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By definition, a factorization algebra is a collection of quasicoherent O–modules Fn on X n , n ≥ 1, satisfying a factorization condition, which essentially means that the fiber of Fn at (x1 , . . , xn ) is isomorphic to ⊗s∈S (F1 )s , where S = {x1 , . . , xn }. For example, j ∗ F2 = j ∗ (F1 F1 ) and ∆∗ F2 = F1 . Intuitively, such a collection may be viewed as an O–module on the Ran space R(X) of all finite non-empty subsets of X. In addition, it is required that F1 has a global section (unit) satisfying natural properties.

KAC, A. RADUL and W. WANG – W1+∞ and WN with central charge N , Comm. Math. Phys. 170 (1995) 337–357. [FR] E. FRENKEL and N. RESHETIKHIN – Towards deformed chiral algebras, Preprint q-alg/9706023. [FK] I. FRENKEL and V. KAC – Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980) 23–66. [FGZ] I. FRENKEL, H. GARLAND and G. ZUCKERMAN – Semi-infinite cohomology and string theory, Proc. Nat. Acad. Sci. A. 83 (1986) 8442–8446. [FLM] I. FRENKEL, J. LEPOWSKY and A.

There is a canonical isomor2 (X 2 , Vr2 ) (see [G], Prop. 1). 4. Beilinson and Drinfeld give a beautiful description of chiral algebras as factorization algebras. By definition, a factorization algebra is a collection of quasicoherent O–modules Fn on X n , n ≥ 1, satisfying a factorization condition, which essentially means that the fiber of Fn at (x1 , . . , xn ) is isomorphic to ⊗s∈S (F1 )s , where S = {x1 , . . , xn }. For example, j ∗ F2 = j ∗ (F1 F1 ) and ∆∗ F2 = F1 . Intuitively, such a collection may be viewed as an O–module on the Ran space R(X) of all finite non-empty subsets of X.

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