CYT and SKT Metrics on Compact Semi-Simple Lie Groups

CYT and SKT Metrics on Compact Semi-Simple
Lie Groups††This paper is a contribution to the Special Issue on Differential
Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon
for his 75th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Bourguignon.html

Anna FINO ab and Gueo GRANTCHAROV b

A. Fino and G. Grantcharov

a) Dipartimento di Matematica “G. Peano”, Università degli studi di Torino,
a) Via Carlo Alberto 10, 10123 Torino, Italy<EMAIL_ADDRESS>
b) Department of Mathematics and Statistics, Florida International University,
b) Miami, FL 33199, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received January 02, 2023, in final form May 11, 2023; Published online May
25, 2023

A Hermitian metric on a complex manifold $(M,I)$ of complex dimension $n$ is
called Calabi–Yau with torsion (CYT) or Bismut–Ricci flat, if the restricted
holonomy of the associated Bismut connection is contained in ${\rm SU}(n)$ and
it is called strong Kähler with torsion (SKT) or pluriclosed if the associated
fundamental form $F$ is $\partial\overline{\partial}$-closed. In the paper we
study the existence of left-invariant SKT and CYT metrics on compact semi-
simple Lie groups endowed with a Samelson complex structure $I$. In
particular, we show that if $I$ is determined by some maximal torus $T$ and
$g$ is a left-invariant Hermitian metric, which is also invariant under the
right action of the torus $T$, and is both CYT and SKT, then $g$ has to be
Bismut flat.

Bismut connection; Hermitian metric

53C55; 53C05; 22E25; 53C30; 53C44

Dedicated to Jean-Pierre Bourguignon
for his 75th birthday

##  1 Introduction

On every Hermitian manifold $(M,I,g)$ there exists a one-parameter family of
Hermitian connections which can be distinguished by their torsion tensor and
coincide with the Levi-Civita connection when $M$ is Kähler. Among them there
are the Chern connection $\nabla^{\rm C}$ on the holomorphic tangent bundle
and the Bismut (or Strominger) connection $\nabla^{\rm B}$, which is also
called by physicists the Kähler with torsion (KT) connection. $\nabla^{\rm B}$
is the unique Hermitian connection whose torsion tensor is totally skew-
symmetric and its torsion $T^{\rm B}$ is characterized by the condition

$g\big{(}T^{\rm B}(X,Y),Z\big{)}={\rm d}F(IX,IY,IZ),$

where $F$ is the fundamental form $F(X,Y)=g(IX,Y)$, and $X$, $Y$, $Z$ are
smooth vector fields. Since $\nabla^{\rm B}$ is a Hermitian connection, its
holonomy is contained in the unitary group ${\rm U}(n)$, where $n$ is the
complex dimension of $M$. If the holonomy of $\nabla^{\rm B}$ is reduced to
${\rm SU}(n)$, the metric $g$ is said to be Calabi–Yau with torsion (shortly
CYT). It is also called Bismut Ricci-flat. This type of geometry in the
physical context was considered first by Strominger [25] and Hull [17]. In
[15], it was conjectured that every compact complex manifold with vanishing
first Chern class admits a CYT metric. Counterexamples to this conjecture
appear in [7]. In [14] it has been shown that every compact complex
homogeneous space with vanishing first Chern class, after an appropriate
deformation of the complex structure, admits a homogeneous CYT metric,
provided that the complex homogeneous space also has an invariant volume form.

SKT metrics, which are called also pluriclosed, are Hermitian metrics whose
fundamental form $F$ is $\partial\overline{\partial}$-closed or equivalently,
whose torsion $3$-form of the Bismut connection is closed. Interest in SKT
manifolds stems from various sources [26]. They occur in physics in the
context of supersymmetric theories, see for example [11, 16, 25]. Secondly,
every conformal class of a hermitian metric on a compact complex surface
contains an SKT metric [12], but this property is no longer true in higher
dimensions. Recently the SKT and CYT metrics have been studied in relation to
the pluriclosed flow and the generalized Ricci flow [2, 4, 9, 10].

Basic examples of SKT manifolds are given by compact Lie groups endowed with a
bi-invariant metric and any of the compatible Samelson’s complex structures
[24]. These SKT metrics are also CYT, but the associated Bismut connection is
flat. Note that a simply connected compact Riemannian manifold admitting a
flat metric connection with closed skew symmetric torsion is isometric to a
product of compact simple Lie groups with bi-invariant metrics (see [10,
Theorem 3.54]), however it is currently not known whether there are other
metrics, which are both SKT and CYT, but with non-flat Bismut connection. Many
of the CYT examples in [14] have non-vanishing curvature of the Bismut
connection. Recently in [22, 23] non-Hermitian examples which are Bismut
Ricci-flat, which have closed torsion form and non-vanishing Bismut curvature
have been constructed on compact homogeneous spaces.

By [24], every compact Lie group G of even dimension admits a complex
structure such that left translations are holomorphic mappings and the complex
structure is determined by a choice of a maximal torus $T$. This result is an
extension of Borel’s theorem which states that the quotient of a compact Lie
group by its maximal torus always has a homogeneous complex structure. Note
that the Samelson’s construction depends on a choice of the maximal torus and
such complex structures are compatible if they are compatible with the metric
restricted to this torus.

More generally, the existence of SKT metrics on the class of Wang C-spaces,
which are defined as compact complex manifolds admitting a transitive action
by a compact Lie group of biholomorphisms and finite fundamental group, was
investigated in [8]. By [8], it turns out that SKT structures could only
appear on a product of a compact Lie group and a Kähler homogeneous C-space
(generalized flag manifold). Note that these spaces admit left-invariant SKT
metrics.

Therefore, in view of the above, it is natural to investigate whether on an
even dimensional compact Lie group with a Samelson complex structure there
exist left-invariant metrics which are both SKT and CYT, but not Bismut flat.

Motivated by this question, in the current paper we study left-invariant SKT
and CYT metrics on compact semi-simple Lie groups endowed with a Samelson’s
complex structure $I$. Using a symmetrization, we show that the existence of
left-invariant SKT or CYT metrics leads to an existence of such metrics which
are left-invariant and right $T$-invariant. As a first consequence we show
that if $g$ is a left-invariant SKT metric compatible with $I$, then the
complex structure $I$ has to be compatible with a bi-invariant metric $g_{0}$.
Then we provide characterizations of left-invariant SKT and CYT metrics which
are also invariant under the right $T$-action. As a consequence, we prove that
on every compact semi-simple Lie group a metric of this type which is both CYT
and SKT must be a bi-invariant one, or equivalently a product of metrics
proportional to the Killing forms on the simple factors. Therefore, in
particular, such metric must be Bismut flat. We also construct an explicit
$5$-parameter family of SKT metrics on ${\rm SO}(9)$, which are $T$-invariant
for the natural choice of the torus $T$.

##  2 Preliminaries on compact semi-simple Lie groups

It was shown independently by Samelson [24] and Wang [28] that every compact
Lie group $G$ of even dimension admits left-invariant complex structures $I$.
Moreover, there exists a maximal toral subgroup $T$ of $G$ that is $I$-complex
and the quotient $G/T$ is a complete flag variety of $G$ with a complex
structure such that the quotient map $\pi\colon G\rightarrow G/T$ is
holomorphic.

We now review the Samelson’s construction of left-invariant complex structures
on compact semi-simple Lie groups. We will follow the conventions of
Alekseevsky–Perelomov [1], and Bordemann–Forger–Romer [5]. Let $G^{\rm c}$ be
the complexification of a compact semi-simple Lie group $G$. Then $G$ may be
regarded as a Lie subgroup of $G^{\rm c}$, and the Lie algebra of $G^{\rm c}$
is the complexification $\mathfrak{g}^{\rm c}$ of the Lie algebra
$\mathfrak{g}$ of $G$.

If we fix a maximal abelian subalgebra $\mathfrak{t}$ of $\mathfrak{g}$, then
$\mathfrak{h}=\mathfrak{t}^{\rm c}$ is a maximal abelian subalgebra of
$\mathfrak{g}^{\rm c}$ and we have the root space decomposition

$\mathfrak{g}^{\rm
c}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{g}_{\alpha},$

where $\Phi$ is the finite subset of nonzero elements of ${\mathfrak{h}}^{*}$
called roots, and each $\mathfrak{g}_{\alpha}$ is the complex $1$-dimensional
subspace of $\mathfrak{g}^{\rm c}$ given by

$\mathfrak{g}_{\alpha}:=\\{X\in{\mathfrak{g}}^{\rm
c}\mid[H,X]=\alpha(H)X,\,\forall H\in\mathfrak{h}\\}.$

Therefore, there exists a system of positive roots, that is a set
$P\subset\Phi$ satisfying $P\cap(-P)=\varnothing$, $P\cup(-P)=\Phi$, and
$\alpha,\beta\in P$, $\alpha+\beta\in\Phi$ $\Rightarrow$ $\alpha+\beta\in P$,
and such that ${\rm{span}}_{\mathbb{C}}(P)=\mathfrak{h}$. We note that $\Phi$
is also invariant under a set of reflections which leads to the classification
of all irreducible root systems. Moreover, the structure of $\Phi$ determines
the structure of $\mathfrak{g}^{\rm c}$ in a sense that
$[\mathfrak{g}_{\alpha},\mathfrak{g}_{\beta}]\subseteq\mathfrak{g}_{\alpha+\beta}$
when $\alpha+\beta\in\Phi$,
$[\mathfrak{h},\mathfrak{g}_{\alpha}]=\mathfrak{g}_{\alpha}$ and
$[\mathfrak{g}_{\alpha},\mathfrak{g}_{-\alpha}]\subseteq\mathfrak{h}$ and the
other commutators vanish.

Among the roots in $P$ there is a set of simple roots, which we denote by
$\alpha_{1},\alpha_{2},\dots,\alpha_{r}$ and forms a basis of the space
$\mathfrak{t}^{*}$. For any $\alpha\in P$, we have
$\alpha=\sum_{i=1}^{r}m_{i}\alpha_{i}$, with $m_{i}$ being a non-negative
integer. The basis $(\alpha_{l})$ also determines a partial order of the set
of positive roots and $r$ is called rank of $\mathfrak{g}^{\rm c}$.

