Differential Equations and Quantum Groups
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A special class of flat logarithmic connections ∇R on ℂn associated with finite complex vector configurations R ⊂ ℂn generating ℂn is considered. The connection ∇R acts on the trivial holomorphic bundle with fiber ℂn and it has logarithmic poles on hyperplanes that are orthogonal to vectors of R with respect to the standard Hermitian form of ℂn. We prove that Veselov’s ∨-conditions for a complex vector configuration R are equivalent to the Frobenius integrability of the connection ∇R. If R is a root system with finite complex reflection group W(R), then R satisfies Veselov’s conditions and ∇R is an integrable connection. In the case of some root systems R, we describe the monodromy representation of the generalized braid group Bn(R) defined by the associated logarithmic connection ∇R on the quotient space ℂn/W(R). These representations are deformations of the standard representations of the corresponding complex reflection groups. They are generalizations of the Burau representations for some complex root systems which were earlier defined by Squier and Givental only for real root systems.
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