# Commutativity equations and their trigonometric solutions

Maali Alkadhem and Misha Feigin

###### Abstract.

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 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.

##  1\. Introduction

A celebrated system of the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations
for a prepotential function $F(x)=F(x^{1},\dots,x^{N})$ has the form

$F_{ijk}g^{kl}F_{lmn}=F_{mik}g^{kl}F_{ljn},$ (1.1)

where

$\displaystyle F_{ijk}=\frac{\partial^{3}F}{\partial x^{i}\partial
x^{j}\partial x^{k}},$

and $G=(g^{kl})$ is a constant symmetric $N\times N$ matrix. These equations
appeared in topological field theories [DVV.1990, Witten.1990] and they are in
the core of Frobenius manifolds theory [Dubrovin.1996]. In these
considerations one normally has the property that the components of the flat
metric $G^{-1}$ can be represented as

$(G^{-1})_{ij}=\sum_{k=1}^{N}e^{k}F_{kij}$ (1.2)

for some vector field $e=\sum_{k=1}^{N}e^{k}(x)\partial_{x^{k}}$ which is the
identity field for the corresponding family of Frobenius algebras. For
example, in the Frobenius manifolds theory one normally has
$e=\partial_{x^{1}}$, which is flat with respect to the metric $G^{-1}$. In
the case of almost dual Frobenius manifold on the space of orbits of a finite
Coxeter group the field $e$ is not constant, it is proportional to the Euler
vector field [Dubrovin2004].

It is also of interest to consider equations (1.1) without the additional
assumption (1.2) which expresses the metric $G^{-1}$ as a linear combination
of the third order derivatives of the prepotential. Indeed, in the case of $G$
being the identity matrix the corresponding equations (1.1) have the form of
the commutativity equations

$F_{i}F_{j}=F_{j}F_{i},$ (1.3)

where $F_{i}$ is the $N\times N$ matrix with matrix entries
$(F_{i})_{kl}=F_{ikl}=\frac{\partial^{3}F}{\partial x^{i}\partial
x^{k}\partial x^{l}}.$

Equations (1.3) appeared in the study of $\mathcal{N}=4$ supersymmetric
mechanical system (see [Wyllard2000]). For a suitable ansatz for the
supercharges the supersymmetry algebra relations are satisfied provided that
equations (1.3) hold. Existence of the identity field or rather, more
specifically, additional relations of the form
$\sum_{i}x^{i}F_{ijk}=-\delta_{jk}$ lead to further superconformal symmetry
(see [Wyllard2000] and also [Bellucci2005], where the relation with the WDVV
equations is emphasized).

In this paper we are interested in commutativity equations (1.3) and the
additional condition of the existence of a vector field
$e=\sum_{i=1}^{N}e^{k}\partial_{x^{k}}$ such that $e(F_{ij})=\delta_{ij}$.
This vector field is the identity vector field for a family of algebras
depending on $x$. One of our main results provides a sufficient condition on
$F$ which ensures that $e$ exists. The components of the field can then be
expressed via determinants of the matrices whose entries are the third order
derivatives of the prepotential $F$. Similarly, for a general constant matrix
$G$ we establish a representation of $G^{-1}$ as a linear combination of the
matrices $F_{i}$ as a consequence of equations (1.1) (see Sections 6 and 7).

There is an interesting class of solutions of the equations (1.1), (1.2)
determined by finite collections $\mathcal{A}$ of vectors. The corresponding
prepotential has the form

$F=\sum_{\alpha\in\mathcal{A}}(\alpha,x)^{2}\log(\alpha,x),\quad x\in V.$
(1.4)

In the case when $\mathcal{A}$ is a root system such solutions of the WDVV
equations appeared in [Martini+Gragert1999]. They are almost dual
prepotentials for the finite group orbit spaces Frobenius manifolds
[Dubrovin2004]. Such solutions also appear in four-dimensional Seiberg–Witten
theory as perturbative parts of the corresponding prepotentials [MMM.2000].
More generally, solutions of the form (1.4) exist for special configurations
of vectors known as $\vee$-systems introduced by Veselov in [Veselov1999].
This class of of solutions was studied further in [Chalykh+Veselov2001,
Misha&Veselov2007, Misha&Veselov2008, Schreiber+Veselov2014]. Thus it was
shown that the class is closed under the operations of taking subsystems and
projections of $\mathcal{A}$, and such solutions have to do with Dubrovin’s
almost duality on the discriminant strata. Connection of these solutions to
the supersymmetric mechanics was explored in [GLK2009]. More generally, one
may also consider solutions of the form (1.4) for the commutativity equations
(1.1) (without extra condition (1.2)). The corresponding (irreducible)
configurations of vectors $\mathcal{A}$ can be shown to be the complex
Euclidean version of $\vee$-systems introduced in [Misha&Veselov2008].

There are also interesting trigonometric solutions of the equations (1.1),
(1.2) of the form

$F=\sum_{\alpha\in\mathcal{A}}c_{\alpha}f((\alpha,x))+Q(x,y),$ (1.5)

where function $f=f(z)$ satisfies $f^{{}^{\prime\prime\prime}}(z)=\cot
z,\,c_{\alpha}\in\mathbb{C}$ 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
[Martini2003] (see also [Shen2019] and [George+Misha2019] for more details).
They appear as almost dual prepotentials for the extended affine Weyl groups
orbit spaces [Dubrovin+Zhang1998, Dubrovin+Strachan+Zhang+Zuo2019], see
[Riley+Strachan2007] for type $A_{N}$. Such solutions also appeared in five-
dimensional Seiberg–Witten theory as perturbative parts of prepotentials
[MMM.2000]. In the case of simply laced root systems these solutions describe
quantum cohomology of resolutions of simple $A,D,E$ singularities [Bryan2008].
Solutions of the form (1.5) for general configurations $\mathcal{A}$ were
initially studied in [Misha2009] where a closely related notion of the
trigonometric $\vee$-system was introduced. Similarly to the rational case, we
showed in [Maali+Misha2021] that this class of solutions is closed under
restrictions and that a subsystem of a trigonometric $\vee$-system is also a
trigonometric $\vee$-system. The restriction procedure for the classical root
systems recovers solutions obtained by Pavlov from reductions of Egorov
hydrodynamic chains [Pavlov2006].

There are also elliptic versions of some of these solutions considered by
Riley and Strachan in [Riley+Strachan2006, Strachan2010].

It appears that solutions of the form (1.5) with $Q=0$ of the WDVV equations
(1.1), (1.3) may also exist. Such a solution for the root system $B_{N}$
appeared in [Martini2003] and it was generalized to $BC_{N}$ in [MGM2020]. The
corresponding metric $G$ is the identity so the commutativity equations (1.3)
hold as well.

Solutions of the form (1.5) with $Q=0$ for the commutativity equations (1.3)
for the root systems $\mathcal{A}=F_{4},G_{2}$ were obtained in
[Maali+Misha2021]. The corresponding multiplicities are Weyl invariant but
they have to satisfy a linear relation. A multi-parameter deformation of the
solution for the root system $BC_{N}$ was also obtained in [MGM2020]. It is
unclear whether there are more Frobenius manifold structures associated with
such solutions.

In this paper we study solutions of the commutativity equations (1.3) of the
form (1.5) with $Q=0$. Thus we give a $\vee$-system version of conditions
which the corresponding configuration of vectors has to satisfy, which we call
a Euclidean trigonometric $\vee$-system (see Section 2). We also show that
restrictions of solutions of the commutativity equations give new solutions
and that a subsystem of a Euclidean trigonometric $\vee$-system is also a
Euclidean trigonometric $\vee$-system (see Sections 3, 5). In Section 4 we
clarify relations of Euclidean trigonometric $\vee$-systems to other versions
of rational and trigonometric $\vee$-systems. All the known irreducible
solutions of the commutativity equations (1.3) of the form (1.5) with $Q=0$
are the non-simply laced root systems $BC_{N},F_{4},G_{2}$ with a relation
between invariant multiplicities as well as restrictions of such solutions to
the intersection of mirrors (in the case of $BC_{N}$ one can also extend
analytically integer parameters defining the restriction). In all these cases
we give an explicit uniform formula for the corresponding identity field $e$
in Section 8. Existence of the identity field implies that we also get new
solutions of WDVV equations (1.1), (1.3) in the case of root system $F_{4}$
and its projections.

##  2\. Commutativity equations and Euclidean trigonometric $\vee$-systems

Let $\mathcal{A}$ be a finite set of non-zero vectors in a Euclidean space
$V\cong\mathbb{C}^{N}$, $N\in\mathbb{N}$, with the bilinear inner product
$(\cdot,\cdot)$. Let $c\colon\mathcal{A}\to\mathbb{C}$ be the (multiplicity)
function. We denote $c_{\alpha}\coloneqq c(\alpha)$ for
$\alpha\in\mathcal{A}$. We assume that $\mathcal{A}$ belongs to a lattice of
rank $N$. For each vector $\alpha\in\mathcal{A}$ let us introduce the set of
its collinear vectors from $\mathcal{A}$:

$\delta_{\alpha}\coloneqq\\{\gamma\in\mathcal{A}\colon\gamma\sim\alpha\\}.$

Let $\delta\subseteq\delta_{\alpha}$ and $\alpha_{0}\in\delta_{\alpha}.$ Then
for any $\gamma\in\delta$ we have $\gamma=k_{\gamma}\alpha_{0}$ for some
$k_{\gamma}\in\mathbb{R}.$ Note that $k_{\gamma}$ depends on the choice of
$\alpha_{0}$ and different choices of $\alpha_{0}$ give rescaled collections
of these parameters. Define
$C_{\delta}^{\alpha_{0}}\coloneqq\displaystyle\sum_{\gamma\in\delta}c_{\gamma}k_{\gamma}^{2}.$
Note that $C_{\delta}^{\alpha_{0}}\neq 0$ if and only if
$C_{\delta}^{\widetilde{\alpha}_{0}}\neq 0$ for any
$\widetilde{\alpha}_{0}\in\delta.$ We define strings (or series) of vectors as
follows (cf. [Misha2009]).

For any $\alpha\in\mathcal{A}$ let us distribute all the vectors in
$\mathcal{A}\setminus\delta_{\alpha}$ into a disjoint union of
$\alpha$-strings

$\mathcal{A}\setminus\delta_{\alpha}=\bigsqcup_{s=1}^{k}\Gamma_{\alpha}^{s},$

where $k\in\mathbb{N}$ depends on $\alpha.$ These stings $\Gamma_{\alpha}^{s}$
are determined by the property that for any $s=1,\dots,k$ and for any two
covectors $\gamma_{1},\gamma_{2}\in\Gamma_{\alpha}^{s}$ one has either
$\gamma_{1}+\gamma_{2}=m\alpha$ or $\gamma_{1}-\gamma_{2}=m\alpha$ for some
$m\in\mathbb{Z}.$ We assume that the strings are maximal, that is if
$\gamma\in\Gamma_{\alpha}^{s}$ for some $s\in\mathbb{N},$ then
$\Gamma_{\alpha}^{s}$ must contain all the covectors of the form
$\pm\gamma+m\alpha\in\mathcal{A}$ with $m\in\mathbb{Z}.$ Note that if for some
$\beta\in\mathcal{A}$ there is no $\gamma\in\mathcal{A}$ such that
$\beta\pm\gamma=m\alpha$ for $m\in\mathbb{Z},$ then $\beta$ itself forms a
single $\alpha$-string.

By replacing some vectors from $\mathcal{A}$ with their opposite ones and
keeping the multiplicity unchanged one can get a new configuration whose
vectors belong to a half-space. We will denote such a system by
$\mathcal{A}_{+}.$ If this system contains repeated vectors $\alpha$ with
multiplicities $c_{\alpha}^{i}$ then we replace them with the single vector
$\alpha$ with multiplicity $c_{\alpha}\coloneqq\sum_{i}c_{\alpha}^{i}.$

Let us now define a Euclidean trigonometric $\vee$-system in analogy with a
trigonometric $\vee$-system [Misha2009].

