#  Projective Hessian and Sasakian manifolds

Pavel Osipov

###### Abstract

The Hessian geometry is the real analogue of the Kähler one. Sasakian geometry
is an odd-dimensional counterpart to the Kähler geometry. In the paper, we
study the connection between projective Hessian and Sasakian manifolds
analogous to the one between Hessian and Kähler manifolds. In particular, we
construct a Sasakian structure on $TM\times\mathbb{R}$ from a projective
Hessian structure on $M$. We are interested in the case of invariant structure
on Lie groups. We define semi-Sasakian Lie groups as a generalization of
Sasakian Lie groups. Then we construct a semi-Sasakian structure on a group
$G\ltimes\mathbb{R}^{n+1}$ for a projective Hessian Lie group $G$. We describe
examples of homogeneous Hessian Lie groups and corresponding semi-Sasakian Lie
groups. The big class of projective Hessian Lie groups can be constructed by
homogeneous regular domains in $\mathbb{R}^{n}$. The groups $\text{SO}(2)$ and
$\text{SU}(2)$ belong to another kind of examples. Using them, we construct
semi-Sasakian structures on the group of the Euclidean motions of the real
plane and the group of isometries of the complex plane.

##  1 Introduction

A flat affine manifold is a differentiable manifold equipped with a flat,
torsion-free connection. Equivalently, it is a manifold equipped with an atlas
such that all translation maps between charts are affine transformations (see
[FGH] or [Sh]). A Hessian manifold is an affine manifold with a Riemannian
metric which is locally equivalent to a Hessian of a function. Any Kähler
metric can be defined as a complex Hessian of a plurisubharmonic function.
Thus, the Hessian geometry is a real analogue of the Kähler one.

A Kähler structure $(I,g^{T})$ on $TM$ can be constructed by a Hessian
structure $(\nabla,g)$ on $M$ (see [Sh]). The correspondence

$\text{r}:\\{\text{Hessian manifolds}\\}\to\\{\text{K\"{a}hler manifolds}\\}$
$\ \ \ \ \ (M,\nabla,g)\ \ \to\ \ (TM,I,g^{T})$

is called the r-map. In particular, this map associates special Kähler
manifolds to special real manifolds (see [AC]). In this case, r-map describes
a correspondence between the scalar geometries for supersymmetric theories in
dimension 5 and 4. See [CMMS] for details on the r-map and supersymmetry.

Hessian manifolds have many different application: in supersymmetry ([CMMS],
[CM], [AC]), in convex programming ([N], [NN]), in the Monge-Ampère Equation
([F1], [F2], [Gu]), in the WDVV equations ([T]).

A Riemannian cone is a Riemannian manifold
$(M\times\mathbb{R}^{>0},t^{2}g_{M}+dt^{2})$, where $t$ is a coordinate on
$\mathbb{R}^{>0}$ and $g_{M}$ is a Riemannian metric on $M$. Riemannian cones
have important applications in supegravity ([ACDM], [ACM], [CDM], [VDMV]).
Geometry and holonomy of pseudo-Riemannian cones are studied in [ACGL] and
[ACL].

The contact geometry is an odd-dimensional counterpart to the symplectic
geometry. A manifold $M$ is contact if and only if there exists a symplectic
form $\omega$ on the cone $M\times\mathbb{R}^{>0}$ satisfying
$\lambda_{q}\omega=q^{2}\omega$, where $\lambda_{q}(m\times t)=m\times qt$.
Moreover, if there exists an $\mathbb{R}^{>0}$-invariant complex structure $I$
on $M\times\mathbb{R}^{>0}$ such that $g=\omega(I\cdot,\cdot)$ is positive
defined, that is, $(M\times\mathbb{R}^{>0},I,\omega)$ is a Kähler manifold
then the manifold $M$ is called Sasakian. Any Sasakian manifold is a
Riemannian cone (see [OV]). Note that our definitions of contact and Sasakian
manifolds are not standard but equivalent. See [AFV] or [BG] for standard
definitions and [ACHK] or [OV] for equivalence of them.

A Hessian cone is a Hessian manifold
$(M\times\mathbb{R}^{>0},\nabla,g=t^{2}g_{M}+dt^{2})$ where $g_{M}$ is a
metric on $M$, $t$ a coordianate on $\mathbb{R}^{>0}$, and $\nabla$ a flat
torsion free connection satisfying

$\nabla\left(t\frac{\partial}{\partial t}\right)=\text{Id}.$

We say that a Riemannian manifold $(M,g)$ is a projective Hessian manifold if
there exists a connection $\nabla$ on $M\times\mathbb{R}^{>0}$ such that
$(M\times\mathbb{R}^{>0},\nabla,g=t^{2}g_{M}+dt^{2})$ is a Hessian cone.

We study a relation between projective Hessian and Sasakian manifolds
analogous to the one between Hessian and Kähler manifolds.