Denote by $B(X,Y)={\rm tr}({\rm ad}_{X}\circ{\rm ad}_{Y})$ the Killing–Cartan
bilinear form of $\mathfrak{g}^{\rm c}$. The restriction of $B$ to
$\mathfrak{t}$ is non-degenerate and $\mathfrak{g}_{\alpha}$ is $B$-orthogonal
to $\mathfrak{g}_{\beta}$ for $\beta\neq-\alpha$. For every $\alpha\in P$ we
choose $(E_{\pm\alpha},H_{\alpha})$ such that
$\mathfrak{g}_{\pm\alpha}={\rm{span}}_{\mathbb{C}}\langle
E_{\pm\alpha}\rangle$, $[E_{\alpha},E_{-\alpha}]=H_{\alpha}$,
$B(E_{\alpha},E_{-\alpha})=1$, $[H,E_{\alpha}]=\alpha(H)E_{\alpha}$ and
$B(H,H_{\alpha})=\alpha(H)$ for all $H\in\mathfrak{t}$.

Note that $\mathfrak{t}={\rm span}_{\mathbb{R}}\langle{\rm
i}H_{\alpha}\rangle$ is the Lie algebra of $T$. Moreover, since all compact
forms of a complex semi-simple Lie algebra are conjugate, we can assume that
$\mathfrak{g}$ is the real span of $X_{\alpha}$, $Y_{\alpha}$, ${\rm
i}H_{\alpha}$, where
$X_{\alpha}=\frac{1}{\sqrt{2}}(E_{\alpha}+E_{-\alpha}),Y_{\alpha}=\frac{1}{\sqrt{2}{\rm
i}}(E_{\alpha}-E_{-\alpha}).$ In particular, $\\{X_{\alpha},Y_{\alpha}\\}$ is
an orthonormal set with respect to $-B$ on $G$.

When the rank $r$ is even, Samelson proved that a choice of $P$ and of a
complex structure on $\mathfrak{t}$ determine a left-invariant complex
structure $I$ on $G$, which is given by $I(E_{\alpha})={\rm i}E_{\alpha}$,
$I(E_{-\alpha})=-{\rm i}E_{-\alpha}$, for every $\alpha\in P$, and by a linear
endomorphism of $\mathfrak{t}$ with square $-{{\rm Id}}$.

Note that if $\mathfrak{g}=\sum_{i=1}^{m}\mathfrak{g}_{i}$, where
$\mathfrak{g}_{i}$ are simple, then
$\mathfrak{t}=\sum_{i=1}^{m}\mathfrak{t}_{i}$ and $B=\sum_{i=1}^{m}B_{i}$. We
denote by $E_{\alpha}^{(i)}$, $E_{-\alpha}^{(i)}$ the corresponding root
vectors in $\mathfrak{g}_{i}$ with respect to $\mathfrak{t}_{i}$ and by
$P_{i}$ the set of positive roots of the simple factor $G_{i}$ with respect to
$\mathfrak{t}_{i}$. We will not use the upper index when there is no
confusion.

######  Remark 2.1.

Let $I$ be a Samelson complex structure on $G$. If $h$ is a left-invariant
$I$-Hermitian metric on $G$ which is also invariant by the right $T$-action on
$G$, then there exists a metric on $G/T$ such that $\pi\colon G\rightarrow
G/T$ is a Riemannian submersion. Since the spaces $\langle
X_{\alpha},Y_{\alpha}\rangle$ are $T$-invariant and irreducible, the metric
$h$ can be written, up to a constant, as $h(X,Y)=-B(\Lambda(X),Y)$, where
$\Lambda$ is a symmetric operator defined by
$\Lambda(E_{\alpha})=\lambda_{\alpha}E_{\alpha}$ for every $\alpha\in P$ and
any appropriate matrix on $\mathfrak{t}$ such that $h$ is positive definite
(see, e.g., [3]).

##  3 Invariant SKT metrics

We can apply the symmetrization with respect to the right $T$-action to prove
that if we have a left-invariant SKT $I$-Hermitian metric, then it is possible
to construct a SKT metric which is right $T$-invariant.

######  Lemma 3.1.

Let $G$ be a compact semi-simple Lie group of even rank endowed with a
Samelson complex structure $I$. Let $g$ be a left-invariant SKT $I$-Hermitian
metric. Then symmetrizing $g$ by the right $T$-action on $G$, we obtain a
Hermitian metric $h$ which is still SKT and also left-invariant and right
$T$-invariant.

###### Proof.

Since $I$ is invariant under the right $T$-action by the aforementioned result
of Wang [28], we can symmetrize $g$. Indeed, let $d\nu$ be the standard volume
on the torus $T$ and denote by $R_{h}$ the right translation by $h\in T$ on
$G$. We can define

$\overline{g}_{x}({\bf u},{\bf v}):=\int_{T}(R_{h}^{*}g)_{x}({\bf u},{\bf
v})\,{\rm d}\nu(h),$

for every $x\in G,{\bf u},{\bf v}\in T_{x}G$. By construction $\overline{g}$
is invariant by the right $T$-action on $G$ and compatible with $I$. Moreover,
since left and right translations commute and right $T$\- translations are
holomorphic, $\overline{g}$ is left-invariant.

Since the SKT condition ${\rm d}{\rm d}^{\rm c}\omega=0$ is linear in $\omega$
and ${\rm d}$, ${\rm d}^{\rm c}$ commute with $R_{h}^{*}$, for every $h\in T$,
$\overline{g}$ is SKT. ∎

Given a Samelson complex structure $I$, we can describe all possible left-
invariant SKT $I$-Hermitian metrics which are also invariant by the right
$T$-action on $G$.

######  Theorem 3.2.

Let $G$ be a compact semi-simple Lie group of even rank endowed with a
Samelson complex structure $I$ and let $g$ be a left-invariant SKT
$I$-Hermitian metric. Then the restriction of $g$ on the fibers of $\pi\colon
G\rightarrow G/T$ coincides with the restriction of a bi-invariant metric on
$G$. As a consequence, the complex structure $I$ has to be compatible with a
bi-invariant metric $g_{0}$.

###### Proof.

By Lemma 3.1, using the symmetrization it is not restrictive to suppose that
$g$ is invariant under the right $T$-action. Suppose that $G$ up to a finite
cover is given by the product $G_{1}\times G_{2}\times\dots\times G_{m}$ of
simple Lie groups $G_{i}$ ($\dim G_{i}>1$). Then
$\mathfrak{t}=\sum_{i=1}^{m}\mathfrak{t}_{i}$, with $\mathfrak{t}_{i}$ the
Cartan subalgebra of the Lie algebra $\mathfrak{g}_{i}$ of every factor
$G_{i}$.

Select a $g$-unitary basis $(\theta_{1},\dots,\theta_{2k})$ of
$\mathfrak{t}^{*}$, such that $I(\theta_{2l-1})=\theta_{2l}$. Then by [13], we
have $F=\sum_{l=1}^{k}\theta_{2l-1}\wedge\theta_{2l}+\pi^{*}(F_{b}),$ where
$F_{b}$ is a $(1,1)$-form of an invariant Hermitian structure on $G/T$. Also
by [13, Section 5],

${\rm d}{\rm d}^{\rm c}F=\pi^{*}{\rm d}{\rm d}^{\rm
c}F_{b}-\sum_{l=1}^{k}\big{(}\omega_{2l-1}^{2}+\omega_{2l}^{2}\big{)},$

where $\omega_{i}={\rm d}\theta_{i}$ are the curvature forms of the connection
on $\pi\colon G\rightarrow G/T$ determined by $\theta_{i}$. This in particular
means that

$\sum_{l=1}^{k}\big{(}[\omega_{2l-1}]^{2}+[\omega_{2l}]^{2}\big{)}=\sum_{i=1}^{2k}[{\rm
d}\theta_{i}]^{2}=0$

in the cohomology $H^{2}(G/T,\mathbb{R})$.

Now, the flag manifold $G/T$ decomposes as $G/T=G_{1}/T_{1}\times\dots\times
G_{m}/T_{m}$ and by the results of Chevalley [6],
$H^{2}(G_{i}/T_{i},\mathbb{R})$ is generated by the (exterior derivatives of
the) simple roots in $\mathfrak{t}_{i}^{*}$, and moreover, there is a unique
quadratic relation among them. This quadratic relation can be written as
$\sum_{j=1}^{s_{i}}\big{[}{\rm d}\widetilde{\theta}^{(i)}_{j}\big{]}^{2}=0$
for a $-B_{i}$-orthonormal basis
$\big{(}\widetilde{\theta}^{(i)}_{1},\dots,\widetilde{\theta}^{(i)}_{s_{i}}\big{)}$
of $\mathfrak{t}_{i}^{*}$, which can be easily checked by direct calculations.
Since $B=\sum_{i=1}^{m}B_{i}$, the basis is also $-B$-orthonormal.

It follows that the only quadratic relations among the cohomology classes in
$H^{2}(G/T,\mathbb{R})$ are given by

$\sum_{i=1}^{m}c_{i}\sum_{j=1}^{s_{i}}\big{[}{\rm
d}\widetilde{\theta}^{(i)}_{j}\big{]}^{2}=0,$ (3.1)

for some constants $c_{1},\dots,c_{m}$.