######  Definition 2.1.

The pair $(\mathcal{A},c)$ is called a Euclidean trigonometric $\vee$-system
if for all $\alpha\in\mathcal{A}$ and for all $\alpha$-strings
$\Gamma_{\alpha}^{s},$ one has the relation

$\sum_{\beta\in\Gamma_{\alpha}^{s}}c_{\beta}(\alpha,\beta)\alpha\wedge\beta=0.$
(2.1)

Consider a function $F$ given by the formula

$F=\sum_{\alpha\in\mathcal{A}}c_{\alpha}f((\alpha,x)),$ (2.2)

where the function $f$ is given by

$f(z)=\frac{1}{6}iz^{3}+\frac{1}{4}Li_{3}(e^{-2iz})$

so that $f^{\prime\prime\prime}(z)=\cot z$. We are interested in
configurations $(\mathcal{A},c)$ such that the commutativity equations

$F_{i}F_{j}=F_{j}F_{i},\quad i,j=1,\dots,N,$ (2.3)

hold, where $F_{i}$ is the $N\times N$ matrix with entries

$(F_{i})_{pq}=F_{ipq}=\frac{\partial^{3}F}{\partial x^{i}\partial
x^{p}\partial x^{q}}.$

The following statement establishes invariance of the commutativity equations
under the action of the group of orthogonal transformations $O(N,\mathbb{C})$.
Summation from 1 to $N$ over repeated indices will be assumed throughout
unless indicated otherwise.

######  Proposition 2.2.

Suppose a function $F=F(x^{1},\dots,x^{N})$ satisfies commutativity equations
(1.3). Let $C=(C_{i}^{k})\in O(N,\mathbb{C})$, and let

$\displaystyle\widetilde{x}^{k}={C}_{i}^{k}x^{i},$ (2.4)

where $\widetilde{x}^{1},\dots,\widetilde{x}^{N}$ is a new coordinates system.
Then $\widetilde{F}(\widetilde{x})=F(x)$ satisfies commutativity equations

$\displaystyle\widetilde{F}_{i}\widetilde{F}_{j}=\widetilde{F}_{j}\widetilde{F}_{i},\quad
i,j=1,\dots,N,$ (2.5)

where
$(\widetilde{F}_{i})_{pq}=\frac{\partial^{3}\widetilde{F}}{\partial\tilde{x}^{i}\partial\tilde{x}^{p}\partial\tilde{x}^{q}}$.

###### Proof.

Since $\partial_{x^{i}}=C_{i}^{k}\partial_{\widetilde{x}^{k}}$, we have
$F_{ijk}={C}_{k}^{\widetilde{k}}{C}_{j}^{\widetilde{j}}{C}_{i}^{\widetilde{i}}\widetilde{F}_{\widetilde{i}\widetilde{j}\widetilde{k}}.$
Then commutativity equations $F_{ijk}F_{klm}=F_{mjk}F_{kli}$ in the new
coordinates take the form

$\displaystyle{C}_{k}^{\widetilde{k}}{C}_{j}^{\widetilde{j}}{C}_{i}^{\widetilde{i}}{C}_{k}^{a}{C}_{l}^{b}{C}_{m}^{d}\widetilde{F}_{\widetilde{i}\widetilde{j}\widetilde{k}}\widetilde{F}_{abd}={C}_{m}^{\widetilde{m}}{C}_{j}^{\widetilde{j}}{C}_{k}^{\widetilde{k}}{C}_{k}^{a}{C}_{l}^{b}{C}_{i}^{d}\widetilde{F}_{\widetilde{m}\widetilde{j}\widetilde{k}}\widetilde{F}_{abd}.$
(2.6)

Now we multiply both sides of equality (2.6) by
$\widehat{C}_{\alpha}^{m}\widehat{C}_{\beta}^{j}\widehat{C}_{\gamma}^{l}\widehat{C}_{\epsilon}^{i}$,
where $\widehat{C}=C^{-1}$ so that
$\widehat{C}_{\alpha}^{k}C_{k}^{\beta}=\delta_{\alpha}^{\beta}.$ We get

$\displaystyle
C_{k}^{\widetilde{k}}C_{k}^{a}\widetilde{F}_{\epsilon\beta\widetilde{k}}\widetilde{F}_{a\alpha\gamma}=C_{k}^{\widetilde{k}}C_{k}^{a}\widetilde{F}_{\alpha\beta\widetilde{k}}\widetilde{F}_{a\gamma\epsilon}.$
(2.7)

For an orthogonal transformation $C$ we have
$C_{k}^{\widetilde{k}}C_{k}^{a}=\delta^{\widetilde{k}a}$. Hence equality (2.7)
reduces to (2.5). ∎

We are going to establish a relation between solutions (2.2) of the
commutativity equations (2.3) and Euclidean trigonometric $\vee$-systems. The
following two lemmas hold.

######  Lemma 2.3.

The commutativity equations (2.3) for the function (2.2) are equivalent to the
identity

$\sum_{\alpha,\beta\in\mathcal{A}}c_{\alpha}c_{\beta}(\alpha,\beta)\cot(\alpha,x)\cot(\beta,x)B_{\alpha,\beta}(a,b)\alpha\wedge\beta=0,$
(2.8)

for all $a,b\in V,$ where
$B_{\alpha,\beta}(a,b)=\alpha\wedge\beta=(\alpha,a)(\beta,b)-(\alpha,b)(\beta,a).$

######  Lemma 2.4.

Suppose that identity (2.8) holds for any $a,b\in V$. Suppose also that
$C_{\delta}^{\alpha_{0}}\neq 0$ for any
$\alpha\in\mathcal{A},\,\delta\subseteq\delta_{\alpha},\,\alpha_{0}\in\delta_{\alpha}.$
Then $\mathcal{A}$ is a Euclidean trigonometric $\vee$-system.

Proofs of Lemmas 2.3 and 2.4 are similar to the proofs of analogous statements
in [Maali+Misha2021] for the case of the trigonometric $\vee$-system (see also
[Maali.2022]).

Note that if $\mathcal{A}$ is a Euclidean trigonometric $\vee$-system then the
left-hand side of identity (2.8) is non-singular. Since all vectors from
$\mathcal{A}$ belong to an $N$-dimensional lattice then the left-hand side of
identity (2.8) is a rational function in suitable exponential variables which
has degree zero and therefore is a constant. In order to find this constant,
by changing some of the vectors from $\mathcal{A}$ to their opposite ones we
can assume that all vectors from $\mathcal{A}$ belong to a half-space, hence
form a positive system $\mathcal{A}_{+}$. Then in an appropriate limit in a
cone $\cot(\alpha,x)\to i$ for all $\alpha\in\mathcal{A}_{+}$ and the identity
(2.8) reduces to

$\sum_{\alpha,\beta\in\mathcal{A}_{+}}c_{\alpha}c_{\beta}(\alpha,\beta)B_{\alpha,\beta}(a,b)\alpha\wedge\beta=0.$

From these considerations we get the following result.

######  Theorem 2.5.

Suppose that a configuration $(\mathcal{A},c)$ satisfies the condition
$C_{\delta}^{\alpha_{0}}\neq 0$ for any
$\alpha\in\mathcal{A},\,\delta\subseteq\delta_{\alpha},\,\alpha_{0}\in\delta_{\alpha}.$
Then the commutativity equations (2.3) for the prepotential (2.2) imply the
following two conditions:

(1) $\mathcal{A}$ is a Euclidean trigonometric $\vee$-system,

(2)
$\sum_{\alpha,\beta\in\mathcal{A}_{+}}c_{\alpha}c_{\beta}(\alpha,\beta)B_{\alpha,\beta}(a,b)\alpha\wedge\beta=0$
for all $a,b\in V$.

Conversely, if a configuration $(\mathcal{A},c)$ satisfies conditions (1), (2)
then commutativity equations (2.3) hold.

Root systems of Weyl groups provide examples of Euclidean trigonometric
$\vee$-systems.

######  Proposition 2.6.

A root system $\mathcal{A}=\mathcal{R}$ with Weyl-invariant multiplicity
function $c$ is a Euclidean trigonometric $\vee$-system.

###### Proof.

Fix $\alpha\in\mathcal{R}.$ Take any $\beta\in\mathcal{R},$ and let
$\gamma=s_{\alpha}\beta=\beta-\frac{2(\alpha,\beta)}{(\alpha,\alpha)}\alpha.$
Since $\frac{2(\alpha,\beta)}{(\alpha,\alpha)}\in\mathbb{Z}$ we get that
$\beta,\gamma\in\Gamma_{\alpha}^{s}$ for some $s.$ We have

$c_{\beta}=c_{\gamma},\quad(\alpha,\beta)=-(\alpha,\gamma),\quad\alpha\wedge\beta=\alpha\wedge\gamma.$

Hence the contribution of $\beta$ and $\gamma$ to the sum in (2.1) cancel each
other.

∎

In general root systems $\mathcal{A}=\mathcal{R}$ with invariant
multiplicities do not satisfy condition $(2)$ in Theorem 2.5. It has been
shown in [George+Misha2019] that this condition is satisfied for root systems
$\mathcal{R}=BC_{N},\,F_{4},\,G_{2}$ with special invariant multiplicities.

Solutions of commutativity equations can be applied to construct
$\mathcal{N}=4$ supersymmetric mechanical systems. Hamiltonians corresponding
to root systems $\mathcal{R}=BC_{N},\,F_{4},\,G_{2}$ were given explicitly in
[George+Misha2019].

##  3\. Subsytems of a Euclidean trigonometric $\vee$-system

Now we consider subsystems of a Euclidean trigonometric $\vee$-system.

######  Definition 3.1.

Let $\mathcal{A}\subset V$ be a finite collection of vectors. A subset
$\mathcal{B}\subseteq\mathcal{A}$ is called a subsystem if

$\mathcal{B}=\mathcal{A}\cap W$

for a linear subspace $W\subseteq V.$ The subsystem $\mathcal{B}$ is called
reducible if $\mathcal{B}$ is a disjoint union of two non-empty subsystems
$\mathcal{B}=\mathcal{B}_{1}\sqcup\mathcal{B}_{2}$. The subsystem
$\mathcal{B}$ is called irreducible if it is not reducible.

If $c$ is a multiplicity function for $\mathcal{A}$ then we will equip a
subsystem $\mathcal{B}\subseteq\mathcal{A}$ with the multiplicity function
which is the restriction of the multiplicity function $c$ on $\mathcal{B}$.

Assume that the linear span $\langle\mathcal{B\rangle}=W.$ We say that the
subsystem $\mathcal{B}$ is non-isotropic if the restriction of the inner
product $(\cdot,\cdot)$ onto $W$ is non-degenerate.

######  Theorem 3.2.

Any non-isotropic subsystem of a Euclidean trigonometric $\vee$-system is also
a Euclidean trigonometric $\vee$-system.

The proof of Theorem 3.2 is similar to the proof in [Maali+Misha2021] of the
analogous statement for the trigonometric $\vee$-system, see also
[Maali.2022].

##  4\. Relation with other types of $\vee$-systems

###  4.1. Relation with trigonometric $\vee$-systems

For a finite subset $\mathcal{A}\subset V$ with a multiplicity function
$c\colon\mathcal{A}\to\mathbb{C},$ consider a bilinear form
$G_{\mathcal{A},c}$ on $V$ given by

$G_{\mathcal{A},c}(x,y)=\sum_{\alpha\in\mathcal{A}}c_{\alpha}(\alpha,x)(\alpha,y),\quad
x,y\in V,$ (4.1)

where $c_{\alpha}\coloneqq c(\alpha)$. Following an analogy with the rational
case (see [Misha&Veselov2008] and subsection 4.2 below), we say that the pair
$(\mathcal{A},c)$ is well-distributed in $V$ if the bilinear form (4.1) is
proportional to the form $(\cdot,\cdot).$ The pair $(\mathcal{A},c)$ is called
a trigonometric $\vee$-system if it satisfies the relations

$\sum_{\beta\in\Gamma_{\alpha}^{s}}c_{\beta}G_{\mathcal{A},c}(\alpha,\beta)\alpha\wedge\beta=0$
(4.2)

for all $\alpha\in\mathcal{A}$ and all $\alpha$-strings $\Gamma_{\alpha}^{s}$.

Now let $(\mathcal{A},c)$ be a Euclidean trigonometric $\vee$-system. Define a
linear operator $M\colon V\to V$ as

$M=\sum_{\beta\in\mathcal{A}}c_{\beta}\beta\otimes\beta,$

that is, $M(u)=\sum_{\beta\in\mathcal{B}}c_{\beta}(\beta,u)\beta$ for any
$u\in V$. The following statement takes place.

######  Lemma 4.1.

Let $(\mathcal{A},c)$ be a Euclidean trigonometric $\vee$-system. Assume that
the linear span $\langle\mathcal{A}\rangle=V.$ Then

  1. (1)

Any $\alpha\in\mathcal{A}$ is an eigenvector of $M$,

  2. (2)

The vector space $V$ can be decomposed as

$V=V_{1}\oplus V_{2}\oplus\dots\oplus V_{k},\quad k\in\mathbb{N},$ (4.3)

where $M|_{V_{i}}=\lambda_{i}I,\,\lambda_{i}\in\mathbb{C},$ and $I$ is the
identity operator, and $\lambda_{i}\neq\lambda_{j}$ for $i\neq j$.

