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V-systems, holonomy Lie algebras and logarithmic vector fields

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posted on 2016-11-18, 16:30 authored by Mikhail V. Feigin, Alexander VeselovAlexander Veselov
It is shown that the description of certain class of representations of the holonomy Lie algebra g Δ associated to hyperplane arrangement is Δ essentially equivalent to the classification of V-systems associated to Δ. The flat sections of the corresponding V-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any V-system is free in Saito's sense and show this for all known V-systems and for a special class of V-systems called harmonic, which includes all Coxeter systems. In the irreducible Coxeter case the potentials of the corresponding gradient vector fields turn out to be Saito flat coordinates, or their one-parameter deformations. We give formulas for these deformations as well as for the potentials of the classical families of harmonic V-systems.


This work was partly supported by the EPSRC (grant EP/J00488X/1) and by the Royal Society/RFBR joint project JP101196/11-01-92612.



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  • Mathematical Sciences

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International Mathematics Research Notices


FEIGIN, M.V. and VESELOV, A.P., 2017. V-systems, holonomy Lie algebras and logarithmic vector fields. International Mathematics Research Notices, 2018 (7), pp.2070–2098.


Oxford University Press (OUP)


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© The Author(s) 2017. Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.