dorsal/arxiv
View SchemaLocalization of $\frak{u}$-modules. II. Configuration spaces and quantum groups
| Authors | M. Finkelberg, V. Schechtman |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9412017 |
| URL | https://arxiv.org/abs/q-alg/9412017 |
Abstract
This paper is a sequel to "Localization of $\frak{u}$-modules. I", hep-th/9411050. We are starting here the geometric study of the tensor category $\cal{C}$ 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 $\cal{C}$ 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.
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"abstract": "This paper is a sequel to \"Localization of $\\frak{u}$-modules. I\",\nhep-th/9411050. We are starting here the geometric study of the tensor category\n$\\cal{C}$ associated with a quantum group (corresponding to a Cartan matrix of\nfinite type) at a root of unity. The main results establish isomorphisms\nbetween homogeneous components of irreducible objects in $\\cal{C}$ and spaces\nof vanishing cycles at the origin of certain Goresky-MacPherson sheaves on\nconfiguration spaces; establish isomorphisms of the stalks at the origin of the\nabove GM sheaves with certain Hochschild complexes (which compute the\nHochschild homology of a certain \"triangular\" subalgebra of our quantum group\nwith coefficients in the coresponding irreducible representation); establish\nthe analogous results for tensor products of irreducibles. In geometry, the\ntensor product of representations corresponds to a \"fusion\" of sheaves on\nconfiguration spaces --- operation defined using the functor of nearby cycles.",
"arxiv_id": "q-alg/9412017",
"authors": [
"M. Finkelberg",
"V. Schechtman"
],
"categories": [
"q-alg",
"alg-geom",
"math.AG",
"math.QA"
],
"title": "Localization of $\\frak{u}$-modules. II. Configuration spaces and quantum groups",
"url": "https://arxiv.org/abs/q-alg/9412017"
},
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