######  Theorem 1.1.

Let $(M,g)$ be a projective Hessian manifold. Then $TM\times\mathbb{R}$ admits
a structure of a Sasakian manifold.

This theorem is closely related to the r-map. Namely, we have a diagram

$\begin{CD}M\times\mathbb{R}^{>0}@>{\text{r}}>{}>T(M\times\mathbb{R}^{>0})\\\
@A{}A{con}A@A{}A{con}A\\\ M@>{}>{}>TM\times\mathbb{R}\end{CD},$

where vertical arrows associate Riemannian cones to the corresponding
Riemannian manifolds. The theorem implies that the Riemannian manifold
$T(M\times\mathbb{R}^{>0})$ with the metric constructed by r-map is actually a
cone over $TM\times\mathbb{R}$.

Then we work with Lie algebras and groups equipped with invariant structures
on them. There are different descriptions of an invariant affine structure on
a Lie algebra $\mathfrak{g}$: a torsion-free flat connection on
$\mathfrak{g}$, an étale affine representation
$\mathfrak{g}\to\mathfrak{aff}(\mathbb{R}^{n})$, where $n$ is dimension of
$\mathfrak{g}$, or a structure of a left symmetric algebra on $\mathfrak{g}$,
that is, a multiplication on $\mathfrak{g}$ satisfying

$XY-YX=[X,Y]$

and

$X(YZ)-(XY)Z=Z(XY)-(ZX)Y$

for any $X,Y,Z\in\mathfrak{g}$ (see [Bu1] or [Bu2]).

Then we adapt r-map and Theorem 1.1. to the case of Lie algebras and groups.

######  Theorem 1.2 ([BD]).

Let $\mathfrak{g}$ be an $n$-dimensional Lie algebra endowed with an affine
structure $\nabla$ and $\eta$ the corresponding étale affine representation.
Then there exists an invariant complex structure on
$\mathfrak{g}\ltimes_{\eta}\mathbb{R}^{n}$.

It is immediately follows from the Theorem 1.2 that if a Lie group $G$ admits
a left invariant affine structure then there exists a left invariant complex
structure on a semidirect product $G\ltimes_{\theta}\mathbb{R}^{n}$. We show
that $\theta$ is a linear part of the corresponding affine action of $G$ and
get the following addition to results from [BD].

######  Theorem 1.3.

Let $G$ be a simply connected Lie group equipped with a left invariant affine
structure, $\theta$ the linear part of the corresponding affine action of $G$.
Then there exists a left invariant integrable complex structure $I$ on the
group

$G\ltimes_{\theta}\mathbb{R}^{n}\simeq TG$

such that $I$ swaps vertical and horizontal tangent subbundles.

Note that a Kähler structure on a the group
$G\ltimes_{\theta^{*}}\left(\mathbb{R}^{n}\right)^{*}$ is constructed by an
invariant Hessian structure on $G$ in [Me]. The corresponding complex
structure on $G\ltimes_{\theta^{*}}\left(\mathbb{R}^{n}\right)^{*}$ depends on
the Hessian metric on $G$. In our case, the complex structure on
$G\ltimes_{\theta}\mathbb{R}^{n}$ depends only on the affine structure on $G$.

A Hessian Lie group $(G,\nabla,g)$ is a Lie group $G$ endowed with a left
invariant affine structure $\nabla$ and a left invariant Hessian metric $g$.

######  Theorem 1.4.

Let $G$ be an $n$-dimensional simply connected Lie group equipped with a left
invariant affine structure $\nabla$ and $\theta$ the linear part of the
corresponding affine action of $G$. Then there exists a left invariant Kähler
metric on $G\ltimes_{\theta}\mathbb{R}^{n}=TG.$

The construction from Theorem 1.4. is a homogeneous analogue of r-map.

For adapting Theorem 1.1 to the case of Lie algebras we introduce definitions
of semi-sasakian Lie algebras and groups.

Let $\mathfrak{g}$ be a $2n+1$-dimensional Lie algebra, $D$ a derivation of
$\mathfrak{g}$, and $\omega$ 2-form on $\mathfrak{g}$ and $\eta$ a 1-form on
$\mathfrak{g}$. If

$d\omega=0,\ \ \ \omega+D^{*}\omega-d\eta=0\ \ \ \text{and}\ \ \
\eta\wedge\omega^{n}\neq 0$

then we say that a collection $(\mathfrak{g},D,\omega,\eta)$ is a semi-contact
Lie algebra. Note that if $D=0$ then we have the pair $(\mathfrak{g},\eta)$
such that $\eta(d\eta)^{n}\neq 0$ i.e. $(\mathfrak{g},\eta)$ is a contact Lie
algebra.

