∨-systems, holonomy lie algebras, and logarithmic vector fields

Feigin, M. V. and Veselov, A. P. (2018) ∨-systems, holonomy lie algebras, and logarithmic vector fields. International Mathematics Research Notices, 2018(7), pp. 2070-2098. (doi: 10.1093/imrn/rnw289)

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It is shown that the description of certain class of representations of the holonomy Lie algebra g associated with hyperplane arrangement is essentially equivalent to the classification of ∨-systems associated with . The flat sections of the corresponding ∨-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any ∨-system is free in Saito’s sense and show this for all known ∨-systems and for a special class of ∨-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 ∨-systems. 1

Item Type:Articles
Additional Information:This work was also partly supported by the EPSRC (grant EP/J00488X/1) to Professor Veselov at the Loughborough University.
Glasgow Author(s) Enlighten ID:Feigin, Professor Misha
Authors: Feigin, M. V., and Veselov, A. P.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:International Mathematics Research Notices
Publisher:Oxford University Press
ISSN (Online):1687-0247
Published Online:08 January 2017
Copyright Holders:Copyright © 2017 The Authors
Publisher Policy:Reproduced under a Creative Commons License
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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
569301From elliptic systems to Frobenius manifolds - 6d theories and AGTMikhail FeiginRoyal Society (ROYSOC)JP101196M&S - MATHEMATICS