The proof of Lemma 4.1 is similar to the proof in [Maali+Misha2021] for the
trigonometric $\vee$-system case (see also [Maali.2022]).

Since $\mathcal{A}\subset V=V_{1}\oplus\dots\oplus V_{k},$ then $\mathcal{A}$
can be represented as a disjoint union

$\mathcal{A}=\mathcal{A}_{1}\sqcup\dots\sqcup\mathcal{A}_{k},$

where $\mathcal{A}_{i}\coloneqq\mathcal{A}\cap V_{i}\subset V_{i}$. The
following two lemmas relate the strings of vectors in $\mathcal{A}$ and its
components $\mathcal{A}_{i}$.

######  Lemma 4.2.

Let $\mathcal{A}$ be a Euclidean trigonometric $\vee$-system. Let
$\alpha\in\mathcal{A}$ be such that $\alpha\in V_{i}$ for some $i$. Consider
an $\alpha$-string $\Gamma_{\alpha}^{s}$ in $\mathcal{A}_{i}$ and let
$\beta\in\Gamma_{\alpha}^{s}.$ Then $\Gamma_{\alpha}^{s}\subset V_{i}$ or
$\Gamma_{\alpha}^{s}\subseteq\\{\pm\beta\\}.$

###### Proof.

For $\beta\in\Gamma_{\alpha}^{s}$ we have two possible cases.

Case (i) $\beta\in V_{i}.$ Then for any $\gamma\in\Gamma_{\alpha}^{s}$ we have
that $\gamma=m\alpha+\varepsilon\beta\in V_{i}$ for some $m\in\mathbb{Z}$ and
$\varepsilon=\pm 1$. Hence $\Gamma_{\alpha}^{s}\subset V_{i}.$

Case (ii) $\beta\notin V_{i}.$ Hence $\beta\in V_{j}$ for some $j\neq i.$ Then
for any $\gamma\in\Gamma_{\alpha}^{s}$ we have that $\gamma\in V_{i}$ or
$\gamma\in V_{j}$ since decomposition (4.3) is the direct sum. Note that
$\gamma\notin V_{i}$ as otherwise we will have
$\beta=m\alpha+\varepsilon\gamma\in V_{i},$ for some $m\in\mathbb{Z}$ and
$\varepsilon=\pm 1,$ which is a contradiction. Note also that $\gamma\notin
V_{j}$ unless $\gamma=\pm\beta$ as otherwise we have
$m\alpha=\beta+\varepsilon\gamma\in V_{j}$ for some $m\in\mathbb{Z}$ and
$\varepsilon=\pm 1$, which is a contradiction. Hence
$\Gamma_{\alpha}^{s}\subseteq\\{\pm\beta\\}$. ∎

######  Lemma 4.3.

Let $\alpha,\beta\in\mathcal{A}_{i}.$ Let
${{}^{\mathcal{A}}\Gamma_{\alpha}^{s}},\,{{}^{\mathcal{A}_{i}}\Gamma_{\alpha}^{t}}$
be the $\alpha$-strings in $\mathcal{A}$ and $\mathcal{A}_{i}$ respectively
containing $\beta.$ Then the set ${{}^{\mathcal{A}}\Gamma_{\alpha}^{s}}$ is
equal to the set ${{}^{\mathcal{A}_{i}}\Gamma_{\alpha}^{t}}$.

###### Proof.

Let $\gamma\in{{}^{\mathcal{A}}\Gamma_{\alpha}^{s}}.$ Then
$\gamma=m\alpha+\varepsilon\beta\in\mathcal{A}_{i},$ for some $m\in\mathbb{Z}$
and $\varepsilon=\pm 1$. Thus
$\gamma\in{{}^{\mathcal{A}_{i}}\Gamma_{\alpha}^{t}}$ by the maximality of
${{}^{\mathcal{A}_{i}}\Gamma_{\alpha}^{t}}$. Hence
${{}^{\mathcal{A}}\Gamma_{\alpha}^{s}}\subseteq{{}^{\mathcal{A}_{i}}\Gamma_{\alpha}^{t}}$.
The opposite inclusion is obvious. Therefore
${{}^{\mathcal{A}_{i}}\Gamma_{\alpha}^{t}}={{}^{\mathcal{A}}\Gamma_{\alpha}^{s}}$.
∎

Note that the operator $M$ is symmetric: $(M(u),v)=(u,M(v))$ for any $u,v\in
V$. Hence its eigenspaces are orthogonal.

######  Proposition 4.4.

We have $(u,v)=0$ for any $u\in V_{i}$ and $v\in V_{j}$ such that $i\neq j$.

The following statement takes place.

######  Lemma 4.5.

Restriction $(\cdot,\cdot)_{i}$ of the bilinear form $(\cdot,\cdot)$ onto the
subspace $V_{i}$ is non-degenerate.

###### Proof.

Suppose that $v\in V_{i}$ satisfies $(v,u)_{i}=0$ for all $u\in V_{i}$. By
Proposition 4.4 we have $(v,u)=0$ for all $u\in V$. Hence $v=0$ since
$(\cdot,\cdot)$ is non-degenerate. ∎

The following statement relates the Euclidean trigonometric $\vee$-systems and
the trigonometric $\vee$-systems.

######  Theorem 4.6.

If $\mathcal{A}$ is a Euclidean trigonometric $\vee$-system then the subsystem
$\mathcal{A}_{i}=\mathcal{A}\cap V_{i}$ is well-distributed in the subspace
$V_{i}$ with the bilinear form $(\cdot,\cdot)_{i}$ for all $i$. Furthermore,
if the bilinear form

$G_{\mathcal{A}_{i},c}(u,v)=\sum_{\alpha\in\mathcal{A}_{i}}c_{\alpha}(\alpha,u)(\alpha,v),\quad
u,v\in V_{i}$

is non-degenerate on $V_{i}$ (equivalently, $G_{\mathcal{A}_{i},c}$ is non-
zero), then $\mathcal{A}_{i}$ is a trigonometric $\vee$-system. Moreover,
$\mathcal{A}$ is a trigonometric $\vee$-system if the form $G_{\mathcal{A},c}$
is non-degenerate.

###### Proof.

By Lemma 4.1 we have $M|_{V_{i}}=\lambda_{i}I$. Hence for any $u\in V_{i}$ and
$v\in V$ we have

$G_{\mathcal{A},c}(u,v)=(M(u),v)=\lambda_{i}(u,v).$ (4.4)

Note also that by Proposition 4.4 we have that

$G_{\mathcal{A}_{i},c}(u,v)=G_{\mathcal{A},c}(u,v).$ (4.5)

Thus the subsystem $\mathcal{A}_{i}$ is well-distributed in the subspace
$V_{i}$.

Let us now assume that $G_{\mathcal{A}_{i},c}$ is non-degenerate on $V_{i}$,
that is $\lambda_{i}\neq 0$. Let $\alpha\in\mathcal{A}_{i}$. Consider an
$\alpha$-string $\Gamma_{\alpha}^{t}$ in $\mathcal{A}_{i}.$ Then by Lemmas
4.2, 4.3 and formulas (4.4), (4.5) we have

$\sum_{\beta\in\Gamma_{\alpha}^{t}}c_{\beta}G_{\mathcal{A}_{i},c}(\alpha,\beta)\alpha\wedge\beta=\lambda_{i}\sum_{\beta\in\Gamma_{\alpha}^{t}}c_{\beta}(\alpha,\beta)\alpha\wedge\beta=0$
(4.6)

since $\mathcal{A}$ is a Euclidean trigonometric $\vee$-system. This proves
that $\mathcal{A}_{i}$ is a trigonometric $\vee$-system. Finally, for
$\alpha\in\mathcal{A}_{i}$ let us consider its $\alpha$-string
${}^{\mathcal{A}}\Gamma_{\alpha}^{s}$ in $\mathcal{A}$. If
${}^{\mathcal{A}}\Gamma_{\alpha}^{s}\subset V_{i}$ then
$\sum_{\beta\in{{}^{\mathcal{A}}\Gamma_{\alpha}^{s}}}c_{\beta}G_{\mathcal{A},c}(\alpha,\beta)\alpha\wedge\beta=0$
by Lemma 4.3 and (4.6). If ${}^{\mathcal{A}}\Gamma_{\alpha}^{s}\not\subset
V_{i}$ then ${}^{\mathcal{A}}\Gamma_{\alpha}^{s}\subseteq\\{\pm\beta\\}$ for
some $\beta\in V_{j},j\neq i$, by Lemma 4.2. Then
$G_{\mathcal{A},c}(\alpha,\beta)=\lambda_{i}(\alpha,\beta)=\lambda_{j}(\alpha,\beta)$
by (4.4). Hence $G_{\mathcal{A},c}(\alpha,\beta)=0$ and the trigonometric
$\vee$-system condition holds. ∎

Let $U\subseteq{V}$ be a linear subspace such that $\langle\mathcal{A}\cap
U\rangle=U$. The following statement takes place.

######  Proposition 4.7.

Let $(\mathcal{A},c)$ be a Euclidean trigonometric $\vee$-system. Then the set
of vectors $\mathcal{A}\cap U$ with the multiplicity function
$c|_{\mathcal{A}\cap U}$ is well-distributed in $U$ or the system
$\mathcal{A}\cap U$ is reducible.

###### Proof.

Define a linear operator $M_{U}\colon U\to U$ by

$M_{U}\coloneqq\sum_{\beta\in\mathcal{A}\cap U}c_{\beta}\beta\otimes\beta.$

Let $\alpha\in\mathcal{A}\cap U$. Let us sum up the Euclidean trigonometric
$\vee$-condition (2.1) over $\alpha$-strings which belong to the subspace $U$.
Then

$\sum_{\beta\in\mathcal{A}\cap
U}c_{\beta}(\beta,\alpha)\beta=M_{U}(\alpha)=\lambda\alpha$

for some $\lambda=\lambda(\alpha).$ Suppose that $\mathcal{A}\cap U$ is
irreducible. Then $\lambda$ does not depend on $\alpha$ and $M_{U}=\lambda I$.
Therefore

$\sum_{\beta\in\mathcal{A}\cap
U}c_{\beta}(\beta,u)(\beta,v)=(M_{U}(u),v)=\lambda(u,v),$

and the pair $(\mathcal{A}\cap U,c|_{\mathcal{A}\cap U})$ is well-distributed.
∎

Suppose that $G_{\mathcal{A},c}$ is non-degenerate and define the vector
$\alpha^{\vee}\in V$ by the relation

$G_{\mathcal{A},c}(\alpha^{\vee},x)=(\alpha,x)$ (4.7)

for all $x\in V$. Now assume that $\alpha\in V_{i}$ in which case we also have
$\alpha^{\vee}\in V_{i}$ for some $i$. Then by Lemma 4.1 we have

$G_{\mathcal{A},c}(\alpha^{\vee},x)=(M_{U}(\alpha^{\vee}),x)=\lambda_{i}(\alpha^{\vee},x).$
(4.8)

Hence from relations (4.7), (4.8) we have that
$\alpha^{\vee}={\lambda_{i}}^{-1}\alpha$. Therefore if the pair
$(\mathcal{A},c)$ satisfies conditions (4.2) then for all $\alpha$-strings
$\Gamma_{\alpha}^{s}$ we have

$\sum_{\beta\in\Gamma_{\alpha}^{s}}c_{\beta}G_{\mathcal{A},c}(\alpha^{\vee},\beta^{\vee})\alpha\wedge\beta=\lambda_{i}^{-2}\sum_{\beta\in\Gamma_{\alpha}^{s}}c_{\beta}G_{\mathcal{A},c}(\alpha,\beta)\alpha\wedge\beta=0.$

These conditions coincide with the definition of the trigonometric
$\vee$-system given in [Misha2009] so the two definitions are equivalent. Let
us now introduce an inner product $\langle\cdot,\cdot\rangle$ on $V$ as

$\langle u,v\rangle\coloneqq G_{\mathcal{A},c}(u^{\vee},v^{\vee}),\quad u,v\in
V.$ (4.9)

The following statement is immediate.

######  Proposition 4.8.

Let $(\mathcal{A},c)$ be a trigonometric $\vee$-system. Then $(\mathcal{A},c)$
is a Euclidean trigonometric $\vee$-system with respect to the bilinear form
(4.9).