The definition of semi-contact Lie algebras is closely related to the
definition of lcs Lie algebras. A locally conformally symplectic (shortly,
lcs) algebra is a $2n$-dimensional Lie algebra $\mathfrak{g}$ endowed with a
2-form $\omega$ and 1-form $\vartheta$ such that

$\Omega^{n}\neq 0,\ \ \ d\vartheta=0,\ \ \text{and}\ \ \
d\Omega=\vartheta\omega\ \ \ \text{(See \cite[cite]{[\@@bibref{}{BD}{}{}]} or
\cite[cite]{[\@@bibref{}{ABP}{}{}]})}.$

According to results of [ABP], a collection $(\mathfrak{g},D,\omega,\eta)$ is
a semi-contact Lie algebra if and if
$(\mathbb{R}\ltimes_{D}\mathfrak{g},\omega+\vartheta\wedge\eta)$ is an lcs Lie
algebra, where $\vartheta(r\ltimes_{D}v)=r$, and any lcs Lie algebra arises by
this way. Thus, there is a one-to-one correspondence between lcs and
semicontact Lie algebras. If $(\mathfrak{g},\Omega,\vartheta)$ is an lcs Lie
algebra then we say that $(G,\Omega,\vartheta)$ is an lcs Lie group.

A semi-contact Lie group is a triple $(G,\theta,\Omega)$, where $G$ is a Lie
algebra $\theta$ be an action of $\mathbb{R}^{>0}$ on $G$ and $\omega$ a
symplectic form on $\mathbb{R}^{>0}\ltimes_{\theta}G$ such that $\omega$ is
invariant with respect to the left action of $G$ and satisfying
$\lambda_{q}^{*}\omega=q^{2}\omega$, where $\lambda_{q}x=(q\ltimes 1)(x)$. If
$\theta=id$ then $G,\Omega$ is a contact Lie group. See [D] for more
information on contact Lie algebras and groups.

######  Theorem 1.5.

Let $\mathfrak{g}$ be a Lie algebra of left invariant vector fields on a Lie
group $G$, $D$ a derivation of $\mathfrak{g}$, $\theta=\exp D$ the
corresponding automorphism of $G$, $\omega$ and $\eta$ left invariant $2$-form
and $1$-form correspondingly. Then the following conditions are equivalent:

  * (i)

$(\mathfrak{g},D,\omega,\eta)$ is a semicontact Lie algebra.

  * (ii)

$\left(\mathbb{R}^{>0}\ltimes_{\theta}G,\Omega=\omega+t^{-1}\eta\wedge
dt,2t^{-1}dt\right)$ is an lcs Lie group.

  * (iii)

$(G,\theta,\hat{\Omega}=t^{2}\omega+tdt\wedge\eta)$ is a semi-contact Lie
group.

A locally conformally Kähler (shortly, lck) Lie algebra is an lcs
$(\mathfrak{h},\Omega,\vartheta)$ Lie algebra endowed with a complex structure
$I$ such that

$g:=\Omega(*,I*)$

is a positive definite symmetric bilinear form (see [HK]). The Lie group
corresponding to an lck Lie algeba is called an lck Lie group. Note that
homogeneous lck manifolds of reductive Lie groups are classified in [ACHK].

A semi-Sasakian Lie algebra is a semi-contact Lie algebra
$(\mathfrak{g},\eta,\omega)$ with a left-invariant integrable almost complex
structure $I$ on $(\mathfrak{g}\ltimes_{D}\mathbb{R})$ such that

$g=\hat{\Omega}(*,I*)$

is a positive definite symmetric bilinear form, where
$\hat{\Omega}=\omega+\vartheta\wedge\eta$. As above, there is a one-to one
correspondence between semi-Sasakian algebras $(\mathfrak{g},D,\omega,\eta)$
and lck algebras
$(\mathbb{R}\ltimes_{D}\mathfrak{g},\omega+\vartheta\wedge\eta)$.

A semi-Sasakian Lie group is a semi-contact Lie group
$(G,\theta,\hat{\Omega})$ with a left invariant complex structure $I$ on
$\mathbb{R}^{>0}\ltimes_{\theta}G$ such that
$(\mathbb{R}^{>0}\ltimes_{\theta}G,\hat{\Omega},I)$ is a Kähler manifold. If
$\theta=id$ then $G$ is called a Sasakian Lie group. Note that in this case
$g=t^{2}g_{G}+dt^{2}$ according to Proposition 3.5.

######  Corollary 1.6.

Let $\mathfrak{g}$ be a Lie algebra of the invariant vector fields on a Lie
group $G$, $D$ the derivation of $\mathfrak{g}$, $\theta=\exp D$ the
corresponding automorphism of $G$, $I$ a complex structure on
$\mathbb{R}\ltimes\mathfrak{g}$, $\omega$ and $\eta$ left invariant $2$-form
and $1$-form correspondingly. Then the following conditions are equivalent:

  * (i)

$(\mathfrak{g},D,\omega,\eta,I)$ is a semi-Sasakian Lie algebra.

