Commutativity equations and their trigonometric solutions

In the theory of Frobenius manifolds and Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations, one normally assumes that Frobenius algebras associated with a solution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>F</mml:mi> </mml:math> have an identity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>e</mml:mi> </mml:math> . Equivalently, the corresponding flat metric can be expressed as a linear combination of the matrices of the third-order derivatives of the pre-potential function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>F</mml:mi> </mml:math> . We show that under certain non-degeneracy conditions, this assumption can be omitted, that is, the identity field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>e</mml:mi> </mml:math> exists automatically without further assumptions. We also study trigonometric solutions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>F</mml:mi> </mml:math> determined by a finite collection of vectors with multiplicities, and we give an explicit formula for the field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>e</mml:mi> </mml:math> for all the known such solutions. The corresponding collections of vectors are given by the non-simply laced root systems or are related to their projections to the intersection of mirrors.