###  4.2. Relation with complex Euclidean $\vee$-systems

Following [Misha&Veselov2008], let us recall the notion of the (rational)
complex Euclidean $\vee$-system. Let $V$ be a complex vector space with a non-
degenerate bilinear form $(\cdot,\cdot).$ Let $\mathcal{A}\subset V$ be a
finite set of vectors. Consider the canonical form

$G_{\mathcal{A}}^{r}(x,y)=\sum_{\alpha\in\mathcal{A}}(\alpha,x)(\alpha,y),\quad
x,y\in V.$

Suppose that $G_{\mathcal{A}}^{r}$ is proportional to the form
$(\cdot,\cdot)$. Let $\pi\subseteq{V}$ be a two-dimensional subspace such that
$\langle\mathcal{A}\cap\pi\rangle=\pi$. $\mathcal{A}$ is said to be a
(rational) complex Euclidean $\vee$-system if for any such $\pi$ the subsystem
$\mathcal{B}=\mathcal{A}\cap\pi$ is reducible or the corresponding form
${G_{\mathcal{B}}^{r}}|_{\pi}$ is proportional to $(\cdot,\cdot)|_{\pi}$.

######  Proposition 4.9.

Let $F=\sum_{\alpha\in\mathcal{A}}(\alpha,x)^{2}\log(\alpha,x)$. Suppose that
$F$ satisfies the commutativity equations and $\mathcal{A}$ is irreducible.
Then $\mathcal{A}$ is a (rational) complex Euclidean $\vee$-system. Moreover,
if for a two-dimensional plane $\pi\subset V$ the subsystem
$\mathcal{A}\cap\pi$ is reducible then the corresponding directions are
orthogonal with respect to the form $(\cdot,\cdot)$.

Proof is similar to [Misha&Veselov2008]. The substitution of $F$ into the
commutativity equations gives the condition

$\sum_{\alpha,\beta\in\mathcal{A}}\frac{(\alpha,\beta)}{(\alpha,x)(\beta,x)}(\alpha\wedge\beta)(\alpha\wedge\beta)=0.$

It implies that for any two-dimensional plane $\pi$

$\sum_{\beta\in\pi\cap{\mathcal{A}}}(\alpha,\beta)\alpha\wedge\beta=0.$

The following statement relates the Euclidean trigonometric $\vee$-system and
the (rational) complex Euclidean $\vee$-system.

######  Proposition 4.10.

Let $(\mathcal{A},c)$ be an irreducible Euclidean trigonometric $\vee$-system.
Then the set of vectors $\sqrt{c_{\alpha}}\alpha,\,\alpha\in\mathcal{A}$, is a
(rational) complex Euclidean $\vee$-system.

###### Proof.

Firstly, since $\mathcal{A}$ is irreducible then by Lemma 4.1 we have
$\mathcal{A}=\mathcal{A}_{1}\subset V=V_{1}$ and $M|_{V_{1}}=\lambda_{1}I.$
Then by Theorem 4.6 we have that $\mathcal{A}_{1}=\mathcal{A}$ is well-
distributed.

Secondly, by Proposition 4.7 we have that any subsystem of $\mathcal{A}$ is
well-distributed or reducible, which implies the statement. ∎

##  5\. Restricted solutions of commutativity equations

In this Section we apply the restriction procedure to a given solution to the
commutativity equations. This gives new solutions of the commutativity
equations.

Let $\mathcal{B}=\mathcal{A}\cap W$ be a subsystem of $\mathcal{A}$ for some
$n$-dimensional linear subspace $W=\langle\mathcal{B}\rangle\subset V.$ Define

$W_{\mathcal{B}}\coloneqq\\{x\in
V\colon(\beta,x)=0\quad\forall\beta\in\mathcal{B}\\}.$

Let $(\cdot,\cdot)_{\mathcal{B}}$ be the restriction of $(\cdot,\cdot)$ on
$W_{\mathcal{B}},$ and assume that it is non-degenerate. Let
$S\subset\mathcal{B}$ be a basis of $W$. Let $f_{1},\dots,f_{n}$ be an
orthonormal basis of the space $W_{\mathcal{B}}$, and let
${\xi=(\xi^{1},\dots,\xi^{n})}$ be the corresponding orthonormal coordinates
in $W_{\mathcal{B}}$. Define
$M_{\mathcal{A}}=V\setminus\bigcup_{\alpha\in\mathcal{A}}\Pi_{\alpha},$ and
$M_{\mathcal{B}}=W_{\mathcal{B}}\setminus\bigcup_{\alpha\in\mathcal{A}\setminus\mathcal{B}}\Pi_{\alpha},$
where $\Pi_{\alpha}=\\{x\in V\colon(\alpha,x)=0\\}.$ The following statement
shows that the class of solutions of commutativity equations corresponding to
Euclidean $\vee$-systems is closed under the restrictions.

######  Theorem 5.1.

Assume that prepotential (2.2) satisfies commutativity equations (2.3).
Suppose that $C_{\delta}^{\alpha_{0}}\neq 0$ for any $\alpha\in
S,\alpha_{0}\in\delta_{\alpha},\,\delta\subseteq\delta_{\alpha}.$ Then the
prepotential

$F_{\mathcal{B}}=\sum_{\alpha\in\mathcal{A}\setminus\mathcal{B}}c_{\alpha}f(({\alpha},\xi)),\quad\xi\in
M_{\mathcal{B}},$ (5.1)

satisfies the commutativity equations

$(F_{\mathcal{B}})_{i}(F_{\mathcal{B}})_{j}=(F_{\mathcal{B}})_{j}(F_{\mathcal{B}})_{i},\quad
i,j=1,\dots,n,$

where $(F_{\mathcal{B}})_{i}$ is the $n\times n$ matrix with entries

$((F_{\mathcal{B}})_{i})_{pq}=(F_{\mathcal{B}})_{ipq}=\frac{\partial^{3}F_{\mathcal{B}}}{\partial\xi^{i}\partial\xi^{p}\partial\xi^{q}}.$

###### Proof.

First for any $u=(u^{1},\dots,u^{N}),\,v=(v^{1},\dots,v^{N})\in V$ let us
consider the vector fields
$\partial_{u}=u^{i}\partial_{x^{i}},\,\partial_{v}=v^{i}\partial_{x^{i}}\in
T_{x}M_{\mathcal{A}}.$ We define the following multiplication on the tangent
space $T_{x}M_{\mathcal{A}}$:

$\partial_{u}\ast\partial_{v}=u^{i}v^{j}F_{ijk}\partial_{x^{k}}.$ (5.2)

It is easy to check that the associativity of the multiplication $\ast$ is
equivalent to the commutativity equations (2.3). From the formula (2.2) we
have

$F_{ijk}=\sum_{\alpha\in\mathcal{A}}c_{\alpha}(\alpha,f_{i})(\alpha,f_{j})(\alpha,f_{k})\cot{(\alpha,x)}.$

Hence multiplication (5.2) takes the form

$\partial_{u}\ast\partial_{v}=\sum_{\alpha\in\mathcal{A}}c_{\alpha}(\alpha,u)(\alpha,v)\cot{(\alpha,x)}\partial_{\alpha}.$
(5.3)

By identifying $V\cong T_{x}V$, we have

$u\ast
v=\sum_{\alpha\in\mathcal{A}}c_{\alpha}(\alpha,u)(\alpha,v)\cot{(\alpha,x)}{\alpha}.$

Consider now a point $x_{0}\in M_{\mathcal{B}}$ and two tangent vectors
$u_{0},v_{0}\in T_{x_{0}}M_{\mathcal{B}}.$ We extend vectors $u_{0}$ and
$v_{0}$ to two local analytic vector fields $u(x),v(x)$ in the neighbourhood
$U$ of $x_{0}$ that are tangent to the subspace $W_{\mathcal{B}}$ at any point
$x\in M_{\mathcal{B}}\cap U$ such that $u_{0}=u(x_{0})$ and $v_{0}=v(x_{0})$.
The proof of the next lemma is similar to the proof of [Maali+Misha2021]*Lemma
4.1 (see also [Misha&Veselov2007]*Lemma 1 for the rational case, and [MGM2020]
for the trigonometric case).

######  Lemma 5.2.

The limit of the product $u(x)\ast v(x)$ exists when vector $x$ tends to
$x_{0}\in M_{\mathcal{B}}$ and it satisfies

$u_{0}\ast
v_{0}=\sum_{\alpha\in\mathcal{A}\setminus\mathcal{B}}c_{\alpha}(\alpha,u_{0})(\alpha,v_{0})\cot(\alpha,x_{0})\alpha.$

In particular, the product $u_{0}\ast v_{0}$ is determined by vectors $u_{0}$
and $v_{0}$ only.

The following lemma holds and it shows that multiplication (5.3) is closed on
the tangent space $T_{x_{0}}M_{\mathcal{B}}$.

######  Lemma 5.3.

Let $u,v\in T_{x_{0}}M_{\mathcal{B}}$ where $x_{0}\in M_{\mathcal{B}}.$ Then
$u\ast v\in T_{x_{0}}M_{\mathcal{B}}.$

The proof of Lemma 5.3 is similar to the proof of [Maali+Misha2021]*Lemma 4.2.
It uses an argument analogues to [Maali+Misha2021] which claims that the
following identity holds for any $a,b\in V$ if $\tan(\alpha,x)=0$:

$\sum_{\beta\in\mathcal{A}\setminus\delta_{\alpha}}c_{\beta}(\alpha,\beta)\cot(\beta,x)B_{\alpha,\beta}(a\otimes
b)\alpha\wedge\beta=0.$

Then for $u,v\in T_{x_{0}}M_{\mathcal{B}}$, $x_{0}\in M_{\mathcal{B}},$ the
product (5.3) takes the form

$\partial_{u}\ast\partial_{v}=\sum_{\alpha\in\mathcal{A}\setminus\mathcal{B}}c_{\alpha}({\alpha},u)({\alpha},v)\cot{({\alpha},x_{0})}\partial_{\alpha}.$

By using the orthonormal basis $f_{1},\dots,f_{n}$ of $W_{\mathcal{B}}$ we
rearrange $\partial_{{\alpha}}$ as

$\partial_{\alpha}=\sum_{k=1}^{n}({\alpha},f_{k})\partial_{f_{k}}.$

Hence for $x_{0}=\xi=\sum_{i=1}^{n}\xi^{i}f_{i}$ we have

$\displaystyle\partial_{f_{i}}\ast\partial_{f_{j}}$
$\displaystyle=\sum_{\alpha\in\mathcal{A}\setminus\mathcal{B}}\sum_{k=1}^{n}c_{\alpha}({\alpha},f_{i})({\alpha},f_{j})({\alpha},f_{k})\cot{({\alpha},\xi)}\partial_{f_{k}}$
$\displaystyle=\sum_{k=1}^{n}\widetilde{F}_{ijk}\partial_{f_{k}},\quad
i,j=1,\dots,n,$ (5.4)

where
$\widetilde{F}(\xi)=\sum_{\alpha\in\mathcal{A}\setminus\mathcal{B}}c_{\alpha}f(\alpha,\xi)=F_{\mathcal{B}}.$
Now multiplication (5) is associative and it is easy to check that its
associativity is equivalent to the commutativity equations

$\widetilde{F}_{i}\widetilde{F}_{j}=\widetilde{F}_{j}\widetilde{F}_{i},\quad
i,j=1,\dots,n.$

∎

As an application of Theorem 5.1 we get the following solutions of the
commutativity equations starting from a solution for the root system $BC_{N}$
[MGM2020]. Let
$q,r,s\in\mathbb{C},\,\underline{m}=(m_{1},\dots,m_{n})\in\mathbb{(C^{\times})}^{n}$.
Suppose

$r=-8s-2q(N-2)$

with $N=\sum_{i=1}^{n}m_{i}$. Define the configuration
$BC_{n}(q,r,s;\underline{m})\subset\mathbb{C}^{n}$ consisting of the following
vectors $\alpha$ with the multiplicities $c_{\alpha}$:

$\displaystyle m_{i}^{-1/2}e_{i},\quad\text{with multiplicity}\quad
rm_{i},\quad 1\leq i\leq n,$ $\displaystyle 2m_{i}^{-1/2}e_{i},\quad\text{with
multiplicity}\quad sm_{i}+\frac{1}{2}qm_{i}(m_{i}-1),\quad 1\leq i\leq n,$
$\displaystyle m_{i}^{-1/2}e_{i}\pm m_{j}^{-1/2}e_{j},\quad\text{with
multiplicity}\quad qm_{i}m_{j},\quad 1\leq i<j\leq n.$ (5.5)

Note that in the case $m_{i}=1$ for all $i=1,\dots,n$ configuration (5)
reduces to the positive half $BC_{n}^{+}$ of the root system $BC_{n}$.
Consider the function

$\widetilde{F}=\sum_{\alpha\in
BC_{n}(q,r,s;\underline{m})}c_{\alpha}f((\alpha,{x})),\quad{x}\in\mathbb{C}^{n}.$
(5.6)

It follows from Theorem 5.1 and it was shown earlier in [MGM2020] that
function (5.6) satisfies the commutativity equations

$\widetilde{F}_{i}\widetilde{F}_{j}=\widetilde{F}_{j}\widetilde{F}_{i}\quad
i,j=1,\dots,n.$

Consider now the positive half $F_{4}^{+}$ of the root system $F_{4}$ given by

$F_{4}^{+}=\\{e_{i}\,(1\leq i\leq 4),\,\,e_{i}\pm e_{j}\,(1\leq i<j\leq
4),\,\,\frac{1}{2}(e_{1}\pm e_{2}\pm e_{3}\pm e_{4})\\}.$ (5.7)

Let $r$ be the multiplicity of short roots and let $q$ be the multiplicity of
long roots. A basis of simple roots consists of
${\alpha_{1}={e_{2}-e_{3}},\alpha_{2}={e_{3}-e_{4}},\alpha_{3}={e_{4}}}$,
${\alpha_{4}=\frac{1}{2}(e_{1}-e_{2}-e_{3}-e_{4})}.$

Up to an orthogonal transformation there are two projected systems in
dimension three and four projected systems in dimension two. Firstly, we give
details of the three-dimensional projections of $F_{4}^{+}$. There are two
different projections of $F_{4}^{+}$ along the root system $A_{1}$. The first
one $(F_{4},A_{1})_{1}$ is obtained by projecting to the hyperplane
$\alpha_{3}=0.$ The second one $(F_{4},A_{1})_{2}$ is obtained by projecting
to the hyperplane $\alpha_{2}=0.$ Hence we have the following three-
dimensional projected systems of the positive root system $F_{4}^{+}$.