Then

$\sum_{i=1}^{2k}[{\rm
d}\theta_{i}]^{2}=\sum_{i=1}^{m}c_{i}\sum_{j=1}^{s_{i}}\big{[}{\rm
d}\widetilde{\theta}^{(i)}_{j}\big{]}^{2}.$ (3.2)

Denote the $-B$-orthonormal basis
$\big{(}\widetilde{\theta}_{1}^{(1)},\dots,\widetilde{\theta}_{s_{1}}^{(1)},\dots,\widetilde{\theta}_{1}^{(m)},\dots,\widetilde{\theta}_{s_{m}}^{(m)}\big{)}$
by $\sigma_{l}$ for $l=1,2,\dots,2k$. Then
$\theta_{i}=\sum_{l=1}^{2k}q_{li}\sigma_{l}$ for a transition matrix
$Q=(q_{ij})$ and we can rewrite $\sum_{i=1}^{2k}[{\rm d}\theta_{i}]^{2}$ as

$\sum_{i=1}^{2k}[{\rm
d}\theta_{i}]^{2}=\sum_{i=1}^{2k}\sum_{l,j=1}^{2k}q_{li}q_{ji}[{\rm
d}\sigma_{l}\wedge{\rm d}\sigma_{j}].$

On the other hand,

$\sum_{i=1}^{m}c_{i}\sum_{j=1}^{s_{i}}\big{[}{\rm
d}\widetilde{\theta}^{(i)}_{j}\big{]}^{2}=c_{1}\sum_{i=1}^{s_{1}}[{\rm
d}\sigma_{i}]^{2}+c_{2}\sum_{i=1}^{s_{2}}[{\rm
d}\sigma_{s_{1}+i}]^{2}+\dots+c_{m}\sum_{i=1}^{s_{m}}[{\rm
d}\sigma_{s_{1}+\dots+s_{m-1}+i}]^{2}.$

Therefore, from the equality (3.2) we have

$\sum_{i=1}^{2k}\sum_{l,j=1}^{2k}q_{li}q_{ji}[{\rm d}\sigma_{l}\wedge{\rm
d}\sigma_{j}]=c_{1}\sum_{i=1}^{s_{1}}[{\rm
d}\sigma_{i}]^{2}+c_{2}\sum_{i=1}^{s_{2}}[{\rm
d}\sigma_{s_{1}+i}]^{2}+\dots+c_{m}\sum_{i=1}^{s_{m}}[{\rm
d}\sigma_{s_{1}+\dots+s_{m-1}+i}]^{2}.$

Using that $[{\rm d}\sigma_{l}\wedge{\rm d}\sigma_{j}]$ are independent, for
every $l\neq j$ we get that

$\sum_{i=1}^{2k}q_{li}q_{ji}=0,\qquad l\neq j,$

and by (3.1)

$\sum_{i=1}^{s_{1}}(q_{li})^{2}=c_{1},\quad\dots,\quad\sum_{i=s_{m-1}}^{s_{m}}(q_{li})^{2}=c_{m},$

i.e., that $QQ^{t}=D$, where $D$ is a diagonal matrix with entries $c_{i}$.
From here we conclude, that $c_{i}>0$ and $Q$ preserves the metric
$-\sum_{i=1}^{m}c_{i}B_{i}$. In particular $\theta_{i}$ are
$-\sum_{i=1}^{m}c_{i}B_{i}$-orthonormal. As the $\theta_{i}$s are $g$\- and
$-\sum_{i=1}^{m}c_{i}B_{i}$-orthonormal, the restriction of $g$ to
$\mathfrak{t}$ coincides with the restriction of a biinvariant metric, the
theorem follows. ∎

######  Remark 3.3.

It is well known that on an even dimensional compact Lie group equipped with a
bi-invariant metric $g_{0}$ there is always a $g_{0}$-compatible complex
structure, which is necessary SKT and also Bismut-flat. But there exist
Samelson’s complex structures which are not compatible with any bi-invariant
metric on $G$ as long as the rank of $G$ is at least 2.

As a consequence of Theorem 3.2, we have the following

######  Corollary 3.4.

Let $G$ be a compact semi-simple Lie group with Samelson’s complex structure
$I$ as above. Let $g$ be a left-invariant $I$-Hermitian metric on $G$, which
is also invariant under the right action of the torus $T$ in the Tits
fibration $G\rightarrow G/T$, where $T$ is a maximal torus in $G$. We can
write the fundamental form $F$ of the metric $g$ as
$F=F|_{\mathfrak{t}}+\sum_{i=1}^{m}\sum_{\alpha\in
P_{i}}\lambda_{\alpha}^{(i)}e_{\alpha}^{(i)}\wedge Ie_{\alpha}^{(i)}$, where
$e_{\alpha}^{(i)}$, $Ie_{\alpha}^{(i)}$ are dual to
$X_{\alpha}^{(i)},Y_{\alpha}^{(i)}$ and
$\lambda_{\alpha}^{(i)}=g(E_{\alpha}^{(i)},E_{-\alpha}^{(i)})$. Then $g$ is
SKT iff $g|_{\mathfrak{t}}=-\sum_{i=1}^{m}\nu_{i}B|_{\mathfrak{t}_{i}}$ and
$\sum_{i=1}^{m}\sum_{\alpha\in
P_{i}}\big{(}\lambda_{\alpha}^{(i)}-\nu_{i}\big{)}e_{\alpha}^{(i)}\wedge
Ie_{\alpha}^{(i)}$ is a linear combination of the basis $\omega_{j}={\rm
d}\alpha_{j}$ of $H^{2}(G/T,\mathbb{R})$, where $\alpha_{j}$ are the simple
roots.

Equivalently, $g$ is SKT iff ${\rm d}\big{(}\sum_{i=1}^{m}\sum_{\alpha\in
P_{i}}\big{(}\lambda_{\alpha}^{(i)}-\nu_{i}\big{)}e_{\alpha}^{(i)}\wedge
Ie_{\alpha}^{(i)}\big{)}=0$ and
$\Lambda|_{\mathfrak{t}}=\sum_{i=1}^{m}\nu_{i}{\rm Id}|_{\mathfrak{t}_{i}}$,
for some constants $\nu_{i}$, $i=1,2,\dots,m$.

###### Proof.

By [13], we can rewrite
$F=\sum_{l=1}^{k}\mu_{l}^{2}\theta_{2l-1}\wedge\theta_{2l}+\pi^{*}(F_{b}),$
where $F_{b}$ is a (1,1)-form of an invariant Hermitian structure on $G/T$ and
$(\theta_{i})$ is a $(-B)$-orthonormal basis of
$\mathfrak{t}^{*}=\sum_{i=1}^{m}\mathfrak{t}_{i}^{*}$, which is also
$g$-orthogonal. Note that $\theta_{i}$ are linear combinations of the simple
roots $\alpha_{i}$ and

$F|_{\mathfrak{t}}=\sum_{l=1}^{k}\mu_{l}^{2}\theta_{2l-1}\wedge\theta_{2l},\qquad\pi^{*}(F_{b})=\sum_{i=1}^{m}\sum_{\alpha\in
P_{i}}\lambda_{\alpha}^{(i)}e_{\alpha}^{(i)}\wedge Ie_{\alpha}^{(i)},$

where $P_{i}$ denotes the set of positive roots of the simple factor $G_{i}$
with respect to $\mathfrak{t}_{i}$. Suppose that
$g|_{\mathfrak{t}}=-\sum_{i=1}^{m}\nu_{i}B|_{\mathfrak{t}_{i}}$ and
$\sum_{i=1}^{m}\sum_{\alpha\in
P_{i}}\big{(}\lambda_{\alpha}^{(i)}-\nu_{i}\big{)}e_{\alpha}^{(i)}\wedge
Ie_{\alpha}^{(i)}$ is a linear combination of the basis $\omega_{j}={\rm
d}\alpha_{j}$ of $H^{2}(G/T,\mathbb{R})$. Using that every $I$-Hermitian bi-
invariant metric is SKT, we have that ${\rm d}{\rm d}^{\rm
c}\big{(}F|_{\mathfrak{t}}+\nu_{i}e_{\alpha}^{(i)}\wedge
Ie_{\alpha}^{(i)}\big{)}=0$. Since ${\rm
d}\big{(}\sum_{i=1}^{m}\sum_{\alpha\in
P_{i}}\big{(}\lambda_{\alpha}^{(i)}-\nu_{i}\big{)}e_{\alpha}^{(i)}\wedge
Ie_{\alpha}^{(i)}\big{)}=0$, then it follows that $g$ is SKT.

Conversely, if $g$ is SKT, then by Theorem 3.2, we have that $I$ is compatible
with $-\sum_{i=1}^{m}\nu_{i}B_{i}$ and
$g|_{\mathfrak{t}}=-\sum_{i=1}^{m}\nu_{i}B_{i}|_{\mathfrak{t}_{i}}$, for some
positive $\nu_{i}$. This implies that $\Lambda$ is of the form
$\sum_{i=1}^{m}\nu_{i}{\rm Id}|_{\mathfrak{t}_{i}}$. Then, the difference
between $F$ and the fundamental form associated to the metric
$-\sum_{i=1}^{m}\nu_{i}B_{i}$ is given by $\sum_{i=1}^{m}\sum_{\alpha\in
P_{i}}\big{(}\lambda_{\alpha}^{(i)}-\nu_{i}\big{)}e_{\alpha}^{(i)}\wedge
Ie_{\alpha}^{(i)}$. As a consequence $\sum_{i=1}^{m}\sum_{\alpha\in
P_{i}}\big{(}\lambda_{\alpha}^{(i)}-\nu_{i}\big{)}e_{\alpha}^{(i)}\wedge
Ie_{\alpha}^{(i)}$ is ${\rm d}{\rm d}^{\rm c}$-closed.

Since $\sum_{i=1}^{m}\sum_{\alpha\in
P_{i}}\big{(}\lambda_{\alpha}^{(i)}-\nu_{i}\big{)}e_{\alpha}^{(i)}\wedge
Ie_{\alpha}^{(i)}$ is the pull-back by $\pi$ of a ${\rm d}{\rm d}^{\rm
c}$-closed $(1,1)$-form on $G/T$ and the Aepply cohomology of $G/T$ is
isomorphic to the Dolbeault cohomology, we have that
$\sum_{i=1}^{m}\sum_{\alpha\in
P_{i}}\big{(}\lambda_{\alpha}^{(i)}-\nu_{i}\big{)}e_{\alpha}^{(i)}\wedge
Ie_{\alpha}^{(i)}$ is $d$-closed, and so $\sum_{i=1}^{m}\sum_{\alpha\in
P_{i}}\big{(}\lambda_{\alpha}^{(i)}-\nu_{i}\big{)}e_{\alpha}^{(i)}\wedge
Ie_{\alpha}^{(i)}$ is a linear combination of $\omega_{i}={\rm d}\alpha_{i}$.
∎

######  Example 3.5.

We will now construct an explicit $5$-parameter family of SKT metrics on the
real compact Lie group ${\rm SO}(9)$ compatible with a Samelson complex
structure. Recall that the complexification of ${\rm SO}(9)$ has rank 4 and
its Lie algebra corresponds to $B_{4}$. Denote by $E_{ij}$ the matrix in
${\mathfrak{so}}(9,\mathbb{C})$, whose unique non-zero entry is 1 at the place
$(i,j)$. Let fix the maximal abelian subalgebra $\mathfrak{t}$ of
${\mathfrak{so}}(9)$ whose complexification $\mathfrak{h}=\mathfrak{t}^{\rm
c}=\operatorname{span}\langle H_{1},H_{2},H_{3},H_{4}\rangle$, where

