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Title: V-systems, holonomy Lie algebras and logarithmic vector fields
Authors: Feigin, Misha V.
Veselov, A.P.
Issue Date: 2017
Publisher: Oxford University Press (OUP)
Citation: FEIGIN, M.V. and VESELOV, A.P., 2017. V-systems, holonomy Lie algebras and logarithmic vector fields. International Mathematics Research Notices, doi: 10.1093/imrn/rnw289
Abstract: 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.
Description: © 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.
Sponsor: This work was partly supported by the EPSRC (grant EP/J00488X/1) and by the Royal Society/RFBR joint project JP101196/11-01-92612.
Version: Published
DOI: https://doi.org/10.1093/imrn/rnw289
10.1093/imrn/rnw289
URI: https://dspace.lboro.ac.uk/2134/23214
Publisher Link: https://doi.org/10.1093/imrn/rnw289
ISSN: 1687-3017
Appears in Collections:Published Articles (Maths)

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