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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Abstract
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. |
| Status: | Published |
| Refereed: | Yes |
| 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: | 1073-7928 |
| 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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