$\displaystyle H_{1}={\rm i}(E_{12}-E_{21}),\qquad\\!H_{2}={\rm
i}(E_{34}-E_{43}),\qquad\\!H_{3}={\rm i}(E_{56}-E_{65}),\qquad\\!H_{4}={\rm
i}(E_{78}-E_{87}).\\!$

The weights for the fundamental representation of the Cartan subalgebra are
$\pm f^{k}$, where

$f^{1}=(1,0,0,0),\quad\dots,\quad f^{4}=(0,0,0,1),$

and the roots are given by $\pm f^{j}\pm f^{k}$, for $k\neq j$ and $\pm
f^{j}$. In particular, the set $P$ of positive roots is the set consisting of
$f^{j}\pm f^{k},$ for $j<k,$ and $f^{j}$ (see, for instance, [27] for more
details). We can choose a basis $(E_{\pm\alpha},H_{\alpha})_{\alpha\in P}$
such that

$\mathfrak{g}_{\pm\alpha}={\rm{span}}_{\mathbb{C}}\langle
E_{\pm\alpha}\rangle,\qquad[E_{\alpha},E_{-\alpha}]=H_{\alpha},\qquad
B(E_{\alpha},E_{-\alpha})=1,\qquad[H,E_{\alpha}]=\alpha(H)E_{\alpha}$

and $B(H,H_{\alpha})=\alpha(H)$ for all $H\in\mathfrak{t}$. The vectors
$(E_{\pm\alpha})$ satisfying the previous conditions are given by

$\displaystyle E_{1}^{+}=\frac{\sqrt{2}}{2}(E_{19}+{\rm i}E_{29}-E_{91}-{\rm
i}E_{92}),\qquad E_{1}^{-}=\frac{\sqrt{2}}{2}(E_{19}-{\rm i}E_{29}-E_{91}+{\rm
i}E_{92}),$ $\displaystyle E_{2}^{+}=\frac{\sqrt{2}}{2}(E_{39}+{\rm
i}E_{49}-E_{93}-{\rm i}E_{94}),\qquad E_{2}^{-}=\frac{\sqrt{2}}{2}(E_{39}-{\rm
i}E_{49}-E_{93}+{\rm i}E_{94}),$ $\displaystyle
E_{3}^{+}=\frac{\sqrt{2}}{2}(E_{59}+{\rm i}E_{69}-E_{95}-{\rm i}E_{96}),\qquad
E_{3}^{-}=\frac{\sqrt{2}}{2}(E_{59}-{\rm i}E_{69}-E_{95}+{\rm i}E_{96}),$
$\displaystyle E_{4}^{+}=\frac{\sqrt{2}}{2}(E_{79}+{\rm i}E_{89}-E_{97}-{\rm
i}E_{98}),\qquad E_{4}^{-}=\frac{\sqrt{2}}{2}(E_{79}-{\rm i}E_{89}-E_{97}+{\rm
i}E_{98}),$
$\displaystyle[E_{1}^{+},E_{2}^{+}],\qquad[E_{1}^{+},E_{3}^{+}],\qquad[E_{1}^{+},E_{4}^{+}],\qquad[E_{2}^{+},E_{3}^{+}],\qquad[E_{2}^{+},E_{4}^{+}],\qquad[E_{3}^{+},E_{4}^{+}],$
$\displaystyle[E_{1}^{-},E_{2}^{-}],\qquad[E_{1}^{-},E_{3}^{-}],\qquad[E_{1}^{-},E_{4}^{-}],\qquad[E_{2}^{-},E_{3}^{-}],\qquad[E_{2}^{-},E_{4}^{-}],\qquad[E_{3}^{-},E_{4}^{-}],$
$\displaystyle[E_{1}^{+},E_{2}^{-}],\qquad[E_{1}^{+},E_{3}^{-}],\qquad[E_{1}^{+},E_{4}^{-}],\qquad[E_{2}^{+},E_{1}^{-}],\qquad[E_{2}^{+},E_{3}^{-}],\qquad[E_{2}^{+},E_{4}^{-}],$
$\displaystyle[E_{3}^{+},E_{1}^{-}],\qquad[E_{3}^{+},E_{2}^{-}],\qquad[E_{3}^{+},E_{4}^{-}],\qquad[E_{4}^{+},E_{1}^{-}],\qquad[E_{4}^{+},E_{2}^{-}],\qquad[E_{4}^{+},E_{3}^{-}].$

Therefore, we have $\mathfrak{t}=\operatorname{span}\langle{\rm i}H_{1},{\rm
i}H_{2},{\rm i}H_{3},{\rm i}H_{4}\rangle$ and ${\mathfrak{so}}(9)$ is the real
span of $(X_{k},Y_{k},{\rm i}H_{j})$, with $k=1,\dots,16$, $j=1,\dots,4$, and

$\displaystyle{\rm i}H_{1}=-(E_{12}-E_{21}),\qquad{\rm
i}H_{2}=-(E_{34}-E_{43}),$ $\displaystyle{\rm
i}H_{3}=-(E_{56}-E_{65}),\qquad{\rm i}H_{4}=-(E_{78}-E_{87}),$ $\displaystyle
X_{1}=E_{19}-E_{91},\qquad Y_{1}=E_{29}-E_{92},\qquad
X_{2}=E_{39}-E_{93},\qquad Y_{2}=E_{49}-E_{94},$ $\displaystyle
X_{3}=E_{59}-E_{95},\qquad Y_{3}=E_{69}-E_{96},\qquad
X_{4}=E_{79}-E_{97},\qquad Y_{4}=E_{89}-E_{98},$ $\displaystyle
X_{5}=\frac{\sqrt{2}}{2}(-E_{13}+E_{24}+E_{31}-E_{42}),\qquad
Y_{5}=\frac{\sqrt{2}}{2}(-E_{14}-E_{23}+E_{41}+E_{32}),$ $\displaystyle
X_{6}=\frac{\sqrt{2}}{2}(-E_{15}+E_{26}+E_{51}-E_{62}),\qquad
Y_{6}=\frac{\sqrt{2}}{2}(-E_{16}-E_{25}+E_{61}+E_{52}),$ $\displaystyle
X_{7}=\frac{\sqrt{2}}{2}(-E_{17}+E_{28}+E_{71}-E_{82}),\qquad
Y_{7}=\frac{\sqrt{2}}{2}(-E_{18}-E_{27}+E_{81}+E_{72}),$ $\displaystyle
X_{8}=\frac{\sqrt{2}}{2}(-E_{35}+E_{46}+E_{53}-E_{64}),\qquad
Y_{8}=\frac{\sqrt{2}}{2}(-E_{36}-E_{45}+E_{63}+E_{54}),$ $\displaystyle
X_{9}=\frac{\sqrt{2}}{2}(-E_{37}+E_{48}+E_{73}-E_{84}),\qquad
Y_{9}=\frac{\sqrt{2}}{2}(-E_{38}-E_{47}+E_{83}+E_{74}),$ $\displaystyle
X_{10}=\frac{\sqrt{2}}{2}(-E_{57}+E_{68}+E_{75}-E_{86}),\qquad
Y_{10}=\frac{\sqrt{2}}{2}(-E_{58}-E_{67}+E_{85}+E_{76}),$ $\displaystyle
X_{11}=\frac{\sqrt{2}}{2}(-E_{13}-E_{24}+E_{31}+E_{42}),\qquad
Y_{11}=\frac{\sqrt{2}}{2}(E_{14}-E_{23}-E_{41}+E_{32}),$ $\displaystyle
X_{12}=\frac{\sqrt{2}}{2}(-E_{15}-E_{26}+E_{51}+E_{62}),\qquad
Y_{12}=\frac{\sqrt{2}}{2}(E_{16}-E_{25}-E_{61}+E_{52}),$ $\displaystyle
X_{13}=\frac{\sqrt{2}}{2}(-E_{17}-E_{28}+E_{71}+E_{82}),\qquad
Y_{13}=\frac{\sqrt{2}}{2}(E_{18}-E_{27}-E_{81}+E_{72}),$ $\displaystyle
X_{14}=\frac{\sqrt{2}}{2}(-E_{35}-E_{46}+E_{53}+E_{64}),\qquad
Y_{14}=\frac{\sqrt{2}}{2}(E_{36}-E_{45}-E_{63}+E_{54}),$ $\displaystyle
X_{15}=\frac{\sqrt{2}}{2}(-E_{37}-E_{48}+E_{73}+E_{84}),\qquad
Y_{15}=\frac{\sqrt{2}}{2}(E_{38}-E_{47}-E_{83}+E_{74}),$ $\displaystyle
X_{16}=\frac{\sqrt{2}}{2}(-E_{57}-E_{68}+E_{75}+E_{86}),\qquad
Y_{16}=\frac{\sqrt{2}}{2}(E_{58}-E_{67}-E_{85}+E_{76}).$

If we denote by $I$ the Samelson complex structure defined by $IH_{1}=H_{2}$,
$IH_{3}=H_{4}$ and by the choice of positive roots, i.e., by $IX_{j}=Y_{j}$,
$j=1,\dots,16$, we get that $({\rm SO}(9),I)$ has the following complex
structure equations

