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Download fileV-systems, holonomy Lie algebras and logarithmic vector fields
journal contribution
posted on 2016-11-18, 16:30 authored by Mikhail V. Feigin, Alexander VeselovAlexander VeselovIt 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.
Funding
This work was partly supported by the EPSRC (grant EP/J00488X/1) and by the Royal Society/RFBR joint project JP101196/11-01-92612.
History
School
- Science
Department
- Mathematical Sciences
Published in
International Mathematics Research NoticesCitation
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.Publisher
Oxford University Press (OUP)Version
- VoR (Version of Record)
Publisher statement
This work is made available according to the conditions of the Creative Commons Attribution 4.0 International (CC BY 4.0) licence. Full details of this licence are available at: http://creativecommons.org/licenses/ by/4.0/Acceptance date
2016-11-08Publication date
2017Notes
© 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.ISSN
1687-3017Publisher version
Language
- en