  * (ii)

$\left(\mathbb{R}^{>0}\ltimes_{\theta}G,\Omega=\omega+t^{-1}\alpha\wedge
dt,2t^{-1}dt,I\right)$ is an lck Lie group.

  * (iii)

$(G,\theta,\hat{\Omega}=t^{2}\omega+tdt\wedge\eta,I)$ is a semi-Sasakian Lie
group.

A Lie algebra $\mathfrak{g}$ is called projective if there is an invariant
affine structure $\nabla$ on $\mathfrak{g}\times\mathbb{R}$ such that

$\nabla_{X}E=\nabla_{E}X=X,$

where $X\in\mathfrak{g}$ and $E\in\mathbb{R}$. A Lie group is called
projective if the corresponding Lie algebra is projective. Note that if $G$ is
a projective Lie group then there is an invariant affine structure on
$G\times\mathbb{R}^{>0}$. A projective Hessian Lie group $(G,g_{G})$ is a
projective Lie group $G$ endowed with a left-invariant Riemannian metric
$g_{G}$ such that $(G\times\mathbb{R}^{>0},\nabla,g=t^{2}g_{G}+dt^{2})$ is a
Hessian cone, where $t$ is a coordinate on $\mathbb{R}^{>0}$ and $\nabla$ the
affine connection on $G\times\mathbb{R}^{>0}$ corresponding to the projective
structure on $G$.

######  Theorem 1.7.

Let $G$ be an $n$-dimensional simply connected projective Hessian Lie group
and $\theta$ the linear part of the corresponding affine representation of
$G\times\mathbb{R}^{>0}$. Then there exists a structure of a semi-Sasakian Lie
group on $G\ltimes_{\theta}\mathbb{R}^{n+1}$. Moreover,
$G\ltimes_{\theta}\mathbb{R}^{n+1}\simeq TG\times\mathbb{R}$.

Note that Theorem 1.7 is a homogeneous analogue of Theorem 1.1.

In section 6, we explain why Theorem 1.4 and Theorem 1.5 generalize the known
construction of homogeneous Kähler domains by a homogeneous convex regular
domains. According to [V], an invariant Hessian metric can be constructed on
any homogeneous convex regular domain. Moreover, a homogeneous Kähler domain
can be associated with a homogeneous convex regular cone (see [Sh]). We can
get the same construction applying Theorem 1.4 to a Lie group acting simply
transitively on a homogeneous regular convex domain. Such group always exists
and it is called a Lie group associated with a homogeneous regular domain. Any
Lie group associated with a homogeneous regular domain is both a Hessian Lie
group and a projective Lie group.

The groups $\text{U}(1)=\text{SO}(2)$ and $\text{SU}(2)$ belong to another
kind of projective Hessian Lie groups. Using them, we can construct a semi-
Sasakian structure on the Euclidean group $\text{E}(2)$ and the group of
isometries of the complex plane $\mathbb{C}^{2}$. Any Sasakian group is
obviously semi-Sasakian. The Sasakian groups of dimension $n\leq 5$ are
classified in [AFV]. Using this classification, we verify that the semi-
Sasakian group $\text{E}(2)$ does not admit a Sasakian structure.

Any Lie group associated with a homogeneous regular domain admits both
structures: Hessian and projective Hessian. The group $\text{SO}(2)$ admits
both structures too. However, not any projective Hessian group is Hessian. The
group $\text{SU}(2)$ is not Hessian just because the sphere $S^{3}$ does not
admit any flat affine structure. However, the group $\text{SU}(2)$ admits an
invariant projective structure, since there is an invariant affine structure
on $\text{SU}(2)\times\mathbb{R}^{>0}$. Thus, the natural question arises:
does the existence of $G$-invariant selfsimilar structure on
$G\times\mathbb{R}^{>0}$ implies the existence of
$G\times\mathbb{R}^{>0}$-invariant Hessian structure on
$G\times\mathbb{R}^{>0}$? The answer is positive when $G$ is Lie group
associated with a homogeneous regular domain or $\text{U}(1)$. We show that
$\text{SU}(2)\times\mathbb{R}^{>0}$ does not admit an invariant Hessian
structure.

##  2 Hessian and Kähler structures

######  Definition 2.1.

A flat affine manifold is a differentiable manifold equipped with a flat,
torsion-free connection. Equivalently, it is a manifold equipped with an atlas
such that all translation maps between charts are affine transformations (see
[FGH]).

######  Definition 2.2.

A Riemannianian metric $g$ on a flat affine manifold $(M,\nabla)$ is called to
be a Hessian metric if $g$ is locally expressed by a Hessian of a function

$g=\text{Hess}\varphi=\nabla d\varphi=\frac{\partial^{2}}{\partial
x^{i}\partial x^{j}}dx^{i}dx^{j},$

where $x^{1},\ldots,x^{n}$ are flat local coordinates. A Hessian manifold
$(M,\nabla,g))$ is a flat affine manifold $(M,\nabla)$ endowed with a Hessian
metric $g$. (see [Sh]).

