Document Zbl 0938.17016 - zbMATH Open

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.

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