$\displaystyle{\rm d}\varphi_{1}=\frac{{\rm
i}}{2}(\varphi_{3}\wedge\overline{\varphi}_{3}+\varphi_{7}\wedge\overline{\varphi}_{7}+\varphi_{8}\wedge\overline{\varphi}_{8}+\varphi_{9}\wedge\overline{\varphi}_{9}+\varphi_{13}\wedge\overline{\varphi}_{13}+\varphi_{14}\wedge\overline{\varphi}_{14}+\varphi_{15}\wedge\overline{\varphi}_{15})$
$\displaystyle\hphantom{{\rm
d}\varphi_{1}=}{}-\frac{1}{2}(\varphi_{4}\wedge\overline{\varphi}_{4}+\varphi_{7}\wedge\overline{\varphi}_{7}+\varphi_{10}\wedge\overline{\varphi}_{10}+\varphi_{11}\wedge\overline{\varphi}_{11}-\varphi_{13}\wedge\overline{\varphi}_{13}+\varphi_{16}\wedge\overline{\varphi}_{16}$
$\displaystyle\hphantom{{\rm d}\varphi_{1}=\frac{{\rm
i}}{2}(}{}+\varphi_{17}\wedge\overline{\varphi}_{17}),$ $\displaystyle{\rm
d}\varphi_{2}=\frac{{\rm
i}}{2}(\varphi_{5}\wedge\overline{\varphi}_{5}+\varphi_{8}\wedge\overline{\varphi}_{8}+\varphi_{10}\wedge\overline{\varphi}_{10}+\varphi_{12}\wedge\overline{\varphi}_{12}-\varphi_{14}\wedge\overline{\varphi}_{14}-\varphi_{16}\wedge\overline{\varphi}_{16}+\varphi_{18}\wedge\overline{\varphi}_{18})$
$\displaystyle\hphantom{{\rm
d}\varphi_{2}=}{}-\frac{1}{2}(\varphi_{6}\wedge\overline{\varphi}_{6}+\varphi_{9}\wedge\overline{\varphi}_{9}+\varphi_{11}\wedge\overline{\varphi}_{11}+\varphi_{12}\wedge\overline{\varphi}_{12}-\varphi_{15}\wedge\overline{\varphi}_{15}-\varphi_{17}\wedge\overline{\varphi}_{17}$
$\displaystyle\hphantom{{\rm d}\varphi_{2}=\frac{{\rm
i}}{2}(}{}-\varphi_{18}\wedge\overline{\varphi}_{18}),$ $\displaystyle{\rm
d}\varphi_{3}=\frac{{\rm
i}}{2}(\varphi_{1}\wedge\varphi_{3}+\overline{\varphi}_{1}\wedge\varphi_{3})-\frac{\sqrt{2}}{2}(\overline{\varphi}_{4}\wedge\varphi_{7}+\varphi_{4}\wedge\varphi_{13}+\overline{\varphi}_{5}\wedge\varphi_{8}+\varphi_{5}\wedge\varphi_{14})$
$\displaystyle\hphantom{{\rm
d}\varphi_{9}=}{}-\frac{\sqrt{2}}{2}(\overline{\varphi}_{6}\wedge\varphi_{9}+\varphi_{6}\wedge\varphi_{15}),$
$\displaystyle{\rm
d}\varphi_{4}=\frac{1}{2}(\varphi_{1}\wedge\varphi_{4}-\overline{\varphi}_{1}\wedge\varphi_{4})+\frac{\sqrt{2}}{2}(\overline{\varphi}_{3}\wedge\varphi_{7}+\varphi_{3}\wedge\overline{\varphi}_{13}-\overline{\varphi}_{5}\wedge\varphi_{10}-\varphi_{5}\wedge\varphi_{16})$
$\displaystyle\hphantom{{\rm
d}\varphi_{9}=}{}-\frac{\sqrt{2}}{2}(\varphi_{6}\wedge\varphi_{17}+\overline{\varphi}_{6}\wedge\varphi_{11})$
$\displaystyle{\rm d}\varphi_{5}=\frac{{\rm
i}}{2}(\varphi_{2}\wedge\varphi_{5}+\overline{\varphi}_{2}\wedge\varphi_{5})+\frac{\sqrt{2}}{2}(\overline{\varphi}_{3}\wedge\varphi_{8}+\varphi_{3}\wedge\overline{\varphi}_{14}+\overline{\varphi}_{4}\wedge\varphi_{10}+\varphi_{4}\wedge\overline{\varphi}_{16})$
$\displaystyle\hphantom{{\rm
d}\varphi_{9}=}{}-\frac{\sqrt{2}}{2}(\overline{\varphi}_{6}\wedge\varphi_{12}+\varphi_{6}\wedge\varphi_{18}),$
$\displaystyle{\rm
d}\varphi_{6}=\frac{1}{2}(\varphi_{2}\wedge\varphi_{6}-\overline{\varphi}_{2}\wedge\varphi_{6})+\frac{\sqrt{2}}{2}(\overline{\varphi}_{3}\wedge\varphi_{9}+\varphi_{3}\wedge\overline{\varphi}_{15}+\overline{\varphi}_{4}\wedge\varphi_{11}+\varphi_{4}\wedge\overline{\varphi}_{17})$
$\displaystyle\hphantom{{\rm
d}\varphi_{9}=}{}+\frac{\sqrt{2}}{2}(\overline{\varphi}_{5}\wedge\varphi_{12}+\varphi_{5}\wedge\overline{\varphi}_{18}),$
$\displaystyle{\rm
d}\varphi_{7}=\frac{1}{2}(\varphi_{1}\wedge\varphi_{7}-\overline{\varphi}_{1}\wedge\varphi_{7})+\frac{{\rm
i}}{2}(\varphi_{1}\wedge\varphi_{7}+\overline{\varphi}_{1}\wedge\varphi_{7})$
$\displaystyle\hphantom{{\rm
d}\varphi_{9}=}{}-\frac{\sqrt{2}}{2}(\varphi_{3}\wedge\varphi_{4}+\varphi_{8}\wedge\varphi_{16}+\varphi_{9}\wedge\varphi_{17})+\frac{\sqrt{2}}{2}(\varphi_{10}\wedge\varphi_{14}+\varphi_{11}\wedge\varphi_{15}),$
$\displaystyle{\rm d}\varphi_{8}=\frac{{\rm
i}}{2}(\varphi_{1}\wedge\varphi_{8}+\overline{\varphi}_{1}\wedge\varphi_{8})+\frac{{\rm
i}}{2}(\varphi_{2}\wedge\varphi_{8}+\overline{\varphi}_{2}\wedge\varphi_{8})$
$\displaystyle\hphantom{{\rm
d}\varphi_{9}=}{}-\frac{\sqrt{2}}{2}(\varphi_{3}\wedge\varphi_{5}-\varphi_{7}\wedge\overline{\varphi}_{16}+\varphi_{9}\wedge\varphi_{18})-\frac{\sqrt{2}}{2}(\varphi_{10}\wedge\varphi_{13}-\varphi_{12}\wedge\varphi_{15}),$
$\displaystyle{\rm d}\varphi_{9}=\frac{{\rm
i}}{2}(\varphi_{1}\wedge\varphi_{9}+\overline{\varphi}_{1}\wedge\varphi_{9})+\frac{1}{2}(\varphi_{2}\wedge\varphi_{9}-\overline{\varphi}_{2}\wedge\varphi_{9})$
$\displaystyle\hphantom{{\rm
d}\varphi_{9}=}{}-\frac{\sqrt{2}}{2}(\varphi_{3}\wedge\varphi_{6}-\varphi_{7}\wedge\overline{\varphi}_{17}-\varphi_{8}\wedge\overline{\varphi}_{18})-\frac{\sqrt{2}}{2}(\varphi_{11}\wedge\varphi_{13}+\varphi_{12}\wedge\varphi_{14}),$
$\displaystyle{\rm
d}\varphi_{10}=\frac{1}{2}(\varphi_{1}\wedge\varphi_{10}-\overline{\varphi}_{1}\wedge\varphi_{10})+\frac{{\rm
i}}{2}(\varphi_{2}\wedge\varphi_{10}+\overline{\varphi}_{2}\wedge\varphi_{10})$
$\displaystyle\hphantom{{\rm
d}\varphi_{18}=}{}-\frac{\sqrt{2}}{2}(\varphi_{4}\wedge\varphi_{5}+\varphi_{7}\wedge\overline{\varphi}_{14}-\varphi_{8}\wedge\overline{\varphi}_{13})-\frac{\sqrt{2}}{2}(\varphi_{11}\wedge\varphi_{18}-\varphi_{12}\wedge\varphi_{17}),$
$\displaystyle{\rm
d}\varphi_{11}=\frac{1}{2}(\varphi_{1}\wedge\varphi_{11}-\overline{\varphi}_{1}\wedge\varphi_{11})+\frac{1}{2}(\varphi_{2}\wedge\varphi_{11}-\overline{\varphi}_{2}\wedge\varphi_{11})$
$\displaystyle\hphantom{{\rm
d}\varphi_{18}=}{}-\frac{\sqrt{2}}{2}(\varphi_{4}\wedge\varphi_{6}+\varphi_{7}\wedge\overline{\varphi}_{15}-\varphi_{9}\wedge\varphi_{13})+\frac{\sqrt{2}}{2}(\varphi_{10}\wedge\overline{\varphi}_{18}-\varphi_{12}\wedge\varphi_{16}),$
$\displaystyle{\rm d}\varphi_{12}=\frac{{\rm
i}}{2}(\varphi_{2}\wedge\varphi_{12}+\overline{\varphi}_{2}\wedge\varphi_{12})+\frac{1}{2}(\varphi_{2}\wedge\varphi_{12}-\overline{\varphi}_{2}\wedge\varphi_{12})$
$\displaystyle\hphantom{{\rm
d}\varphi_{18}=}{}-\frac{\sqrt{2}}{2}(\varphi_{5}\wedge\varphi_{6}+\varphi_{8}\wedge\overline{\varphi}_{15}-\varphi_{9}\wedge\overline{\varphi}_{14})-\frac{\sqrt{2}}{2}(\varphi_{10}\wedge\overline{\varphi}_{17}-\varphi_{11}\wedge\overline{\varphi}_{16}),$
$\displaystyle{\rm d}\varphi_{13}=\frac{{\rm
i}}{2}(\varphi_{1}\wedge\varphi_{13}+\overline{\varphi}_{1}\wedge\varphi_{13})-\frac{1}{2}(\varphi_{1}\wedge\varphi_{13}-\overline{\varphi}_{1}\wedge\varphi_{13})$
$\displaystyle\hphantom{{\rm
d}\varphi_{18}=}{}-\frac{\sqrt{2}}{2}(\varphi_{3}\wedge\overline{\varphi}_{4}+\varphi_{8}\wedge\overline{\varphi}_{10}+\varphi_{9}\wedge\overline{\varphi}_{11})-\frac{\sqrt{2}}{2}(\varphi_{14}\wedge\overline{\varphi}_{16}+\varphi_{15}\wedge\overline{\varphi}_{17}),$
$\displaystyle{\rm d}\varphi_{14}=\frac{{\rm
i}}{2}(\varphi_{1}\wedge\varphi_{14}+\overline{\varphi}_{1}\wedge\varphi_{14})-\frac{{\rm
i}}{2}(\varphi_{2}\wedge\varphi_{14}+\overline{\varphi}_{2}\wedge\varphi_{14})$
$\displaystyle\hphantom{{\rm
d}\varphi_{18}=}{}-\frac{\sqrt{2}}{2}(\varphi_{3}\wedge\overline{\varphi}_{5}-\varphi_{7}\wedge\overline{\varphi}_{10}+\varphi_{9}\wedge\overline{\varphi}_{12})+\frac{\sqrt{2}}{2}(\varphi_{13}\wedge\varphi_{16}-\varphi_{15}\wedge\overline{\varphi}_{18}),$
$\displaystyle{\rm d}\varphi_{15}=\frac{{\rm
i}}{2}(\varphi_{1}\wedge\varphi_{15}+\overline{\varphi}_{1}\wedge\varphi_{15})-\frac{1}{2}(\varphi_{2}\wedge\varphi_{15}-\overline{\varphi}_{2}\wedge\varphi_{15})$
$\displaystyle\hphantom{{\rm
d}\varphi_{18}=}{}-\frac{\sqrt{2}}{2}(\varphi_{3}\wedge\overline{\varphi}_{6}-\varphi_{7}\wedge\overline{\varphi}_{11}-\varphi_{8}\wedge\overline{\varphi}_{12})+\frac{\sqrt{2}}{2}(\varphi_{13}\wedge\varphi_{17}+\varphi_{14}\wedge\varphi_{18}),$
$\displaystyle{\rm
d}\varphi_{16}=\frac{1}{2}(\varphi_{1}\wedge\varphi_{16}-\overline{\varphi}_{1}\wedge\varphi_{16})-\frac{{\rm
i}}{2}(\varphi_{2}\wedge\varphi_{16}+\overline{\varphi}_{2}\wedge\varphi_{16})$
$\displaystyle\hphantom{{\rm
d}\varphi_{18}=}{}-\frac{\sqrt{2}}{2}(\varphi_{4}\wedge\overline{\varphi}_{5}+\varphi_{7}\wedge\overline{\varphi}_{8}+\varphi_{11}\wedge\overline{\varphi}_{12})-\frac{\sqrt{2}}{2}(\overline{\varphi}_{13}\wedge\overline{\varphi}_{14}+\varphi_{17}\wedge\overline{\varphi}_{18}),$
$\displaystyle{\rm
d}\varphi_{17}=\frac{1}{2}(\varphi_{1}\wedge\varphi_{17}-\overline{\varphi}_{1}\wedge\varphi_{17})-\frac{1}{2}(\varphi_{2}\wedge\varphi_{17}-\overline{\varphi}_{2}\wedge\varphi_{17})$
$\displaystyle\hphantom{{\rm
d}\varphi_{18}=}{}-\frac{\sqrt{2}}{2}(\varphi_{4}\wedge\overline{\varphi}_{6}+\varphi_{7}\wedge\overline{\varphi}_{9}-\varphi_{10}\wedge\overline{\varphi}_{12})-\frac{\sqrt{2}}{2}(\overline{\varphi}_{13}\wedge\varphi_{15}-\varphi_{16}\wedge\varphi_{18}),$
$\displaystyle{\rm d}\varphi_{18}=\frac{{\rm
i}}{2}(\varphi_{2}\wedge\varphi_{18}+\overline{\varphi}_{2}\wedge\varphi_{18})-\frac{1}{2}(\varphi_{2}\wedge\varphi_{18}-\overline{\varphi}_{2}\wedge\varphi_{18})$
$\displaystyle\hphantom{{\rm
d}\varphi_{18}=}{}-\frac{\sqrt{2}}{2}(\varphi_{5}\wedge\overline{\varphi}_{6}+\varphi_{8}\wedge\overline{\varphi}_{9}+\varphi_{10}\wedge\overline{\varphi}_{11})-\frac{\sqrt{2}}{2}(\overline{\varphi}_{14}\wedge\varphi_{15}+\overline{\varphi}_{16}\wedge\varphi_{17}).$