Let $U$ be an open chart on a flat affine manifold $M$, functions
$x^{1},\ldots,x^{n}$ be affine coordinates on $U$, and
$x^{1},\ldots,x^{n},y^{1},\ldots,y^{n}$ be the corresponding coordinates on
$TU$. Define the complex structure $I$ by $I(\frac{\partial}{\partial
x^{i}})=\frac{\partial}{\partial y^{i}}$. Corresponding complex coordinates
are given by $z^{i}=x^{i}+\sqrt{-1}y^{i}$. The complex structure $I$ does not
depend on a choice of flat coordinates on $U$. Thus, in this way, we get a
complex structure on the $TM$.

Let $\pi:TM\to M$ be a natural projection. Consider a Riemannian metric $g$
given locally by

$f_{i,j}dx^{i}dx^{j}.$

Define the Hermitian metric $g^{T}$ on $TM$ by

$\pi^{*}f_{i,j}dz^{i}d\bar{z}^{j}.$

######  Proposition 2.3 ([Sh], [AC]).

Let $M$ be a flat affine manifold, $g$ and $g^{T}$ as above. Then the
following conditions are equivalent:

  * (i)

$g$ is a Hessian metric.

  * (ii)

$g^{T}$ is a Kähler metric.

Moreover, if $g=\text{Hess}\varphi$ locally then $g^{T}$ is equal to a complex
Hessian

$g=\text{Hess}_{\mathbb{C}}(4\pi^{*}\varphi).$

######  Definition 2.4.

The metric $g^{T}$ is called Kähler metric associated with $g$. The
correspondence that associates the Kähler manifold $(TM,g^{T})$ with a Hessian
manifold $(M,g)$ is called r-map (see [AC]).

######  Proposition 2.5.

Let $(M,g)$ be a Hessian manifold, $I$ the corresponding complex structure on
$TM$, and $\pi:TM\to M$ a projection. Then the associated Kähler metric
$g^{T}$ equals

$h(X,Y)=\pi^{*}g(X,Y)+\pi^{*}g(IX,IY)+\sqrt{-1}\pi^{*}g(IX,Y)-\sqrt{-1}\pi^{*}g(X,IY).$
(2.1)

###### Proof.

If

$g=f_{i,j}dx^{i}dx^{j}$ (2.2)

then

$g^{T}=\pi^{*}f_{i,j}dz^{i}d\bar{z}^{j}=\pi^{*}f_{i,j}d(x^{i}+\sqrt{-1}y^{i})d(x^{j}-\sqrt{-1}y^{j})=$
$=\pi^{*}f_{i,j}dx^{i}dx^{j}+\pi^{*}f_{i,j}dy^{i}dy^{j}+\sqrt{-1}\pi^{*}f_{i,j}dy^{i}dx^{j}-\sqrt{-1}\pi^{*}f_{i,j}dx^{i}dy^{j}.$
(2.3)

It is enough to check the identity

$h(X,Y)=g^{T}(X,Y)$

on pairs of basis vectors. For any $i\in\\{1,\ldots,n\\}$ we have

$\pi^{*}g\left(\frac{\partial}{\partial y^{i}},\ldots\right)=0,$

moreover,

$I\frac{\partial}{\partial x^{i}}=\frac{\partial}{\partial y^{i}}.$

Hence, by (2.1),

$h\left(\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial
x^{j}}\right)=\pi^{*}g\left(\frac{\partial}{\partial
x^{i}},\frac{\partial}{\partial x^{j}}\right)$

and, by (2.2),

$\pi^{*}g\left(\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial
x^{j}}\right)=f_{i,j}.$

On the other hand, by (2.3),

$g^{T}\left(\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial
x^{j}}\right)=f_{i,j}.$

Thus, we get

$g^{T}\left(\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial
x^{j}}\right)=h\left(\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial
x^{j}}\right).$

Checking for the pairs $\left(\frac{\partial}{\partial
x^{i}},\frac{\partial}{\partial y^{j}}\right)$ and
$\left(\frac{\partial}{\partial y^{i}},\frac{\partial}{\partial y^{j}}\right)$
is similar. ∎

##  3 Projective Hessian and Sasakian manifolds

######  Definition 3.1.

A radiant manifold $(C,\nabla,\xi)$ is a flat affine manifold $(C,\nabla)$
endowed with a vector field $\xi$ satisfying

$\nabla\xi=\text{Id}.$ (3.1)

Equivalently, it is a manifold equipped with an atlas such that all
translation maps between charts are linear transformations (see e.g. [Go]).

