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Factorizable sheaves and quantum groups. (English) Zbl 0938.17016

Lecture Notes in Mathematics. 1691. Berlin: Springer. x, 287 p. (1998).
Since the appearance of quantum groups, there were several attempts to provide “geometric” or “topological” constructions of these mathematical objects. The book under review belongs to this trend; it can be considered as a continuation of [V. Schechtman and A. Varchenko, Quantum groups and homology of local systems, in: Algebraic geometry and analytic geometry, ICM-90 Satell. Conf. Proc., 182-197 (1991; Zbl 0760.17014)].
Let \(A\) be a finite Cartan matrix, let \(l\) be a positive integer satisfying suitable technical hypothesis with respect to \(A\); let \({\mathbf k}\) be a field such that char \({\mathbf k}\) is not divisible by \(l\), and contains a primitive \(l\)-th root of 1 \(\zeta\). Let \({\mathfrak u}\) be the small quantum group (also called Frobenius-Lusztig kernel by some authors) attached to \(A\), \({\mathbf k}\), \(\zeta\). There is a triangular decomposition \({\mathfrak u}= {\mathfrak u}^+ {\mathfrak u}^0{\mathfrak u}^-\); \({\mathfrak u}^0\) is the group algebra of the group-like elements of \({\mathfrak u}\). Let \(\mathcal C\) be the category of finite dimensional \({\mathfrak u}\) where \({\mathfrak u}^0\) “acts by powers of \(\zeta\)”. By results of Kazhdan, Lusztig and the authors, \(\mathcal C\) is a rigid balanced tensor category, or ribbon category in the terminology of Turaev.
The main result of this book is the identification of the ribbon category \(\mathcal C\) with a category arising from the topology of configuration spaces. Specifically, the authors introduce the notion of “factorizable sheaf”; this is a compatible collection of perverse sheaves over configuration spaces, in a suitable sense. The category \(\mathcal{FS}\) of factorizable sheaves has a structure of tensor category and the authors show that \(\mathcal{FS}\) is isomorphic to \(\mathcal{C}\) as tensor category. Then they offer global versions of this result, placing sheaves into points of an arbitrary algebraic curve.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
55N33 Intersection homology and cohomology in algebraic topology
14H10 Families, moduli of curves (algebraic)
17B35 Universal enveloping (super)algebras
18B10 Categories of spans/cospans, relations, or partial maps
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)

Citations:

Zbl 0760.17014
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