# Factorizable Sheaves and Quantum Groups by Roman Bezrukavnikov By Roman Bezrukavnikov

The booklet is dedicated to the geometrical development of the representations of Lusztig's small quantum teams at roots of cohesion. those representations are discovered as a few areas of vanishing cycles of perverse sheaves over configuration areas. As an program, the bundles of conformal blocks over the moduli areas of curves are studied. The e-book is meant for experts in workforce representations and algebraic geometry.

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Ark]. 1. /, A c X , x C Na. All definitions and results concerning the category E given above and below, have the obvious versions for the category G. For M E C, define M v E G as follows: (MV)a = (M_a)* (the dual vector space); the action of the operators 0i, ei being the transpose of their action on M. 2. Let us call an object M E d u - - (resp. u +-) good if it admits a filtration whose 9 u_ ~) (resp. md~_-0 successive quotients have the form md,,>o(M (resp. M" E d -<~ (cf.

Equipped with these complementary structures, the category 5c8 becomes a braided tensor category. 6. 1. Let us fix a finite set J, and consider the space D J. Inside this space, let us consider the subspaces D J = D J N R J and D J+ = D J • R J0. Let 7 / b e the set (arrangement) of all real hyperplanes in D J of the form Hj : t3 = 0 or Hi,j,, : tj, = tj,,. An edge L of the arrangement 7/ is a subspace of D J which is a non-empty intersection ('/H of some hyperplanes from 7/. We denote by L ~ the complement L - U L', the union over all edges L' c L of smaller dimension.

For example, we have a unique smallest facet 0 - - the origin. For each j C J, we have a positive one-dimensional facet /:) given by the equations tj, = 0 (j' r j ) ; tj >_ O. Let us choose a point wF on each positive facet F . We call a flag a sequence of embedded positive facets F : F0 C . . Fp; we say t h a t F starts from F0. ~Ib such a flag we assign the simplex A F the convex hull of the points wFo,. -. , wFp. 24 To each positive facet F we assign the following two spaces: DF = U/kF, the union over all flags F starting from F , and SF = U AF,, the union over all flags F ~ starting from a facet which properly contains F .