######  Proposition 3.2 ([Go]).

Let $t$ be a coordinate on $\mathbb{R}^{>0}$ and
$(M\times\mathbb{R}^{>0},\nabla,t\frac{\partial}{\partial t})$ a radiant
manifold. Consider a natural action of $\mathbb{R}^{>0}$ on
$M\times\mathbb{R}^{>0}$. Then the connection $\nabla$ is
$\mathbb{R}^{>0}\text{-invariant}$.

######  Definition 3.3.

A Hessian cone is a Hessian manifold
$(M\times\mathbb{R}^{>0},\nabla,g=t^{2}g_{M}+dt^{2})$ such that
$(M\times\mathbb{R}^{>0},\nabla,t\frac{\partial}{\partial t})$ is a radiant
manifold, where $t$ is a coordinate on $\mathbb{R}^{>0}$. We say that a
Riemannian manifold $(M,g_{M})$ is a projective Hessian manifold if there
exists a connection $\nabla$ on $M\times\mathbb{R}^{>0}$ such that
$(M\times\mathbb{R}^{>0},\nabla,g=t^{2}g_{M}+dt^{2})$ is a Hessian cone.

######  Definition 3.4.

A Sasakian manifold is a Riemannian manifold $(M,g_{M})$ such that the cone
metric $t^{2}g_{M}+dt^{2}$ on $M\times\mathbb{R}^{>0}$ is Kähler with respect
to a dilation invariant complex structure.

######  Proposition 3.5 ([OV]).

Let $(M\times\mathbb{R}^{>0},g,I)$ be a Kähler manifold. For any
$q\in\mathbb{R}^{>0}$ consider a map $\lambda_{q}:M\times\mathbb{R}^{>0}\to
M\times\mathbb{R}^{>0}$ defined by $\lambda_{q}(m\times t)=m\times qt.$ If
$\lambda_{q}^{*}g=q^{2}g$ then $g=t^{2}g_{M}+dt^{2}$ and $(M,g_{M})$ is a
Sasakian manifold.

There exists a decomposition

$T(M\times\mathbb{R}^{>0})=TM\times
T\mathbb{R}^{>0}=TM\times\mathbb{R}^{>0}\times\mathbb{R}.$

If $M\times\mathbb{R}^{>0}$ possess a Hessian structure then, by Proposition
2.3, $T(M\times\mathbb{R}^{>0})$ admits a Kähler structure.

######  Proposition 3.6.

Let $(M\times\mathbb{R}^{>0},\nabla,g)$ be a Hessian cone and $g^{T}$ the
associated Kähler metric on $T(M\times\mathbb{R}^{>0})$ . Consider
$T(M\times\mathbb{R}^{>0})=TM\times\mathbb{R}\times\mathbb{R}^{>0}$ as a cone
over $TM\times\mathbb{R}$. Then for any $q\in\mathbb{R}^{>0}$ we have
$\mu_{q}^{*}g=q^{2}g$, where the map
$\mu_{q}:TM\times\mathbb{R}\times\mathbb{R}^{>0}\to
TM\times\mathbb{R}\times\mathbb{R}^{>0}$ is defined by $\mu_{q}(m\times
s\times t)=m\times s\times qt.$

###### Proof.

We have the commutative diagram

$\begin{CD}T(M\times\mathbb{R}^{>0})@>{\mu_{q}}>{}>T(M\times\mathbb{R}^{>0})\\\
@V{}V{\pi}V@V{}V{\pi}V\\\
M\times\mathbb{R}^{>0}@>{\lambda_{q}}>{}>M\times\mathbb{R}^{>0}\end{CD},$

where $\mu_{q}$ and $\lambda_{q}$ are multiplications of the coordinate on
$\mathbb{R}^{>0}$ by $q$. By Proposition 2.5, we have

$g^{T}(X,Y)=\pi^{*}g(X,Y)+\pi^{*}g(IX,IY)+\sqrt{-1}\pi^{*}g(IX,Y)-\sqrt{-1}\pi^{*}g(X,IY).$
(3.2)

Since the diagram is commutative, it follows that

$\mu_{q}^{*}\pi^{*}=\pi^{*}\lambda_{q}^{*}.$ (3.3)

Moreover, $g$ is a cone metric. Hence,

$\lambda_{q}^{*}g=q^{2}g.$ (3.4)

It follows from (3.2), (3.3), and (3.4) that

$\mu_{q}^{*}g^{T}(X,Y)=q^{2}g^{T}(X,Y).$

∎

######  Theorem 3.7.

Let $(M,g_{M})$ be a projective Hessian manifold. Then $TM\times\mathbb{R}$
admits a structure of a Sasakian manifold.