  * •

The projected system $(F_{4},A_{1})_{1}$ consists of the following vectors:

$\displaystyle e_{i},\quad\text{with multiplicity}\quad r+2q,\quad 1\leq i\leq
3,$ $\displaystyle e_{i}\pm e_{j},\quad\text{with multiplicity}\quad q,\quad
1\leq i<j\leq 3,$ $\displaystyle\frac{1}{2}(e_{1}\pm e_{2}\pm
e_{3}),\quad\text{with multiplicity}\quad 2r.$

  * •

The projected system $(F_{4},A_{1})_{2}$ consists of the following vectors
(after doing a change of variables and renaming vectors):

$\displaystyle e_{1},e_{2},\quad\text{with multiplicity}\quad r,$
$\displaystyle\sqrt{2}e_{3},\quad\text{with multiplicity}\quad q,$
$\displaystyle\frac{\sqrt{2}}{2}e_{3},\quad\text{with multiplicity}\quad 2r,$
$\displaystyle e_{1}\pm e_{2},\quad\text{with multiplicity}\quad q,$
$\displaystyle\frac{1}{2}(e_{1}\pm e_{2}),\quad\text{with multiplicity}\quad
2r,$ $\displaystyle
e_{1}\pm\frac{\sqrt{2}}{2}e_{3},\,e_{2}\pm\frac{\sqrt{2}}{2}e_{3},\quad\text{with
multiplicity}\quad 2q,$ $\displaystyle\frac{1}{2}(e_{1}\pm
e_{2}\pm\sqrt{2}e_{3}),\quad\text{with multiplicity}\quad r.$

Secondly, we give details of the two-dimensional projections of $F_{4}^{+}$.
There are two different projections of $F_{4}^{+}$ along the root system
$A_{2}$. The first one $(F_{4},A_{2})_{1}$ is obtained by projecting to the
plane $\alpha_{1}=\alpha_{2}=0.$ The second one $(F_{4},A_{2})_{2}$ is
obtained by projecting to the plane $\alpha_{3}=\alpha_{4}=0.$ There is also a
projected configuration $(F_{4},A_{1}^{2})$ along the subsystem $A_{1}\times
A_{1}$ to the plane $\alpha_{1}=\alpha_{3}=0$, and there is a projected
configuration $(F_{4},B_{2})$ along the subsystem $B_{2}$ to the plane
$\alpha_{2}=\alpha_{3}=0$. These configurations have the following explicit
form

  * •

The configuration $(F_{4},A_{2})_{1}$ consists of vectors $\alpha$ with the
corresponding multiplicities $c_{\alpha}$ given as follows:

$\displaystyle e_{1},\quad\text{with multiplicity}\quad r,$
$\displaystyle\frac{1}{\sqrt{3}}e_{2},\quad\text{with multiplicity}\quad 3r,$
$\displaystyle\frac{2}{\sqrt{3}}e_{2},\quad\text{with multiplicity}\quad 3q,$
$\displaystyle e_{1}\pm\frac{1}{\sqrt{3}}e_{2},\quad\text{with
multiplicity}\quad 3q,$
$\displaystyle\frac{1}{2}(e_{1}\pm\frac{1}{\sqrt{3}}e_{2}),\quad\text{with
multiplicity}\quad 3r,$
$\displaystyle\frac{1}{2}(e_{1}\pm\frac{3}{\sqrt{3}}e_{2}),\quad\text{with
multiplicity}\quad r.$

  * •

The configuration $(F_{4},A_{1}^{2})$ consists of vectors $\alpha$ with the
corresponding multiplicities $c_{\alpha}$ given as follows:

$\displaystyle e_{1},\quad\text{with multiplicity}\quad r+2q,$
$\displaystyle\sqrt{2}e_{2},\quad\text{with multiplicity}\quad q,$
$\displaystyle\frac{1}{2}e_{1},\quad\text{with multiplicity}\quad 4r,$
$\displaystyle\frac{\sqrt{2}}{2}e_{2},\quad\text{with multiplicity}\quad
2(r+2q),$ $\displaystyle e_{1}\pm\frac{1}{\sqrt{2}}e_{2},\quad\text{with
multiplicity}\quad 2q,$
$\displaystyle\frac{1}{2}(e_{1}\pm\frac{2}{\sqrt{2}}e_{2}),\quad\text{with
multiplicity}\quad 2r.$

Configurations $(F_{4},A_{2})_{2}$ and $(F_{4},B_{2})$ are equivalent to the
root systems $G_{2}$ and $BC_{2}$, respectively, the corresponding solutions
of the commutativity equations were found in [George+Misha2019]. Theorem 5.1
gives new solutions of the commutativity equations which are listed in the
next statement.

######  Theorem 5.4.

Let $({\mathcal{A}},c)$ be one of the configurations
$(F_{4},A_{1})_{1},\,(F_{4},A_{1})_{2},\,(F_{4},A_{2})_{1},\,(F_{4},A_{1}^{2})$
described above. Then the function
$F=\sum_{\alpha\in{\mathcal{A}}}c_{\alpha}f((\alpha,x))$ satisfies the
commutativity equations, where $x\in\mathbb{C}^{3}$ for the first two
configurations and $x\in\mathbb{C}^{2}$ for the last two configurations and
parameters $r,q$ satisfy the condition $r=-2q$ or $r=-4q$.

##  6\. Commutativity equations and WDVV equations

In this Section we investigate the relation between the commutativity
equations and the WDVV equations. Let $V\cong\mathbb{C}^{N},N\geq 2$. Let
$F=F(x^{1},\dots,x^{N})$ be a function on $V$. We recall that it was proven in
[Martini+Gragert1999] (see also [MMM.2000]) that the generalized WDVV
equations

${F}_{i}{F}_{j}^{-1}{F}_{k}={F}_{k}{F}_{j}^{-1}{F}_{i},\quad i,j,k=1,\dots,N$

can be written equivalently in the form

${F}_{i}B^{-1}{F}_{j}={F}_{j}B^{-1}{F}_{i},\quad i,j=1,\dots,N,$ (6.1)

where $B$ is any non-degenerate matrix of the form
$B=\sum_{i=1}^{N}A^{k}F_{k}$ for some functions $A^{k}$. If the matrix $B$
happens to be a multiple of the identity matrix then WDVV equations (6.1)
reduce to the commutativity equations

${F}_{i}{F}_{j}={F}_{j}{F}_{i},\quad i,j=1,\dots,N.$ (6.2)

A natural question to investigate is when there exists such a linear
combination $B$. We give an answer in this Section.

Let us assume that a function $F=F(x^{1},\dots,x^{N})$ satisfies the
commutativity equations (6.2). Let us denote by $[F_{i},F_{j}]_{(a,b)}$ the
$(a,b)$-entry of the commutator $[F_{i},F_{j}]$ or, more explicitly,

$[F_{i},F_{j}]_{(a,b)}=\sum_{m=1}^{N}(F_{iam}F_{jbm}-F_{ibm}F_{jam}).$ (6.3)

The equality $[F_{i},F_{k}]_{(i,j)}=0$ implies that

$\displaystyle F_{ijk}F_{iii}=\sum_{m=1}^{N}F_{ijm}F_{ikm}-\sum_{m\neq
i}F_{iim}F_{jkm}.$ (6.4)

Observe the equality of matrix entries
$[F_{a},F_{b}]_{(i,j)}=[F_{i},F_{j}]_{(a,b)}$. Introduce the notation

$[F_{i},F_{j}]_{(a,b)}^{\\{m\\}}=F_{iam}F_{jbm}-F_{ibm}F_{jam},$

where there is no summation over $m$ in the right-hand side. Define a matrix
$B=(B_{ij})_{i,j=1}^{N}$ with the entries given as a linear combination of the
third order derivatives of $F$:

$B_{ij}=\sum_{k=1}^{N}A^{k}F_{kij},$ (6.5)

for some functions $A^{k}=A^{k}(x^{1},\dots,x^{N})$. Now we will investigate
when there exists such a combination $B$ so that equations (6.2) imply
equations (6.1). For that it is sufficient to deduce that the matrix $B$ is
proportional to the identity.

Fix $i_{0}\in\mathbb{N},\,1\leq i_{0}\leq N$. Let $P$ be the $(N-1)\times N$
matrix $P=(F_{i_{0}ij}),\,1\leq i,j\leq N,\,i\neq i_{0}$. Define

$A^{k}=(-1)^{k+1}\det P_{k},$ (6.6)

where the matrix $P_{k}$ is obtained from the matrix $P$ by removing its
$k$-th column.

The following statement takes place.

######  Lemma 6.1.

For any function $F=F(x^{1},\dots,x^{N})$ which satisfies the commutativity
equations (6.2) the following relation holds

$\det\begin{pmatrix}F_{ai_{0}r}&F_{bi_{0}r}&F_{ci_{0}r}\\\
F_{ai_{0}t}&F_{bi_{0}t}&F_{ci_{0}t}\\\
F_{art}&F_{brt}&F_{crt}\end{pmatrix}=-\sum_{m\neq
t}\det\begin{pmatrix}F_{ai_{0}r}&F_{bi_{0}r}&F_{ci_{0}r}\\\
F_{ai_{0}m}&F_{bi_{0}m}&F_{ci_{0}m}\\\ F_{arm}&F_{brm}&F_{crm}\end{pmatrix},$

where $1\leq a,b,c\leq N,\,1\leq r<t\leq N$, and $r,t\neq i_{0}$.

###### Proof.

By applying the first row expansion and the commutativity equations we get

$\displaystyle\det\begin{pmatrix}F_{ai_{0}r}&F_{bi_{0}r}&F_{ci_{0}r}\\\
F_{ai_{0}t}&F_{bi_{0}t}&F_{ci_{0}t}\\\
F_{art}&F_{brt}&F_{crt}\end{pmatrix}=F_{ai_{0}r}\det\begin{pmatrix}F_{bi_{0}t}&F_{ci_{0}t}\\\
F_{brt}&F_{crt}\end{pmatrix}-F_{bi_{0}r}\det\begin{pmatrix}F_{ai_{0}t}&F_{ci_{0}t}\\\
F_{art}&F_{crt}\end{pmatrix}$
$\displaystyle+F_{ci_{0}r}\det\begin{pmatrix}F_{ai_{0}t}&F_{bi_{0}t}\\\
F_{art}&F_{brt}\end{pmatrix}=F_{ai_{0}r}[F_{i_{0}},F_{r}]_{(b,c)}^{\\{t\\}}-F_{bi_{0}r}[F_{i_{0}},F_{r}]_{(a,c)}^{\\{t\\}}+F_{ci_{0}r}[F_{i_{0}},F_{r}]_{(a,b)}^{\\{t\\}}$
$\displaystyle=-F_{ai_{0}r}\sum_{m\neq
t}[F_{i_{0}},F_{r}]_{(b,c)}^{\\{m\\}}+F_{bi_{0}r}\sum_{m\neq
t}[F_{i_{0}},F_{r}]_{(a,c)}^{\\{m\\}}-F_{ci_{0}r}\sum_{m\neq
t}[F_{i_{0}},F_{r}]_{(a,b)}^{\\{m\\}}$ $\displaystyle=-\sum_{m\neq
t}\det\begin{pmatrix}F_{ai_{0}r}&F_{bi_{0}r}&F_{ci_{0}r}\\\
F_{ai_{0}m}&F_{bi_{0}m}&F_{ci_{0}m}\\\ F_{arm}&F_{brm}&F_{crm}\end{pmatrix}.$

∎

The following statement takes place.

