Notes on Factorizable Sheaves

  • 1 . 9 .
  • Published 2008


1.1. Configuration spaces. Let Λ be a lattice and Λ ⊂ Λ a sub-semigroup, isomorphic to (Z≥0)k. Let X be an algebraic curve (smooth, but not necessarily complete). For λ ∈ Λ, we will denote by X the algebraic variety classifying Λ-valued divisors D := Σλi ·xi with xi 6= xj and λi ∈ Λ. By definition, X = pt. For a marked point x0 ∈ X and an arbitrary λ ∈ Λ, let X x0 be the ind-scheme classifying Λ-valued divisors Σλi · xi as above but with a weaker condition, namely, that λi ∈ Λ for xi 6= x0. Let us denote by addλ1,λ2 either of the maps X1 ×X2 → X12 and X1 ×X2 x0 → X λ1+λ2 x0 . The map addλ1,λ2 is finite. For a perverse sheaves F1 on X λ1 and F2 on X2 or X2 x0 , we shall denote by F1 ? F2 the perverse sheaf (addλ1,λ2)!(F1 F2) ' (addλ1,λ2)∗(F1 F2) on X12 (or X12 x0 ). Let (X1 ×X)disj ⊂ X1 ×X2 and (X1 ×X2 x0 )disj ⊂ X λ1 ×X2 x0 the open subschemes, corresponding to pairs of divisors (D1, D2) with the condition that the support of D1 does not intersect the support of D2 in the former case, and is also disjoint from {x0} in the latter case. 1.2. The construction. Let A be a Λ-graded Hopf algebra. For the following construction we will work over the ground field C and we take X to be the affine line A. (For the construction to work for any X, we need that the antipode on A be involutive.) We will assume that A ' C and that its graded components are finite-dimensional.

Cite this paper

@inproceedings{2008NotesOF, title={Notes on Factorizable Sheaves}, author={1 . 9 .}, year={2008} }