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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Additional resources for Factorizable Sheaves and Quantum Groups
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~_
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 .