######  Lemma 6.2.

Suppose that a function $F=F(x^{1},\dots,x^{N})$ satisfies the commutativity
equations (6.2). Let $1\leq r<t\leq N,\,r,t\neq i_{0}$. Let $N\times N$ matrix
$Q$ be obtained from the matrix $P$ by inserting the $i_{0}$-th row
$R_{i_{0}}=(F_{1rt},\dots,F_{Nrt})$. Then $\det Q=0$.

###### Proof.

Let $D=\det Q$. Let $R_{i}$ denote the $i$-th row in the matrix $Q$. Let us
perform the Laplace expansion of $D$ along the rows

$\displaystyle R_{r}$
$\displaystyle=(F_{i_{0}1r},F_{i_{0}2r},\dots,F_{i_{0}Nr}),$ $\displaystyle
R_{t}$ $\displaystyle=(F_{i_{0}1t},F_{i_{0}2t},\dots,F_{i_{0}Nt}),$

and the row $R_{i_{0}}$. For subsets $S,T\subset[N]\coloneqq\\{1,\dots,N\\}$
we define $Q_{ST}$ to be the submatrix of $Q$ defined by deleting rows $S$ and
columns $T$. Let $I=\\{r,t,i_{0}\\},\,J=\\{a,b,c\\}$ for some $1\leq a<b<c\leq
N.$ By applying the Laplace expansion to $Q$ we get

$D=\varepsilon\sum_{J}\sigma_{J}\det
Q_{IJ}\det\begin{pmatrix}F_{ai_{0}r}&F_{bi_{0}r}&F_{ci_{0}r}\\\
F_{ai_{0}t}&F_{bi_{0}t}&F_{ci_{0}t}\\\ F_{art}&F_{brt}&F_{crt}\end{pmatrix},$
(6.7)

where $\varepsilon=\pm 1$ is determined by the relative order of $r,t$ and
$i_{0}$, and $\sigma_{J}=(-1)^{s}$ with $s=\sum_{i\in I}i+\sum_{j\in J}j$. Now
by Lemma 6.1 the determinant (6.7) can be rewritten as

$\displaystyle D$ $\displaystyle=-\varepsilon\sum_{J}\sigma_{J}\det
Q_{IJ}\Big{(}\sum_{m\neq
t}\det\begin{pmatrix}F_{ai_{0}r}&F_{bi_{0}r}&F_{ci_{0}r}\\\
F_{ai_{0}m}&F_{bi_{0}m}&F_{ci_{0}m}\\\
F_{arm}&F_{brm}&F_{crm}\end{pmatrix}\Big{)}$ $\displaystyle=-\sum_{m\neq
t}\det Q^{(m)},$

where the matrix $Q^{(m)}$ is obtained from the matrix $Q$ by replacing rows
$R_{i_{0}}$ and $R_{t}$ as follows:

$\displaystyle R_{i_{0}}\to\widetilde{R}_{i_{0}}=(F_{1rm},\dots,F_{Nrm}),$
$\displaystyle R_{t}\to\widetilde{R}_{t}=(F_{i_{0}1m},\dots,F_{i_{0}Nm}).$

Note that the row $\widetilde{R}_{t}$ is equal to the $m$-th row of the matrix
$Q^{(m)}$. Hence $\det D=0$. ∎

The following statement takes place.

######  Proposition 6.3.

Assume that the function $F=F(x^{1},\dots,x^{N})$ satisfies the commutativity
equations (6.2). Assume also that the rank of the matrix $P$ is $N-1$. Then
matrix $B$ with the entries given by formulae (6.5), (6.6) is diagonal.

###### Proof.

Let us assume that $i_{0}=1$, the general case can be dealt with similarly.
Consider the system of linear equations

$B_{1m}=\sum_{k=1}^{N}A^{k}F_{1km}=0,$ (6.8)

for some functions $A^{k}=A^{k}(x^{1},\dots,x^{N})$. The system (6.8)
represents a homogeneous system of $N-1$ linear equations in variables
$A^{k}$. The assumption that the rank of the matrix $P=(F_{1ij})$, where
$2\leq i\leq N,\,1\leq j\leq N$, is $N-1$ implies that the system (6.8) has a
non-trivial solution which is unique up to proportionality.

Now, fix $2\leq s\leq N$. The direct substitution of the functions $A^{k}$
given by formula (6.6) into the right-hand side of relation (6.8) gives a row
expansion of the determinant of a matrix with the repeated rows, hence the
equation $B_{1s}=0$ is satisfied. Note also that $A^{k}\neq 0$ for some $k$
since the rank of the matrix $P$ is $N-1$.

Now we will show that off-diagonal entries $B_{rt}=0,$ where $2\leq r<t\leq
N$. In order to do so, we add a row corresponding to the non-diagonal entry
$B_{rt}$ to the coefficient matrix $P$ of the linear system (6.8) and we will
show that the resulting matrix is singular. This will imply the existence of a
non-trivial solution to the resulting system of $N$ equations. Indeed, as the
first $N-1$ equations have a unique solution given by (6.6) up to
proportionality, it also has to solve the last equation. Thus we consider
equations (6.8) together with

$B_{rt}=\sum_{k=1}^{N}A^{k}F_{krt}=0$ (6.9)

as a system of linear equations for functions $A^{k}$. Its coefficient matrix
is singular by Lemma 6.2. This proves the statement.

∎

The following statement gives a further property of the matrix $B$.

######  Proposition 6.4.

Under the assumptions of Proposition 6.3 we have

$B_{11}=B_{pp}$

for all $1\leq p\leq N$.

###### Proof.

Let us assume that $i_{0}=1$, the general case can be dealt with similarly.
Let us first consider the case $p=2$. Since the matrix $P$ has rank $N-1$,
this implies that there exists some $q\,\,(1\leq q\leq N)$ such that
$F_{12q}\neq 0.$ Following the idea of the proof of Proposition 6.3, let us
consider the following set of homogeneous equations:

$\displaystyle B_{1m}=0,\quad 2\leq m\leq N,$ $\displaystyle
F_{12q}(B_{11}-B_{22})=0.$ (6.10)

It is sufficient to show that the coefficient matrix $M$ corresponding to
equations (6) considered as linear equations for $A^{k}$ is singular. We have

$M=\begin{pmatrix}F_{112}&F_{212}&\cdots&F_{N12}\\\
F_{113}&F_{213}&\cdots&F_{N13}\\\ \vdots&\vdots&\ddots&\vdots\\\
F_{11N}&F_{21N}&\cdots&F_{N1N}\\\
F_{12q}(F_{111}-F_{122})&F_{12q}(F_{211}-F_{222})&\cdots&F_{12q}(F_{N11}-F_{N22})\end{pmatrix}.$

Let $D=\det M$. From the identity (6.4) we have

$F_{12q}F_{111}=\sum_{m=1}^{N}F_{12m}F_{1qm}-\sum_{m=2}^{N}F_{11m}F_{2qm}.$
(6.11)

Similarly, we have

$F_{12q}F_{222}=\sum_{m=1}^{N}F_{12m}F_{2qm}-\sum_{m\neq 2}F_{1qm}F_{22m}.$
(6.12)

Let $R_{i}$ denote the $i^{th}$ row in the matrix $M$. We have

$\displaystyle R_{i}$
$\displaystyle=(F_{11(i+1)},F_{21(i+1)},\dots,F_{N1(i+1)}),\quad 1\leq i\leq
N-1,$ $\displaystyle R_{N}$ $\displaystyle=(r_{N1},r_{N2},\dots,r_{NN}),$

where

$\displaystyle r_{N1}$ $\displaystyle=\sum_{m\neq 2}F_{1qm}F_{12m}-\sum_{m\neq
1}F_{11m}F_{2qm},$ $\displaystyle r_{N2}$ $\displaystyle=\sum_{m\neq
2}F_{1qm}F_{22m}-\sum_{m\neq 1}F_{12m}F_{2qm},$ $\displaystyle r_{Nk}$
$\displaystyle=F_{12q}(F_{k11}-F_{k22}),\quad 3\leq k\leq N$

by applying formulas (6.11), (6.12). Now let us perform the following row
operation on the matrix $M$ and let $\widetilde{M}$ be the resulting matrix:

$R_{N}\rightarrow\widetilde{R}_{N}=R_{N}-F_{11q}R_{1}+\sum_{i=2}^{N}F_{2qi}R_{i-1}.$

Let $\widetilde{r}_{Nk}$ be the $k^{th}$ element in the row
$\widetilde{R}_{N}$ of the matrix $\widetilde{M}$. We have

$\displaystyle\widetilde{r}_{N1}$ $\displaystyle=\sum_{m\neq
1,2}F_{1qm}F_{12m},$ $\displaystyle\widetilde{r}_{N2}$
$\displaystyle=\sum_{m\neq 1,2}F_{1qm}F_{22m},$
$\displaystyle\widetilde{r}_{Nk}$
$\displaystyle=\sum_{m=1}^{N}F_{2qm}F_{1km}-F_{12q}F_{22k}-F_{11q}F_{12k},\quad
3\leq k\leq N.$

Let $S_{2m}=(F_{12m},F_{22m},\dots,F_{N2m}),$ where $3\leq m\leq N$. By Lemma
6.2 row $S_{2m}$ is a linear combination of the rows of the matrix $P$.
Therefore one can add the row $S_{2m}$ to the last row of the matrix
$\widetilde{M}$ without changing its determinant $D$. Let us add the rows
$-F_{1qm}S_{2m},\,m=3,\dots,N$, consecutively to the last row of
$\widetilde{M}$. The last row
$(\widehat{r}_{N1},\widehat{r}_{N2},\dots,\widehat{r}_{NN})$ of the resulting
matrix has the form

$\displaystyle\widehat{r}_{N1}=0,\quad\widehat{r}_{N2}=0,$
$\displaystyle\widehat{r}_{Nk}=\sum_{m=1}^{N}F_{2qm}F_{1km}-\sum_{m=1}^{N}F_{1qm}F_{2km}=-[F_{1},F_{2}]_{(q,k)},\quad
3\leq k\leq N$

by formula (6.3). It follows from the commutativity equations that $D=0$. This
proves that $B_{11}=B_{22}$. Similarly, one can prove that $B_{11}=B_{pp}$ for
all $p$. ∎

As a corollary of Propositions 6.3 and 6.4 the following statement takes
place.

######  Theorem 6.5.

Under the assumptions of Proposition 6.3 the matrix $B$ given by formulas
(6.5), (6.6) is proportional to the identity matrix.

We also have the following result.

######  Proposition 6.6.

Under the assumptions of Proposition 6.3 suppose also that there exists a non-
degenerate linear combination $G=\eta^{k}F_{k}$ for some functions
$\eta^{k},(1\leq k\leq N)$. Then matrix $B$ given by formulas (6.5), (6.6) is
a non-zero multiple of the identity matrix.

###### Proof.

From Theorem 6.5 we know that the matrix $B$ is proportional to the identity
matrix. It remains to show that $B$ is not the zero matrix. Let
$B_{ij}=A^{k}F_{ijk}=h\delta_{ij}$ for some function $h=h(x)$. Assume that
$h=0$. Then $A^{k}F_{ijk}=0$. Hence $\eta^{l}A^{k}F_{ljk}=0$, which means that
the non-zero vector $(A^{1},\dots,A^{N})$ belongs to the kernel of the form
$G$ (cf. a similar argument in [Misha&Veselov2008]). Therefore $G$ is
degenerate, which contradicts the assumption. Hence $h\neq 0$ and the
statement follows. ∎

The following theorem is a corollary of Theorem 6.5 and Proposition 6.6, and
it explains that a function $F$ satisfying the commutativity equations also
solves the WDVV equations.

######  Theorem 6.7.

Under the assumptions of Proposition 6.3 suppose that there exists a non-
degenerate linear combination $G=\eta^{k}F_{k}$ for some functions $\eta^{k}$.
Then $F$ is a solution of WDVV equations (6.1) where the matrix $B$ is given
by formulas (6.5), (6.6).

######  Remark 6.8.

Note that under the assumptions of Theorem 6.7, function $F$ also satisfies
the generalized WDVV equations

${F}_{i}{F}_{j}^{-1}{F}_{k}={F}_{k}{F}_{j}^{-1}{F}_{i},\quad i,j,k=1,\dots,N$

provided that matrices $F_{j}$ are non-degenerate. Indeed these equations
follow from equations (6.1) by the result from [Martini+Gragert1999] (see also
[MMM.2000]). It also follows that $F$ satisfies the WDVV equations

${F}_{i}G^{-1}{F}_{j}={F}_{j}G^{-1}{F}_{i},\quad i,j=1,\dots,N$

for any non-degenerate linear combination $G=a^{i}F_{i}$.