By a direct computation, we have that the left-invariant metric
$F=\sum_{k=1}^{18}{\rm i}b_{k}\varphi_{k}\wedge\overline{\varphi}_{k}$, with
$b_{k}>0$ for every $k$, is SKT if and only if

$\displaystyle F=\frac{{\rm
i}}{2}(b_{16}-b_{17}+b_{18})(\varphi_{1}\wedge\overline{\varphi}_{1}+\varphi_{2}\wedge\overline{\varphi}_{2})+\frac{{\rm
i}}{2}(b_{6}+b_{15}-b_{16}+b_{17}-b_{18})\varphi_{3}\wedge\overline{\varphi}_{3}$
$\displaystyle\hphantom{F=}{}+\frac{{\rm
i}}{2}(b_{6}-b_{16}+2b_{17}-b_{18})\varphi_{4}\wedge\overline{\varphi}_{4}+\frac{{\rm
i}}{2}(b_{6}+b_{17}-b_{16})\varphi_{5}\wedge\overline{\varphi}_{5}+b_{6}\varphi_{6}\wedge\overline{\varphi}_{6}$
$\displaystyle\hphantom{F=}{}+\frac{{\rm
i}}{2}(2b_{6}+b_{15}-3b_{16}+4b_{17}-3b_{18})\varphi_{7}\wedge\overline{\varphi}_{7}\\!+\frac{{\rm
i}}{2}(2b_{6}+b_{15}-3b_{16}-2b_{18}+3b_{17})\varphi_{8}\wedge\overline{\varphi}_{8}\\!$
$\displaystyle\hphantom{F=}{}+\frac{{\rm
i}}{2}(2b_{6}+b_{15}-2b_{16}+2b_{17}-2b_{18})\varphi_{9}\wedge\overline{\varphi}_{9}+\frac{{\rm
i}}{2}(2b_{6}-3b_{16}+4b_{17}-2b_{18})\varphi_{10}\wedge\overline{\varphi}_{10}$
$\displaystyle\hphantom{F=}{}+\frac{{\rm
i}}{2}(3b_{17}-2b_{18}+2b_{6}-2b_{16})\varphi_{11}\wedge\overline{\varphi}_{11}+\frac{{\rm
i}}{2}(2b_{6}-2b_{16}+2b_{17}-b_{18})\varphi_{12}\wedge\overline{\varphi}_{12}$
$\displaystyle\hphantom{F=}{}+\frac{{\rm
i}}{2}(b_{15}+b_{16}-2b_{17}+b_{18})\varphi_{13}\wedge\overline{\varphi}_{13}+\frac{{\rm
i}}{2}(b_{15}+b_{16}-b_{17})\varphi_{14}\wedge\overline{\varphi}_{14}$
$\displaystyle\hphantom{F=}{}+b_{15}\varphi_{15}\wedge\overline{\varphi}_{15}+b_{16}\varphi_{16}\wedge\overline{\varphi}_{16}+b_{17}\varphi_{17}\wedge\overline{\varphi}_{17}+b_{18}\varphi_{18}\wedge\overline{\varphi}_{18}.$

So we obtain a $5$-parameter family of left-invariant $I$-Hermitian SKT
metrics which are also right $T$-invariant. By the results in the next section
(Corollary 5.1) this family describes all possible such metrics. Moreover,
$F=F_{0}$, where $F_{0}$ is the fundamental form associated to $-B$, if and
only if $b_{15}=b_{16}=b_{17}=b_{18}=b_{6}=1$. Since

$\displaystyle
F=(b_{16}-b_{17}+b_{18})F_{0}+\frac{1}{2}(b_{6}+b_{15}-2b_{16}+2b_{17}-2b_{18}){\rm
d}(\varphi_{1}+\overline{\varphi}_{1})$
$\displaystyle\hphantom{F=}{}+\frac{{\rm
i}}{2}(b_{6}-2b_{16}+3b_{17}-2b_{18}){\rm
d}(\overline{\varphi}_{1}-\varphi_{1})+\frac{1}{2}(b_{6}-2b_{16}+2b_{17}-b_{18}]){\rm
d}(\varphi_{2}+\overline{\varphi}_{2})$
$\displaystyle\hphantom{F=}{}+\frac{1}{2}(b_{6}-b_{16}+b_{17}-b_{18}){\rm
d}(\overline{\varphi}_{2}-\varphi_{2}),$

the Aeppli class of $F$ coincides with the one of $F_{0}$. Note that by doing
similar computations as in [20, 21] (see also [19]) it is possible to describe
the minimal model for the Dolbeault cohomology of $({\rm SO}(9),I)$.
Calculation for the Aeppli cohomology of the compact simple Lie groups of rank
two is given in [4].

##  4  Invariant CYT metrics

We first mention an analog of Lemma 3.1 for CYT metrics on compact semi-simple
Lie groups:

######  Proposition 4.1.

Let $I$ be a Samelson’s complex structure on a compact semi-simple Lie group
$G$ and $g$ be a left-invariant CYT $I$-Hermitian metric. Then there is a
left-invariant CYT $I$-Hermitian metric on $(G,I)$ which is also right
$T$-invariant.

###### Proof.

We use the same idea as in [7] combined with the symmetrization of Lemma 3.1.
By Koszul [18], the Ricci form of the Chern connection for every left-
invariant $I$-Hermitian metric on $(G,I)$, is the exterior derivative of the
half-sum of all positive roots, i.e.,

$\rho^{\rm Ch}=\frac{1}{2}{\rm i}\sum_{\alpha\in P}{\rm d}\alpha.$

Since the Bismut Ricci form $\rho^{\rm B}$ is related to the Chern Ricci form
by the relation $\rho^{\rm B}=\rho^{\rm Ch}+{\rm d}(\delta F)$, where $F$
denotes the fundamental form associated to the Hermitian metric, to prove the
proposition it is enough to find a right $T$-invariant $I$-Hermitian metric
$h$ whose fundamental form $F$ satisfies

$\delta F=-\frac{1}{2}{\rm i}\sum_{\alpha\in P}\alpha=\sigma,$

where $\delta=-\star_{h}d\star_{h}$ denotes the codifferential with respect to
$h$ and $\sigma$ is a left-invariant $1$-form. This follows from the fact that
${\rm d}(\delta F-\sigma)=0$ leads to $\delta F=\sigma$ because on a compact
semi-simple Lie group a closed left-invariant $1$-form has to be trivial.
Since $\star_{h}F=-\frac{1}{(n-1)!}F^{n-1}$, where $2n$ is the real dimension
of $G$, the previous condition is equivalent to ${\rm
d}F^{n-1}=(n-1)!\star_{h}(\sigma)=-I\sigma\wedge F^{n-1}$. Now if we
symmetrize $F^{n-1}$ with respect to the right $T$-action as in Lemma 3.1 we
obtain a right $T$-invariant $(n-1,n-1)$-form which is also positive, so it is
of the type $\tilde{F}^{n-1}$ for a right $T$-invariant form $\tilde{F}$.
Since $\sigma$ and $I\sigma$ belong to $\mathfrak{t}^{*}$ and the ${\rm
Ad}(T)$-action on $\mathfrak{t}$ is trivial, then ${\rm
d}\tilde{F}^{n-1}=-I\sigma\wedge\tilde{F}^{n-1}$. Since $F$ and $\tilde{F}$
coincide on $\mathfrak{t}$, the proof is completed. ∎

Let $g$ be a left-invariant $I$-Hermitian metric on $G$ which is also
invariant by the right $T$-action. Then we can write
$g(\cdot,\cdot)=-B(\Lambda(\cdot),\cdot)$, where $\Lambda$ is a positive-
definite Hermitian matrix on $\mathfrak{t}$ and diagonal on
$\sum_{\alpha}\mathfrak{g}_{\alpha}$. In particular,
$g(E_{\alpha},E_{-\alpha})=\lambda_{\alpha}$ and $g(E_{\alpha},E_{\beta})=0$,
for every $\alpha\neq-\beta$.

