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Arrangements, KZ systems and Lie algebra homology. (English) Zbl 0953.17011

Bruce, Bill (ed.) et al., Singularity theory. Proceedings of the European singularities conference, Liverpool, UK, August 18-24, 1996. Dedicated to C.T.C. Wall on the occasion of his 60th birthday. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 263, 109-130 (1999).
The author introduces Knizhnik-Zamolodchikov systems (KZ systems) associated to hyperplane complements in projective space. A KZ system is given by the data: \(\mathbb P^n\) a projective space, \(C\) a finite collection of hyperplanes in this space, \(F\) a trivial vector bundle over the complement \(U=\mathbb P^n\setminus (\bigcup_{H\in C} H)\) of the arrangement, and a flat logarithmic connection \(E\) on \(F\). A flat local section of \(F\) composed with a linear form on \(F\) is called a hypergeometric function. In this framework the basic example of the classical Knizhnik-Zamolodchikov equations associated to a finite-dimensional Lie algebra is included. V. V. Schechtman and A. N. Varchenko [Invent. Math. 106, 139-194 (1991; Zbl 0754.17024)] established a connection between the KZ equations and the cohomology of local systems on the complement of hyperplane arrangements. The article under review aims of presenting this approach in a simplified and self-contained way. Twisted coefficients and higher direct images of KZ systems are studied. The example of a KZ system appearing in the theory of root systems of Kac-Moody algebras is studied in detail.
For the entire collection see [Zbl 0919.00048].

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B55 Homological methods in Lie (super)algebras
14N20 Configurations and arrangements of linear subspaces
33C80 Connections of hypergeometric functions with groups and algebras, and related topics

Citations:

Zbl 0754.17024
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