1994/12/31 by Michael Finkelberg, Finkelberg, M., Vadim Schechtman +1
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.q-alg/9412017
openalex publication_date 1995/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is a sequel to "Localization of \fraku-modules. I", hep-th/9411050. We are starting here the geometric study of the tensor category \calC associated with a quantum group (corresponding to a Cartan matrix of finite type) at a root of unity. The main results establish isomorphisms between homogeneous components of irreducible objects in \calC and spaces of vanishing cycles at the origin of certain Goresky-MacPherson sheaves on configuration spaces; establish isomorphisms of the stalks at the origin of the above GM sheaves with certain Hochschild complexes (which compute the Hochschild homology of a certain "triangular" subalgebra of our quantum group with coefficients in the coresponding irreducible representation); establish the analogous results for tensor products of irreducibles. In geometry, the tensor product of representations corresponds to a "fusion" of sheaves on configuration spaces --- operation defined using the functor of nearby cycles.