Let $F(\cdot,\cdot)=g(I\cdot,\cdot)$ be the fundamental form associated to the
Hermitian structure $(I,g)$. We can calculate the codifferential $\delta F$
using the formula in [12]:

$I\circ\delta F(X)=\frac{1}{2}\sum_{l=1}^{n}{\rm d}F(e_{l},Ie_{l},X),$

where $n$ is the complex dimension of $G$ and $(e_{i})$ is a $I$-adapted
orthonormal basis of $(G,I,g)$, such that $Ie_{2l-1}=e_{2l}$, $l=1,\dots,n$.
We can choose as $I$-adapted orthonormal basis $(e_{i})$ of $(G,I,g)$

$\left(\frac{X_{\alpha}}{\sqrt{-\lambda_{\alpha}}},\frac{Y_{\alpha}}{\sqrt{-\lambda_{\alpha}}},H_{k}\right),$

where $(H_{k})$ is an orthonormal basis of $\mathfrak{t}$ such that
$IH_{2l-1}=H_{2l}$. Because

${\rm d}F(X,Y,Z)=F([X,Y],Z)+F([Y,Z],X)+F([Z,X],Y)$

for every left-invariant $2$-form $F$ on $G$ and every $X,Y,Z\in\mathfrak{g}$,
we get that the sum

$\sum_{l=1}^{n}{\rm d}F(e_{l},Je_{l},X)=\sum_{\alpha\in
P}\frac{1}{\lambda_{\alpha}}{\rm
d}F(X_{\alpha},Y_{\alpha},X)+\sum_{l=1}^{r/2}{\rm d}F(H_{2l-1},H_{2l},X)$

has contributions only from

$\displaystyle{\rm d}{\rm d}F(X_{\alpha},Y_{\alpha},Z)$ $\displaystyle={\rm
i}{\rm d}F{\rm d}(E_{\alpha},E_{-\alpha},Z)={\rm i}{\rm
d}([E_{\alpha},E_{-\alpha}],Z)$ $\displaystyle={\rm
i}g(I([E_{\alpha},E_{-\alpha}]),Z)=-{\rm i}g(H_{\alpha},IZ)$
$\displaystyle={\rm i}B(\Lambda(H_{\alpha}),IZ)={\rm i}B(H_{\alpha},\Lambda
IZ)={\rm i}\alpha(\Lambda IZ)$

for every $Z\in\mathfrak{h}$.

Then we get

$\delta F\circ I=-\frac{1}{2}{\rm i}\sum_{\alpha\in
P}\frac{1}{\lambda_{\alpha}}(\alpha\circ\Lambda)\circ I.$

By the above, the Ricci form $\rho^{\rm B}$ is given by

$\rho^{\rm B}=\frac{1}{2}{\rm i}\sum_{\alpha\in P}\bigg{(}{\rm
d}\alpha-\frac{1}{\lambda_{\alpha}}{\rm d}(\alpha\circ\Lambda)\bigg{)}.$

Therefore $g$ is CYT if and only $\sum_{\alpha\in P}\big{(}{\rm
d}\alpha-\frac{1}{\lambda_{\alpha}}{\rm d}(\alpha\circ\Lambda)\big{)}=0$.
Using this characterization we have:

######  Theorem 4.2.

Let $G$ be a compact semi-simple Lie group with a Samelson’s complex structure
determined by the set $P$ of positive roots associated to a maximal torus $T$
of $G$. Let $g$ be a left-invariant $I$-Hermitian metric on $G$, which is also
invariant under the right action of the torus $T$ in the Tits fibration
$G\rightarrow G/T$, where $T$ is a maximal torus in $G$. We can write the
fundamental form $F$ of the metric $g$ as
$F=F|_{\mathfrak{t}}+\sum_{i=1}^{m}\sum_{\alpha\in
P_{i}}\lambda_{\alpha}^{(i)}e_{\alpha}\wedge Ie_{\alpha}^{(i)}$, where
$e_{\alpha}^{(i)}$, $Ie_{\alpha}^{(i)}$ are dual to $X_{\alpha}$, $Y_{\alpha}$
and $\lambda_{\alpha}^{(i)}=g(E_{\alpha},E_{-\alpha})$. Then $g$ is CYT iff
$\sum_{\alpha\in P}\big{(}{\rm d}\alpha-\frac{1}{\lambda_{\alpha}}{\rm
d}(\alpha\circ\Lambda)\big{)}=0$, where $\Lambda$ is the positive operator
such that $g|_{\mathfrak{t}}(\cdot,\cdot)=-B(\Lambda(\cdot),\cdot)$.

##  5 CYT and SKT invariant metrics

As a consequence of Corollary 3.4 and Theorem 4.2, we can get the explicit
equations to determine all left-invariant and right $T$-invariant CYT (or SKT)
metrics on $(G,I).$ In particular, we can prove that if $g$ is both SKT and
CYT, then $g$ has to be Bismut flat.

######  Corollary 5.1.

Let $G$ be a compact semi-simple Lie group with Samelson’s complex structure
$I$ determined by the set $P$ of positive roots associated to a maximal torus
$T$ of $G$. Let $g$ be a left-invariant $I$-Hermitian metric on $G$, which is
also invariant under the right action of the torus $T$ in the Tits fibration
$G\rightarrow G/T$. Let $\alpha_{1},\dots,\alpha_{r}$ be the choice of simple
roots and $p$ be the cardinality of $P$. Then

  1. $(i)$

$g$ is CYT if and only if for every $\alpha\in P$ there exist $p$ positive
real numbers $\lambda_{\alpha}>0$, $\alpha\in P$, satisfying the $r$ equations

$\sum_{\alpha\in
P}\Bigg{(}x_{\alpha}^{i}-\sum_{j=1}^{r}\frac{x_{\alpha}^{j}}{\lambda_{\alpha}}\Lambda^{i}_{j}\Bigg{)}=0,\qquad
i=1,\dots,r,$ (5.1)

where $x_{\alpha}^{j}$ denotes the coefficient of $\alpha$ with respect to the
simple root $\alpha_{j}$ and $\Lambda_{i}^{j}$ are the entries of $\Lambda$ in
the basis $(\alpha_{1},\dots,\alpha_{r})$.

  2. $(ii)$

$g$ is SKT if and only if there exist $r$ real numbers $\tilde{N}_{j}$ and $p$
positive real numbers $\tilde{\lambda}_{\alpha}>0$, $\alpha\in P$, satisfying
the $p$ equations

$\tilde{\lambda}_{\alpha}-1-\sum_{j=1}^{r}\tilde{N}_{j}B(H_{\alpha_{j}},H_{\alpha})=0,\qquad\alpha\in
P.$ (5.2)

  3. $(iii)$

If $g$ is both SKT and CYT, then $g$ is Bismut flat.

###### Proof.

From Theorem 4.2, it follows that $g$ is CYT if and only if there exist $p$
positive real number $\lambda_{\alpha}$, $\alpha\in P,$ such that

$\sum_{\alpha\in P} \bigg{(}{\rm d}\alpha-\frac{1}{\lambda_{\alpha}}{\rm
d}(\alpha\circ\Lambda)\bigg{)}=0.$

Therefore, since $\alpha=\sum_{i=1}^{r}x_{\alpha}^{i}\alpha_{i}$, with
$x_{\alpha}^{i}$ positive integers and $\Lambda_{i}^{j}$ the entries of
$\Lambda$, we have that the CYT condition is equivalent to

$\sum_{\alpha\in
P}\Bigg{(}x_{\alpha}^{i}-\sum_{j=1}^{r}\frac{x_{\alpha}^{j}}{\lambda_{\alpha}}\Lambda^{i}_{j}\Bigg{)}{\rm
d}\alpha_{i}=0.$

Since the forms ${\rm d}\alpha_{i}$, $i=1,\dots,r$, are linearly independent,
we get the non-linear system (5.1) of $r$ equations in the $p$ variables
$\lambda_{\alpha}$, $\alpha\in P$, and (i) follows.