###### Proof.

Define a Kähler structure $(g^{T},I)$ on
$T(M\times\mathbb{R}^{>0})=TM\times\mathbb{R}\times\mathbb{R}^{>0}$ as above.
By proposition 3.2, the connection $\nabla$ is $\mathbb{R}^{>0}$ invariant.
Therefore, the constructed by $\nabla$ complex structure $I$ is
$\mathbb{R}^{>0}$-invariant. Then, by Proposition 3.4 and Proposition 3.5,
$TM\times\mathbb{R}$ admits a structure of a Sasakian manifold. ∎

##  4 Affine representations and flat torsion free connections on Lie groups

The group of affine transformations $\text{Aff}(\mathbb{R}^{n})$ is given by
the matriсes of the form

$\begin{pmatrix}A&a\\\ 0&1\end{pmatrix}\in\text{GL}(\mathbb{R}^{n+1}),$

where $A\in\text{GL}(\mathbb{R}^{n})$, and $a\in\mathbb{R}^{n}$ is a column
vector. The corresponding Lie algebra $\mathfrak{aff}(\mathbb{R}^{n})$ is
given by matrices of the form

$\begin{pmatrix}A&a\\\ 0&0\end{pmatrix}\in\mathfrak{gl}(\mathbb{R}^{n+1}).$

The commutator of $\mathfrak{aff}(\mathbb{R}^{n})$ is equal to

$\left[\begin{pmatrix}A&a\\\ 0&0\end{pmatrix},\begin{pmatrix}B&b\\\
0&0\end{pmatrix}\right]=\begin{pmatrix}[A,B]&A(b)-B(a)\\\ 0&0\end{pmatrix}.$

Algebra $\mathfrak{aff}(\mathbb{R}^{n})$ is the semidirect product
$\mathfrak{gl}(\mathbb{R}^{n})\ltimes\mathbb{R}^{n}$, where the commutator is
given by

$[A\ltimes a,B\ltimes b]=[A,B]\ltimes(Ab-Ba).$

Group $\text{Aff}(\mathbb{R}^{n})$ is the semidirect product
$\text{GL}(\mathbb{R}^{n})\ltimes\mathbb{R}^{n}$, where multiplication is
given by

$(A\ltimes a)(B\ltimes B)=AB\ltimes(a+Ab).$

######  Definition 4.1.

An affine representation is called étale if there exists a point
$x\in\mathbb{R}^{n}$ such that the orbit of $x$ is open and the stabilizer of
$x$ is discrete.

######  Theorem 4.2 ([Bu1] or [Bu2]).

Let $G$ be a Lie group. There is a correspondence between left invariant
torsion-free flat connections and étale affine representations. Moreover, if
the connection is complete then the corresponding étale affine representation
acts simply transitive on $\mathbb{R}^{n}$.

###### Proof.

Choosing a basis, identify $\mathfrak{g}$ with $\mathbb{R}^{n}$. Then consider
$\nabla_{X}$ as a linear endomorphism of $\mathbb{R}^{n}$ for any
$X\in\mathfrak{g}$. The corresponding to $\nabla$ étale representation is
given by

$\eta:\mathfrak{g}\to\mathfrak{aff}(\mathbb{R}^{n}),$

$\eta(X)=\begin{pmatrix}\nabla_{X}&X\\\
0&0\end{pmatrix}\in\mathfrak{aff}(\mathbb{R}^{n})\subset\mathfrak{gl}(\mathbb{R}^{n+1})$
(4.1)

(see [Bu1] or [Bu2] for details). ∎

For any $X\in\mathfrak{g}$ we can consider $\nabla_{X}$ on the Lie algebras
$\mathfrak{g}$ as a linear automorphism of the vector space $\mathfrak{g}$.
Thus, the linear automorphism

$\exp\nabla_{X}=\text{id}+\frac{\nabla_{X}}{1!}+\frac{\nabla_{X}\nabla_{X}}{2!}+\frac{\nabla_{X}\nabla_{X}\nabla_{X}}{3!}+\ldots$

is well defined.

######  Proposition 4.3.

Let $\nabla$ be a left invariant flat torsion-free connection on a simply
connected Lie group $G$ and $\tau$ the corresponding étale affine
representation of $G$. If $X\in\mathfrak{g}$ then the linear part of
$\tau(\exp X)$ is equal to $\exp\nabla_{X}$.

###### Proof.

By (4.1), we have

$\tau(\exp X)=\exp\begin{pmatrix}\nabla_{X}&X\\\
0&0\end{pmatrix}=\begin{pmatrix}\exp\nabla_{X}&(\exp\nabla_{X})(X)\\\
0&1\end{pmatrix}.$

Hence, the linear part of $\tau(\exp X)$ is equal to $\exp\nabla_{X}$. ∎

Fix notations: let $\mathfrak{g}$ be a Lie algebra; $G$ the corresponding
simply connected Lie group; $\eta$ an étale affine representation
$\mathfrak{g}\to\mathfrak{aff}$; $\theta$ the linear part of the corresponding
affine representation of $G$; $i$ an identification
$\mathfrak{g}\to\mathbb{R}^{n}$.

