism g(R) → g is an algebra morphism φ : g(R) → g such that φY (R) ⊂ R(g). By the universal property of g(R), the map φ → φ ◦ Y is a bijection onto the set of morphisms ψ : R → g[[z ±1 ]] such that ψ(R) is local and ψ(R) = φY (R) ⊂ R(g). Since R(g) is local, these ψs are just the vertex Lie algebra morphisms R → R(g).

Hence L(1) is diagonalizable. The above two identities also show that L(1) is a dilatation operator. 9 Griess Algebra We show that the Virasoro vectors are the idempotents of the Griess algebra. Let R be a vertex Lie algebra of CFT-type and kˆ ∈ R0 , kˆ = 0. Then R2 with ab := a0 b is an algebra with a bilinear form defined by (a, b)kˆ = a2 b. The algebra R2 is called the Griess algebra. Remark. Let R be a vertex Lie algebra of CFT-type such that R1 = 0. Then ab is commutative and ( , ) is invariant and symmetric.

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