Let denote by $P_{i}$ the set of positive roots of the simple factor $G_{i}$
with respect to $\mathfrak{t}_{i}$. To prove the second part of the corollary
we use that by Corollary 3.4 the SKT condition is equivalent to the existence
of $r$ real numbers $N_{i}$, $i=1,\dots,r$ and $p$ positive numbers
$\lambda_{\alpha}$, $\alpha\in P$, such that

$\sum_{i=1}^{m}\sum_{\alpha\in
P_{i}}\big{(}\lambda_{\alpha}^{(i)}-\nu_{i}\big{)}e_{\alpha}^{(i)}\wedge
Ie_{\alpha}^{(i)}=\sum_{i=1}^{r}N_{i}\,{\rm d}\alpha_{i}$

and $\Lambda=\sum_{i=1}^{m}\nu_{i}{\rm Id}|_{\mathfrak{t}_{i}}$. Since

${\rm d}\alpha_{j}=\sum_{i=1}^{m}\sum_{\alpha\in
P_{i}}B\big{(}H_{\alpha_{j}},H_{\alpha}^{(i)}\big{)}e_{\alpha}^{(i)}\wedge
Ie_{\alpha}^{(i)},\qquad j=1,\dots,r,$

we obtain

$\sum_{i=1}^{m}\sum_{\alpha\in
P_{i}}\big{(}\lambda_{\alpha}^{(i)}-\nu_{i}\big{)}e_{\alpha}^{(i)}\wedge
Ie_{\alpha}^{(i)}=\sum_{i=1}^{m}\sum_{j=1}^{r}N_{j}\sum_{\alpha\in
P_{i}}B\big{(}H_{\alpha_{j}},H_{\alpha}^{(i)}\big{)}e_{\alpha}^{(i)}\wedge
Ie_{\alpha}^{(i)}.$

Therefore, using that the previous equality holds on every simple factor
$\mathfrak{g}_{i}$ and rescaling $\lambda_{\alpha}^{(i)}$ and $N_{i}$ by
$\nu_{i}$ we get the linear system (5.2) of $p$ equations in the $p+r$
variables $\tilde{N}_{i}$, $i=1,\dots,r,$ and $\tilde{\lambda}_{\alpha}$,
$\alpha\in P.$

Now to prove the last part of the theorem suppose that $g$ is both CYT and
SKT. Then using that $\Lambda=\sum_{i=1}^{m}\nu_{i}{\rm
Id}|_{\mathfrak{t}_{i}}$ and rescaling $\lambda_{\alpha}$ in (5.1) in the same
way as for the SKT case we have that the equations for the CYT condition
become

$\sum_{\alpha\in
P}x_{\alpha}^{i}\Bigg{(}1-\sum_{j=1}^{r}\frac{1}{\tilde{\lambda}_{\alpha}}\Bigg{)}=0,\qquad
i=1,\dots,r.$

Let

$P=\\{\alpha_{1},\dots,\alpha_{r},\alpha_{r+1},\dots,\alpha_{p}\\},$

where $\alpha_{1},\dots,\alpha_{r}$ are the simple roots. Imposing the SKT and
CYT condition we get the following non-linear system of $p+r$ equations in the
variables $\tilde{\lambda}_{j}$, $j=1,\dots,p$, $\tilde{N}_{i}$,
$i=1,\dots,r$,

$\begin{cases}\displaystyle\tilde{\lambda}_{j}-1-\sum_{i=1}^{r}\tilde{N}_{i}B(H_{\alpha_{i}},H_{\alpha_{j}})=0,\qquad
j=1,\dots,p,\\\
\displaystyle\sum_{j=1}^{p}x_{\alpha_{j}}^{i}\bigg{(}1-\frac{1}{\tilde{\lambda}_{j}}\bigg{)}=0,\qquad
i=1,\dots,r.\end{cases}$ (5.3)

Note that the solution $\tilde{N}_{i}=0$, $i=1,\dots,r$,
$\tilde{\lambda}_{j}=1$, $j=1,\dots,p$, corresponds to a Bismut flat metric.
Since $\tilde{\lambda}_{j}\neq 0$, for every $j=1,\dots,p$, if we multiply

$\sum_{j=1}^{p}x_{\alpha_{j}}^{i}\bigg{(}1-\frac{1}{\tilde{\lambda}_{j}}\bigg{)}=0,\qquad
i=1,\dots,r,$

by the product of all the $\tilde{\lambda}_{j}$,we get

$\sum_{j=1}^{p}x_{\alpha_{j}}^{i}\tilde{\lambda}_{1}\cdots\hat{\tilde{\lambda_{j}}}\cdots\tilde{\lambda}_{p}\big{(}\tilde{\lambda}_{j}-1\big{)}=0,\qquad
i=1,\dots,r,$

where $\hat{\tilde{\lambda_{j}}}$ denotes that the element
$\tilde{\lambda}_{j}$ is removed in the product. We know that

$\tilde{\lambda}_{j}-1=\sum_{k=1}^{r}\tilde{N}_{k}B(H_{\alpha_{k}},H_{\alpha_{j}})$

for every $j=1,\dots,p$. Therefore

$\sum_{j=1}^{p}x_{\alpha_{j}}^{i}\tilde{\lambda}_{1}\cdots\hat{\tilde{\lambda_{j}}}\cdots\tilde{\lambda}_{p}\Bigg{(}\sum_{k=1}^{r}\tilde{N}_{k}B(H_{\alpha_{k}},H_{\alpha_{j}})\Bigg{)}=0,\qquad
i=1,\dots,r,$

which can be viewed as a homogeneous system of $r$ equations in the $r$
variables $\tilde{N}_{i}$, $i=1,\dots,r$. The $r\times r$ associated matrix
$M=M\big{(}\tilde{\lambda}_{1},\dots,\tilde{\lambda}_{p}\big{)}$ has
$(i,k)$-th entry given by

$\sum_{j=1}^{p}x_{\alpha_{j}}^{i}\tilde{\lambda}_{1}\cdots\hat{\tilde{\lambda_{j}}}\cdots\tilde{\lambda}_{p}B(H_{\alpha_{k}},H_{\alpha_{j}}).$

We can rewrite $B(H_{\alpha_{k}},H_{\alpha_{j}})$, $k=1,\dots,r$,
$j=1,\dots,p,$ in terms of $B(H_{\alpha_{k}},H_{\alpha_{j}})$ with
$k,j\in\\{1,\dots,r\\}$, using that $B(H,H_{\alpha})=\alpha(H)$ for every
$H\in\mathfrak{t}$. Indeed, if $\alpha=\sum_{l=1}^{r}x_{\alpha}^{l}\alpha_{l}$
we have,

$B(H_{\alpha_{i}},H_{\alpha})=\alpha(H_{\alpha_{i}})=\sum_{l=1}^{r}x_{\alpha}^{l}\alpha_{l}(H_{\alpha_{i}})=\sum_{l=1}^{r}x_{\alpha}^{l}B(H_{\alpha_{i}},H_{\alpha_{l}}),$

for every $i=1,\dots,r$. As consequence the $(i,k)$-th entry of the matrix
$M=M\big{(}\tilde{\lambda}_{1},\dots,\tilde{\lambda}_{p}\big{)}$ can be
written as

$\displaystyle\sum_{j=1}^{p}x_{\alpha_{j}}^{i}\tilde{\lambda}_{1}\cdots\hat{\tilde{\lambda_{j}}}\cdots\tilde{\lambda}_{p}B(H_{\alpha_{k}},H_{\alpha_{j}})=\sum_{j=1}^{p}x_{\alpha_{j}}^{i}\tilde{\lambda}_{1}\cdots\hat{\tilde{\lambda_{j}}}\cdots\tilde{\lambda}_{p}B(H_{\alpha_{j}},H_{\alpha_{k}})$
$\displaystyle\hphantom{\sum_{j=1}^{p}x_{\alpha_{j}}^{i}\tilde{\lambda}_{1}\cdots\hat{\tilde{\lambda_{j}}}\cdots\tilde{\lambda}_{p}B(H_{\alpha_{k}},H_{\alpha_{j}})}{}=\sum_{j=1}^{p}x_{\alpha_{j}}^{i}\tilde{\lambda}_{1}\cdots\hat{\tilde{\lambda_{j}}}\cdots\tilde{\lambda}_{p}\sum_{l=1}^{r}x_{\alpha_{j}}^{l}B(H_{\alpha_{l}},H_{\alpha_{k}})$
$\displaystyle\hphantom{\sum_{j=1}^{p}x_{\alpha_{j}}^{i}\tilde{\lambda}_{1}\cdots\hat{\tilde{\lambda_{j}}}\cdots\tilde{\lambda}_{p}B(H_{\alpha_{k}},H_{\alpha_{j}})}{}=B\Bigg{(}\sum_{l=1}^{r}\sum_{j=1}^{p}x_{\alpha_{j}}^{i}\tilde{\lambda}_{1}\cdots\hat{\tilde{\lambda_{j}}}\cdots\tilde{\lambda}_{p}x_{\alpha_{j}}^{l}H_{\alpha_{l}},H_{\alpha_{k}}\Bigg{)},$

where $i,k=1,\dots,r$. Since

$\sum_{j=1}^{p}x_{\alpha_{j}}^{i}\tilde{\lambda}_{1}\cdots\hat{\tilde{\lambda_{j}}}\cdots\tilde{\lambda}_{p}x_{\alpha_{j}}^{l}=\tilde{\lambda}_{1}\cdots\tilde{\lambda}_{p}\sum_{j=1}^{p}x_{\alpha_{j}}^{i}\frac{1}{\tilde{\lambda}_{j}}x_{\alpha_{j}}^{l},$

it will be sufficient to show, for every $\tilde{\lambda}_{j}$, the
invertibility of the matrix $L$, whose $(i,l)$-entry is given

$\sum_{j=1}^{p}x_{\alpha_{j}}^{i}\tilde{\lambda}_{1}\cdots\hat{\tilde{\lambda_{j}}}\cdots\tilde{\lambda}_{p}x_{\alpha_{j}}^{l}=\tilde{\lambda}_{1}\cdots\tilde{\lambda}_{p}\sum_{j=1}^{p}x_{\alpha_{j}}^{i}\frac{1}{\tilde{\lambda}_{j}}x_{\alpha_{j}}^{l}.$

Since the $r\times p$ matrix $(x_{\alpha}^{i})$ has rank $r$, the $r$ vectors

$v_{1}=\big{(}x^{1}_{\alpha_{1}},\dots,x^{1}_{\alpha_{p}}\big{)},\quad\dots,\quad
v_{r}=\big{(}x^{r}_{\alpha_{1}},\dots,x^{r}_{\alpha_{p}}\big{)},$

are linearly independent. Using that for every $j=1,\dots,p,$
$\tilde{\lambda}_{j}>0$, we get that
$\sum_{j=1}^{p}\frac{1}{\tilde{\lambda}_{j}}x_{\alpha_{j}}^{i}x_{\alpha_{j}}^{l}$
can be viewed as a positive definite scalar product on $\mathbb{R}^{p}$ with
weights $\frac{1}{\tilde{\lambda}_{j}}$ of the two vectors $v_{i}$ and
$v_{l}$. Therefore, the matrix $L$ is invertible and $\tilde{N}_{i}=0$, for
every $i=1,\dots,r$. By (5.3), it follows that $\tilde{\lambda}_{j}=1$, for
every $j=1,\dots,p$. ∎

### Acknowledgements

Anna Fino is partially supported by Project PRIN 2017 “Real and complex
manifolds: Topology, Geometry and Holomorphic Dynamics”, by GNSAGA (Indam) and
by a grant from the Simons Foundation (#944448). Gueo Grantcharov is partially
supported by a grant from the Simons Foundation (#853269). We would like to
thank the anonymous referees for the helpful comments and suggestions as well
as pointing out a confusing statement in the earlier version of the proof of
Theorem 3.2.

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