##### What's Hot

We consider commutativity equations \$F_i F_j =F_j F_i\$ for a function \$F(x^1,
\dots, x^N),\$ where \$F_i\$ is a matrix of the third order derivatives \$F_{ikl}\$.
We show that under certain non-degeneracy conditions a solution \$F\$ satisfies
the WDVV equations. Equivalently, the corresponding family of Frobenius
algebras has the identity field \$e\$. We also 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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