##  7\. Existence of the identity field

In this Section we define a natural multiplication on the tangent plane
$T_{x}V$ associated with a solution $F$ of the commutativity equations. We
find the identity vector field of this multiplication and establish that it is
proportional to the vector field $A^{k}\partial_{x^{k}}$, where functions
$A^{k}$ are given by formula (6.6). Thus we will express the identity vector
field in terms of $F$.

For any functions $u=(u^{1},\dots,u^{N}),\,v=(v^{1},\dots,v^{N})\colon V\to
V$, consider vector fields
$\partial_{u}=u^{i}\partial_{x^{i}},\,\partial_{v}=v^{i}\partial_{x^{i}}\in\Gamma(TV)$.
Let us define the following multiplication on the tangent space $T_{x}V$ for
generic $x\in V\colon$

$\partial_{u}\ast\partial_{v}=u^{i}v^{j}F_{ijk}\partial_{x^{k}}.$ (7.1)

Note that multiplication (7.1) defines a commutative associative algebra on
$T_{x}V$ if $F$ satisfies commutativity equations (6.2).

Consider a vector field

$e=e^{k}\partial_{x^{k}},$ (7.2)

where $e^{k}=e^{k}(x^{1},\dots,x^{N})$ are some functions. Consider an
$N\times N$ matrix $B=(B_{ij})_{i,j=1}^{N}$ given by

$B_{ij}=e(F_{ij})=e^{k}F_{ijk},\quad i,j=1,\dots,N.$ (7.3)

######  Proposition 7.1.

The following statements are equivalent:

(1) The matrix $B$ with entries given by (7.3) is equal to the identity
matrix,

(2) The vector field $e$ given by formula (7.2) is the identity vector field
of the multiplication (7.1).

###### Proof.

From relations (7.1), (7.2) and (7.3) we have

$\displaystyle
e\ast\partial_{v}=e^{i}v^{j}\partial_{x^{i}}\ast\partial_{x^{j}}=e^{i}v^{j}F_{ijk}\delta^{kl}\partial_{x^{l}}=B_{jk}v^{j}\delta^{kl}\partial_{x^{l}}.$
(7.4)

Let us firstly assume that $B_{jk}=\delta_{jk}$. Then relation (7.4) reduces
to

$\displaystyle e\ast\partial_{v}=v^{j}\partial_{x^{j}}=\partial_{v}.$

That is statement (2) follows from (1).

Secondly, assume that $e$ is the identity vector field of the multiplication
(7.1). Then from relation (7.4) we have

$\displaystyle
e\ast\partial_{v}=B_{jk}v^{j}\partial_{x^{k}}=\partial_{v}=v^{j}\partial_{x^{j}}.$

This implies that $B_{jk}=\delta_{jk}$, that is statement (1) holds. ∎

Proposition 7.1 allows us to reformulate Theorem 6.5 as follows.

######  Theorem 7.2.

Under the assumptions of Proposition 6.3 there exists the identity vector
field $e=e^{k}\partial_{x^{k}}$ for the multiplication (7.1). It has the form
$e^{k}=h^{-1}A^{k}$, where $A^{k}$ is given by formula (6.6) and
$h=A^{k}F_{kii}$ (for any $i=1,\dots,N$).

Now we are going to generalize Theorem 6.7 to the case of an arbitrary
constant metric $g$ in place of the standard metric $\delta_{ij}$. Thus we
start with equations of the form

$F_{ij\alpha}g^{\alpha\beta}F_{\beta kl}=F_{kj\alpha}g^{\alpha\beta}F_{\beta
il},quad1\leq i,j,k,l\leq N.$ (7.5)

We will show that matrix $(g_{\alpha\beta})_{\alpha,\beta=1}^{N}$ can be
represented as a linear combination of the matrices $F_{i}$ under some non-
degeneracy assumptions.

######  Theorem 7.3.

Let $F=F(x^{1},\dots,x^{N})$ be a function on $V$ which satisfies equations
(7.5) for some constant symmetric non-degenerate matrix $(g^{\alpha\beta})$.
Define new coordinates

$\displaystyle y^{i}=\widehat{C}_{j}^{i}x^{j},$ (7.6)

where the matrix $\widehat{C}=(\widehat{C}^{i}_{j})$ satisfies the relations
$\widehat{C}_{i}^{\alpha}\widehat{C}_{j}^{\beta}g^{ij}=\delta^{\alpha\beta}$,
where $1\leq\alpha,\beta\leq N$. Let $\widetilde{F}(y)=F(x)$. Suppose that
there exists $i_{0},1\leq i_{0}\leq N$, such that the matrix
$(\widetilde{F}_{i_{0}ij}(y))$ has rank $N-1$, where
$\widetilde{F}_{i_{0}ij}=\frac{\partial^{3}\widetilde{F}}{\partial
y^{i_{0}}y^{i}y^{j}}$ and $1\leq i,j\leq N,\,i\neq i_{0}$. Then there exists a
unique vector field $e=e^{k}(x)\partial_{x^{k}}$ such that

$e(F_{lm})=e^{k}F_{klm}=g_{lm},$

where $(g_{lm})$ is the inverse matrix for $(g^{\alpha\beta})$.

###### Proof.

Let $C=\widehat{C}^{-1}=(C_{k}^{i})$. Then $x^{i}=C_{j}^{i}y^{j}$. We also
have

$\partial_{x^{j}}=\widehat{C}^{i}_{j}\partial_{y^{i}},\quad\partial_{y^{j}}={C}^{i}_{j}\partial_{x^{i}}.$
(7.7)

From (7.7) we have the following relations:

$F_{j}(x)=\widehat{C}^{i}_{j}\partial_{y^{i}}\widetilde{F}(y)=\widehat{C}^{i}_{j}\widetilde{F}_{i}(y).$

Hence,

$\displaystyle
F_{pjk}(x)=\widehat{C}^{m}_{p}\widehat{C}^{i}_{j}\widehat{C}^{l}_{k}\widetilde{F}_{mil}(y).$
(7.8)

By multiplying relation (7.8) by $C_{a}^{p}C_{b}^{j}C_{c}^{k}$ we get

$\displaystyle\widetilde{F}_{abc}(y)=C^{p}_{a}C^{j}_{b}C^{k}_{c}F_{pjk}(x).$

By applying relation (7.8) we rewrite equation (7.5) as

$\displaystyle\widehat{C}_{i}^{p}\widehat{C}_{j}^{q}\widehat{C}_{\alpha}^{r}\widetilde{F}_{pqr}g^{\alpha\beta}\widehat{C}_{\beta}^{a}\widehat{C}_{k}^{b}\widehat{C}_{l}^{d}\widetilde{F}_{abd}=\widehat{C}_{k}^{s}\widehat{C}_{j}^{q}\widehat{C}_{\alpha}^{r}\widetilde{F}_{sqr}g^{\alpha\beta}\widehat{C}_{\beta}^{a}\widehat{C}_{i}^{b}\widehat{C}_{l}^{d}\widetilde{F}_{abd}.$
(7.9)

It follows from the relation
$\widehat{C}^{\alpha}_{i}\widehat{C}^{\beta}_{j}g^{ij}=\delta^{\alpha\beta}$
that

$\widehat{C}^{l}_{a}\widehat{C}^{m}_{b}\delta_{lm}=g_{ab}.$ (7.10)

Hence we reduce equation (7.9) to

$\displaystyle\widehat{C}_{i}^{p}\widehat{C}_{k}^{b}\widetilde{F}_{pqr}\widetilde{F}_{rbd}=\widehat{C}_{k}^{s}\widehat{C}_{i}^{b}\widetilde{F}_{sqr}\widetilde{F}_{rbd}.$
(7.11)

By multiplying equation (7.11) by $C_{n}^{i}C_{m}^{k}$ we get

$\displaystyle\widetilde{F}_{nqr}\widetilde{F}_{rmd}=\widetilde{F}_{mqr}\widetilde{F}_{rnd},$

that is $\widetilde{F}_{m}$ and $\widetilde{F}_{n}$ commute. Now since the
rank of the matrix $(\widetilde{F}_{i_{0}ij})$ is $N-1$, it follows by Theorem
6.5 that there exists a unique vector field $e=e^{j}(y)\partial_{y^{j}}$ such
that

$e(\widetilde{F}_{\alpha\beta}(y))=e^{j}(y)\widetilde{F}_{j\alpha\beta}(y)=\delta_{\alpha\beta}.$

From relation (7.8) we have

$C_{j}^{i}\partial_{x^{i}}F_{ab}(x)=\widehat{C}_{a}^{l}\widehat{C}_{b}^{m}\widetilde{F}_{jlm}(y).$

This equation implies that

$e(F_{ab}(x))=e^{j}C_{j}^{i}\partial_{x^{i}}F_{ab}(x)=e^{j}\widetilde{F}_{jlm}(y)\widehat{C}_{a}^{l}\widehat{C}_{b}^{lm}=\delta_{lm}\widehat{C}_{a}^{l}\widehat{C}_{b}^{m}=g_{ab}$

by relation (7.10). This proves the theorem. ∎

######  Remark 7.4.

We note that the maximality of the rank of the matrix $P$ is sufficient but
not necessary for the existence of the identity field. Indeed, in the case of
two-dimensional Frobenius manifold consider the function $F$ given by

$F(t^{1},t^{2})=\frac{1}{2}(t^{1})^{2}t^{2}+f(t^{2})$ (7.12)

with some function $f(t^{2})$. We have equation $F_{1}GF_{2}=F_{2}GF_{1}$,
where the matrix entries $(F_{i})_{kl}=\frac{\partial^{3}F}{\partial
t^{i}\partial t^{k}\partial t^{l}}$, and

$G=G^{-1}=\begin{pmatrix}0&1\\\ 1&0\\\ \end{pmatrix}=F_{1}.$ (7.13)

Now let

$\displaystyle C=\begin{pmatrix}-\frac{i}{2}&\frac{1}{2}\\\ i&1\\\
\end{pmatrix},\quad\widehat{C}=\begin{pmatrix}i&-\frac{i}{2}\\\
1&\frac{1}{2}\\\ \end{pmatrix}$

be the matrices of the change of variables such that

$\displaystyle t^{i}={C}_{j}^{i}x^{j},\quad x^{i}=\widehat{C}_{j}^{i}t^{j},$
(7.14)

where $(x^{1},x^{2})$ is a new coordinate system and the matrix $G=(g^{ij})$
satisfies the relation
$\widehat{C}_{i}^{\alpha}\widehat{C}_{j}^{\beta}g^{ij}=\delta^{\alpha\beta}$.
Let $\widetilde{F}(x)=F(t)$. Then we have

$\displaystyle\widetilde{F}_{112}$
$\displaystyle=\frac{\partial^{3}\widetilde{F}}{\partial x^{1}\partial
x^{1}\partial x^{2}}=\frac{1}{4}-f^{{}^{\prime\prime\prime}}(ix^{1}+x^{2}),$
$\displaystyle\widetilde{F}_{122}$
$\displaystyle=\frac{\partial^{3}\widetilde{F}}{\partial x^{1}\partial
x^{2}\partial x^{2}}=-\frac{i}{4}+if^{{}^{\prime\prime\prime}}(ix^{1}+x^{2}).$

Note that the matrix $(\widetilde{F}_{112},\widetilde{F}_{122})$ has rank zero
if $f(t^{2})=\frac{1}{24}(t^{2})^{3}=\frac{1}{24}(ix^{1}+x^{2})^{3}$.
Nonetheless $e=\partial_{t^{1}}=i\partial_{x^{1}}+\partial_{x^{2}}$ is the
identity field.

######  Remark 7.5.

The maximality of the rank of the matrix $P$ condition may be satisfied in the
case of a family of non-semisimple algebras. An example is given by
prepotential (7.12) with $f=0$, as it follows from considerations in Remark
7.4.

##  8\. Identity field for non-simply laced root systems and their projections

We are going to relate the identity field for a solution of commutativity
equations and a restriction of such a solution. Firstly, we have the following
statement.

######  Lemma 8.1.

Let $F(x)$ be a function and $e=e^{k}\partial_{x^{k}}$ be a vector field such
that $e(F_{ij})=\delta_{ij}$, where $1\leq i,j\leq N$. Let
$\widetilde{x}^{k}=C^{k}_{i}x^{i}$ for a matrix $C=(C^{k}_{i})\in
O(N,\mathbb{C})$. Let $\widetilde{F}(\widetilde{x})=F(x)$. Then
${e}(\widetilde{F}_{\mu\nu})=\delta_{\mu\nu}$, where $1\leq\mu,\nu\leq N$.

