Vsystems, holonomy Lie algebras and logarithmic vector fields
Abstract
It is shown that the description of certain class of representations of the holonomy Lie algebra associated to hyperplane arrangement $\Delta$ is essentially equivalent to the classification of $\vee$systems associated to $\Delta.$ The flat sections of the corresponding $\vee$connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any $\vee$system is free in Saito's sense and show this for all known $\vee$systems and for a special class of $\vee$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 oneparameter deformations. We give formulas for these deformations as well as for the potentials of the classical families of harmonic $\vee$systems.
 Publication:

arXiv eprints
 Pub Date:
 July 2014
 DOI:
 10.48550/arXiv.1409.2424
 arXiv:
 arXiv:1409.2424
 Bibcode:
 2014arXiv1409.2424F
 Keywords:

 Mathematics  Representation Theory;
 Mathematical Physics;
 53D45;
 20F55
 EPrint:
 21 pages, slightly revised version, to appear in IMRN