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By Rosellen M.

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Example text

Let R be a conformal algebra. Then T R is a two-sided ideal with respect to [ , ]. If R satisfies conformal skew-symmetry then [ , ] on R/T R is skew-symmetric. Since [aλ b] and a ⊗ b → ζ ab [b−λ−T a] are λ-brackets, we obtain: Lemma. Let R be a conformal algebra and S ⊂ R a subset. If any a, b ∈ S satisfy conformal skew-symmetry then so do any a, b ∈ K[T ]S. Proposition. Let g be a skew-symmetric algebra. If a(z), b(z) ∈ g[[z ±1 ]] are local then they satisfy conformal skew-symmetry. Proof. 5 (ii), (iii) imply (i) δ(z, w).

The map p → pa a is obviously surjective. It is injective since its composition with the map F¯g → g, a(z) → a−1 , is injective. ✷ Note that F¯g = K[∂z ]Fg iff a(z)t b(z) ∈ K[∂z ]Fg for any a(z), b(z) ∈ Fg . 3 the Proposition applies to the local Lie algebras ˜g, ˆg, C(B), Witt, Vir, 1-Vir, and 2-Vir. 6 The Lie Bracket of a Vertex Lie Algebra We prove that any vertex Lie algebra has a natural Lie bracket [ , ]Lie . 4 The Borcherds Lie Algebra β at λ(t) → dλ : E[[λ]] → E[[α, β]], 31 at (β (t+1) − α(t+1) ).

Here we only remark that the space Endv (E) of fields is an unbounded cont t−i formal subalgebra of gl(E)[[z ±1 ]] since a(w)t b(w) = . 1. 3 Examples of λ-Brackets of Distributions We calculate the λ-brackets of any a(z), b(z) ∈ Fg for affine, Clifford, and N =0, 1, 2 Virasoro Lie algebras. We always have a(z)i b(z) ∈ K[∂z ]Fg . Let g be a K-graded Lie algebra and a(z), b(z) ∈ g[[z ±1 ]]H be homogeneous. The “constant” weak commutator formula [an , bm ] = cn+m is equivalent to a(z)λ b(z) = c(z).

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