Frobenius algebras and root systems: the trigonometric case

Shen, Dali

doi:10.1007/s11005-021-01466-2

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2022

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“…where function f = f (z) satisfies f ′′′ (z) = cot z, c α ∈ C and Q is a cubic polynomial depending on the additional variable y. Solutions of this form for reduced root systems and Weyl-invariant multiplicities were obtained by Hoevenaars and Martini in [19] (see also [24] and [4] for more details). They appear as almost dual prepotentials for the extended affine Weyl groups orbit spaces [10,12], see [22] for type A N .…”

Section: Introductionmentioning

confidence: 99%

Commutativity equations and their trigonometric solutions

Alkadhem¹,

Feigin²

2022

Preprint

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We consider commutativity equations F i F j = F j F i for a function F (x 1 , . . . , x N ), where F i is a matrix of the third order derivatives F ikl . We show that under certain nondegeneracy conditions a solution F satisfies the WDVV equations. Equivalently, the corresponding family of Frobenius algebras has the identity field e.We study trigonometric solutions F determined by a finite collection of vectors with multiplicities, and we give an explicit formula for e for all the known such solutions. The corresponding collections of vectors are given by non-simply laced root systems or are related to their projections to the intersection of mirrors.

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