Hyperplane arrangements and holonomy equations. (English) Zbl 0848.18004

De Concini, C.; Procesi, C.

Sel. Math., New Ser. 1, No. 3, 495-535 (1995).

In this beautifully written paper, the authors study the solutions of the Knizhnik-Zamolodchikov equation associated to some hyperplane arrangements and made an important contribution to the subject. Firstly, a convenient framework for hyperplane arrangements was set up to handle the combinatorics. Then, using this framework, they recovered some of the fundamental results of Drinfeld’s theory, for example the Drinfeld associatior in quantum group. In the case of the hyperplane arrangements associated to a root system, they gave a description of the monodromy representation which can be regarded as an extension of the result of A. Kono for the root system \(A_n\). Finally, they explained how the universal Vassiliev invariants constructed by M. Kontsevich can be viewed as values of certain monodromy. Many of the above topics can be found in separate treatments by others and often in some abstract categorical framework. The advantage of the authors’ approach is that it presents these topics from a uniform and geometric viewpoint.

Reviewer: R.Lee (New Haven)

MSC:

18D10	Monoidal, symmetric monoidal and braided categories (MSC2010)
17B37	Quantum groups (quantized enveloping algebras) and related deformations
57M25	Knots and links in the \(3\)-sphere (MSC2010)

Keywords:

Knizhnik-Zamolodchikov equation; hyperplane arrangements; Drinfeld associatior; quantum group; root system; universal Vassiliev invariants; monodromy

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