###### Proof.

We have

$e^{k}F_{ijk}=\delta_{ij}.$ (8.1)

By relation (2.4) we have
$\partial_{x^{i}}=C_{i}^{k}\partial_{\widetilde{x}^{k}}$. Hence we have

$\displaystyle
F_{ijk}={C}_{k}^{\widetilde{k}}{C}_{j}^{\widetilde{j}}{C}_{i}^{\widetilde{i}}\widetilde{F}_{\widetilde{i}\widetilde{j}\widetilde{k}}.$
(8.2)

Then by formula (8.2) relation (8.1) can be written as

$\displaystyle{e}^{k}{C}_{k}^{\widetilde{k}}{C}_{j}^{\widetilde{j}}{C}_{i}^{\widetilde{i}}\widetilde{F}_{\widetilde{i}\widetilde{j}\widetilde{k}}=\delta_{ij}.$
(8.3)

Let $\widehat{C}=C^{-1}=(\widehat{C}^{i}_{j})$. Multiply equality (8.3) by
$\widehat{C}_{\mu}^{i}\widehat{C}_{\nu}^{j}$. We get

$\displaystyle{e}^{k}{C}_{k}^{\widetilde{k}}\widetilde{F}_{\mu\nu\widetilde{k}}=\widehat{C}_{\mu}^{i}\widehat{C}_{\nu}^{i}=\delta_{\mu\nu}$
(8.4)

since $\widehat{C}\in O(N,\mathbb{C})$. Hence equality (8.4) becomes

$\displaystyle{e}^{k}{C}_{k}^{\widetilde{k}}\widetilde{F}_{\mu\nu\widetilde{k}}=\delta_{\mu\nu}.$
(8.5)

Note that ${e}={e}^{k}C_{k}^{\widetilde{k}}\partial_{x^{\widetilde{k}}}$. We
have by relation (8.5) that ${e}(\widetilde{F}_{\mu\nu})=\delta_{\mu\nu}$ as
required. ∎

Let $\mathcal{B}=\mathcal{A}\cap W$ be a subsystem of $\mathcal{A}$ for some
$n$-dimensional linear subspace $W=\langle\mathcal{B}\rangle\subseteq V.$ Let

$W_{\mathcal{B}}\coloneqq\\{x\in
V\colon(\beta,x)=0\quad\forall\beta\in\mathcal{B}\\}.$

Let $f_{1},\dots,f_{n}$ be an orthonormal basis of the space
$W_{\mathcal{B}}$, and let ${\xi^{1},\dots,\xi^{n}}$ be the corresponding
orthonormal coordinates in $W_{\mathcal{B}}$. Let us extend the orthonormal
basis in $W_{\mathcal{B}}$ to an orthonormal basis
$f_{1},\dots,f_{n},f_{n+1},\dots,f_{N}$ in $V$ and let
${\xi^{1},\dots,\xi^{n},\xi^{n+1},\dots,\xi^{N}}$ be the corresponding
orthonormal coordinates in $V$. The following statement takes place.

######  Proposition 8.2.

Let a function $F$ be given by formula (2.2). Let $e=e(z),\,z\in V$, be a
vector field such that $e(F_{ij})=\delta_{ij}$ for all $i,j=1,\dots,N$, where
$F_{ij}=\frac{\partial^{2}F}{\partial x^{i}\partial x^{j}}$. Let
$\widehat{F}(\xi^{1},\dots,\xi^{N})=F(x^{1},\dots,x^{N})$. Suppose that
$e(\xi)\in T_{{\xi}}W_{\mathcal{B}}$ for a generic ${\xi}\in W_{\mathcal{B}}$.
Let $\widetilde{e}={e}|_{W_{\mathcal{B}}}\in\Gamma(T_{\ast}W_{\mathcal{B}})$.
Then $\widetilde{e}((F_{\mathcal{B}})_{ij})=\delta_{ij},$ where
$(F_{\mathcal{B}})_{ij}=\frac{\partial^{2}F_{\mathcal{B}}}{\partial\xi^{i}\partial\xi^{j}}$
and function $F_{\mathcal{B}}$ is given by formula (5.1).

###### Proof.

Let $C=(C^{k}_{i})\in O(N,\mathbb{C})$ be such that $\xi^{k}=C_{i}^{k}x^{i}$.
By Proposition 8.1 we have ${e}(\widehat{F}_{ij})=\delta_{ij}$, where
$\widehat{F}_{ij}=\frac{\partial^{2}\widehat{F}}{\partial\xi^{i}\partial\xi^{j}}$,
and $i,j=1,\dots,N$. Hence
$\widetilde{e}(\widehat{F}_{ij}|_{W_{\mathcal{B}}})=\delta_{ij},\,1\leq
i,j\leq n$, which implies the statement since
$\widehat{F}|_{W_{\mathcal{B}}}=F_{\mathcal{B}}$ and
$\widehat{F}_{ij}|_{W_{\mathcal{B}}}=(F_{\mathcal{B}})_{ij}$. ∎

In the next proposition we give a formula for the identity field for the
multiplication (7.1) corresponding to the root system $F_{4}$, see
[Maali.2022] for a proof.

######  Proposition 8.3.

The matrix $B=h^{-1}\sum_{k=1}^{4}B^{k}F_{k}$ is the identity matrix in
dimension four in the following cases:

  * •

$F$ has the form (2.2) corresponding to $\mathcal{A}=F_{4}^{+}$ with the
condition $r=-2q$, $q\neq 0$, where

$\displaystyle B^{k}=\sin{x^{k}}\Big{(}\cos{x^{k}}(-1+\sum_{i\neq
k}\cos{2x^{i}})-2\prod_{i\neq k}\cos{x^{i}}\Big{)},\quad k=1,2,3,4,$
$\displaystyle h(x)=6q+\frac{1}{2}\sum_{\alpha\in
F_{4}^{+}}c_{\alpha}\cos{(2\alpha,x)}.$

  * •

$F$ has the form (2.2) corresponding to $\mathcal{A}=F_{4}^{+}$ with the
condition $r=-4q$, $q\neq 0$, where

$\displaystyle B^{k}=\sin{x^{k}}\Big{(}\cos{x^{k}}+2\prod_{i\neq
k}\cos{x^{i}}\Big{)},\quad k=1,2,3,4,$ $\displaystyle
h(x)=-q\Big{(}6+\sum_{i=1}^{4}\cos{2x^{i}}+8\prod_{i=1}^{4}\cos{x^{i}}\Big{)}.$

Solutions of the WDVV equations corresponding to the root system $BC_{n}$ and
its deformation $BC_{n}(q,r,s;\underline{m})$ were found in [MGM2020]. In the
case of the root system $F_{4}$ and its projections we get new solutions of
the WDVV equations.

######  Theorem 8.4.

Function (2.2) corresponding to $\mathcal{A}=F_{4}^{+}$ or any of its
$3$-dimensional projections $(F_{4},A_{1})_{1},\,(F_{4},A_{1})_{2}$ satisfies
WDVV equations (6.1) if $r=-2q$ or $r=-4q$, $q\neq 0$.

###### Proof.

It was proven in [George+Misha2019] that function (2.2) for the collection
$\mathcal{A}=F_{4}^{+}$ satisfies commutativity equations (6.2) if $r=-2q$ or
$r=-4q$. For $\mathcal{A}=F_{4}^{+}$ the statement follows by Proposition 8.3.
It is easy to see that for the three-dimensional restrictions
$\mathcal{A}=(F_{4},A_{1})_{1,2}$ the assumptions of Proposition 8.2 hold. The
statement follows. ∎

Now we give the identity vector field for all the non-simply laced root
systems as well as their projections. In the case of root system $F_{4}$ it
can be checked that the components of the identity field given by the next
theorem are equal to $h^{-1}B^{k}$ given by Proposition 8.3 (see
[Maali.2022]).

######  Theorem 8.5.

Let function $F$ be given by (2.2). Consider a vector field $e$ given by

$e=c_{0}H^{-1}\sum_{\alpha\in\mathcal{A}}\bar{c}_{\alpha}\sin(2(\alpha,x))\partial_{\alpha}$
(8.6)

for some $c_{0},\,\bar{c}_{\alpha}\in\mathbb{C}$ and

$H=H_{0}+\sum_{\alpha\in\mathcal{A}}\bar{c}_{\alpha}\sin^{2}(\alpha,x)$

for some $H_{0}\in\mathbb{C}$. Then $e(F_{ij})=\delta_{ij}$ if

  1. (1)

$\mathcal{A}=F_{4}^{+}$ given by formula (5.7) or $\mathcal{A}$ is one of the
projections $(F_{4},A_{1})_{1},\,(F_{4},A_{1})_{2}$,
$(F_{4},A_{2})_{1},\,(F_{4},A_{2})_{2},\,(F_{4},B_{2}),\,(F_{4},A_{1}^{2})$,
and

$r=-2q\neq 0,\quad c_{0}=-\frac{1}{4q},\quad
H_{0}=0,\quad\bar{c}_{\alpha}={c}_{\alpha}\,\forall\alpha\in\mathcal{A},$

  2. (2)

$\mathcal{A}$ is the same as in (1) and

$r=-4q\neq 0,\quad c_{0}=\frac{1}{4q},\quad
H_{0}=36q,\quad\bar{c}=c_{\alpha}|_{q=0}\,\forall\alpha\in\mathcal{A},$

  3. (3)

$\mathcal{A}=G_{2}^{+}$ and

$p=-3q\neq 0,\quad c_{0}=-\frac{1}{9q},\quad
H_{0}=0,\quad\bar{c}_{\alpha}={c}_{\alpha}\,\forall\alpha\in G_{2}^{+},$

where $q$ is the multiplicity of the long roots
$\sqrt{3}e_{1},\frac{1}{2}(\sqrt{3}e_{1}\pm 3e_{2})$ and $p$ is the
multiplicity of the short roots $e_{2},\frac{1}{2}(\sqrt{3}e_{1}\pm e_{2})$,

  4. (4)

$\mathcal{A}$ is the same as in $\eqref{it3}$, and

$p=-9q\neq 0,\quad c_{0}=\frac{1}{9q},\quad
H_{0}=27q,\quad\bar{c}=c_{\alpha}|_{q=0}\,\forall\alpha\in G_{2}^{+},$

  5. (5)

$\mathcal{A}=BC_{n}(q,r,s;\underline{m}),q\neq 0,n\geq 2,$ and

$r=-8s-2q(\sum_{i=1}^{n}m_{i}-2),\quad c_{0}=-\frac{1}{4q},\quad
H_{0}=\frac{r(2s-q)}{q},\quad\bar{c}_{\alpha}={c}_{\alpha}|_{q=s=0}\,\forall\alpha\in
BC_{n}(q,r,s;\underline{m}).$

  6. (6)

$\mathcal{A}=BC_{1}^{+}$ with $c_{\pm e_{1}}=r,c_{\pm 2e_{1}}=s$ and

$c_{0}=-\frac{1}{2(r+8s)},\quad
H_{0}=-\frac{r(r+4s)}{r+8s},\quad\bar{c}_{e_{1}}=r,\,\bar{c}_{2e_{1}}=0.$

Theorem 8.5 follows from the identity

$\sum_{\alpha,\beta\in\mathcal{A}}\bar{c}_{\alpha}c_{\beta}(\alpha,\beta)(\beta,u)(\beta,v)\sin(2\alpha,x)\cot(\beta,x)=c_{0}^{-1}H(u,v)$
(8.7)

for any $u,v\in V$ for each case specified in Theorem 8.5. By Proposition 8.2
it is sufficient to establish identity (8.7) for the case when $\mathcal{A}$
is a (non-simply laced) root system. Indeed it is easy to see that the vector
field $e$ given by (8.6) for $\mathcal{A}=F_{4},\,BC_{N}$ satisfies the
condition $e|_{W}\in\Gamma(T_{\ast}W)$ for any intersection of mirrors $W$. It
is also clear that the restricted vector field $e|_{W}$ has the form (8.6) for
the corresponding projections of the root system $\mathcal{A}$. A case by case
proof of the identity (8.7) for $\mathcal{A}=F_{4}$ and $\mathcal{A}=G_{2}$ is
contained in [Maali.2022]; see [MGM2020] for $\mathcal{A}=BC_{N}$. We are not
aware of a uniform proof of Theorem 8.5 or the identity (8.7).

## Acknowledgments

We thank I. Strachan, O. Mokhov, A.P. Veselov, E. Ferapontov and G. Cotti for
useful discussions and comments. The work of M.A was funded by Department of
Mathematics, College of Science and Humanities, Imam Abdulrahman Bin Faisal
University, P.O. Box 12020 Jubail Industrial City 31961, Saudi Arabia.

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