Define an almost complex structure $I$ on
$\mathfrak{g}\ltimes_{\eta}\mathbb{R}^{n}$ by the rule

$I(X\ltimes_{\eta}Y)=-i^{-1}Y\ltimes_{\eta}iX.$

######  Theorem 4.4 ([CO] or [BD]).

Let $\mathfrak{g}$, $\eta$, $I$ be as above. Then the almost complex structure
$I$ is integrable.

######  Definition 4.5.

The algebra $\mathfrak{g}\ltimes_{\eta}\mathbb{R}^{n}$ is called associated
with the connection $\nabla$ and denoted by $\mathfrak{g_{\nabla}}$.

######  Proposition 4.6.

Let $\mathfrak{g},\nabla,\theta,\eta$ be as above and $G_{\nabla}$ the simply
connected Lie group corresponded to $\mathfrak{g}_{\nabla}$. Then

$G_{\nabla}=G\ltimes_{\theta}\mathbb{R}^{n}.$

Moreover, there is an identification $TG=G\ltimes_{\theta}\mathbb{R}^{n}$ such
that fields of the form $0\oplus I\mathfrak{g}\subset T(TG)$ are vertical,
that is, they lie in $\ker d\pi$, where $\pi:TG\to G$ is a projection.

###### Proof.

By the definition, $\mathfrak{g}_{\nabla}$ is isomorphic to
$\mathfrak{g}\ltimes_{\eta}\mathbb{R}^{n}$. The corresponding Lie group is
equal to the semidirect product $G\ltimes\mathbb{R}^{n}$ with respect to the
action such that the element $\exp X$ acts on $\mathbb{R}^{n}$ by
$\exp\nabla_{X}$. By Proposition 4.6, this action equals $\theta$. Thus, the
corresponding Lie group $G_{\nabla}$ is equal to
$G\ltimes_{\theta}\mathbb{R}^{n}$

Using the trivialization of the tangent bundle of $G$ by the flat connection
$\nabla$, we identify $TG$ with $G\times\mathbb{R}^{n}$. Define the
multiplication by

$(g_{1}\times X_{1})(g_{2}\times
X_{2})=(g_{1}g_{2})\times(X_{1}+\theta(g_{1})(X_{2})).$

This multiplication is equal to the multiplication on
$G_{\nabla}=G\ltimes_{\theta}\mathbb{R}^{n}$. Thus, the group $TG$ with this
multiplication is isomorphic to $G_{\nabla}$. Moreover, the left invariant
fields corresponding to the subalgebra $0\oplus I\mathfrak{g}$ are actually
vertical.

∎

######  Theorem 4.7.

Let $G$ be a simply connected Lie group equipped with a left invariant affine
structure, $\mathfrak{g}$ the corresponding Lie algebra, and $\theta$ the
linear part of the corresponding affine action of $G$. Then there exists a
left invariant integrable complex structure $I$ on the group

$G\ltimes_{\theta}\mathbb{R}^{n}\simeq TG$

defined in Proposition 4.6 such that $I$ swaps vertical and horizontal tangent
subbundles.

###### Proof.

The theorem follows from Theorem 4.4 and Proposition 4.6. ∎

##  5 Semi-sasakian Lie algebras and groups

######  Definition 5.1.

A locally conformally symplectic (shortly, lcs) algebra is a $2n$-dimensional
Lie algebra $\mathfrak{g}$ endowed with a 2-form
$\omega\in\Lambda^{2}\mathfrak{g}^{*}$ and 1-form
$\vartheta\in\mathfrak{g}^{*}$ such that

$\Omega^{n}\neq 0,\ \ \ d\vartheta=0,\ \ \text{and}\ \ \
d\Omega=\vartheta\wedge\omega.$

If $(\mathfrak{g},\Omega,\vartheta)$ is a lcs Lie algebra then we say that
$(G,\Omega,\vartheta)$ is lcs Lie group. Here we consider $\Omega$ and
$\vartheta$ as tensors on left invariant vector fields.

######  Proposition 5.2 ([ABP]).

Any lcs Lie algebra $(\mathfrak{h},\Omega,\vartheta)$ takes a form
$(\mathbb{R}\ltimes_{D}\mathfrak{g},\Omega=\omega+\vartheta\wedge\eta)$, where
$\mathfrak{g}$ is a Lie algebra, $D$ a derivation of $\mathfrak{g}$,
$\vartheta$ a 1-form on $\mathbb{R}\ltimes_{D}\mathfrak{g}$ defined by
$\vartheta(r\ltimes_{D}v)=r$, $\omega$ and $\eta$ are $G$-invariant 2-form and
1-form correspondingly satisfying

$d^{\mathfrak{g}}\omega=0,\ \ \ \omega+D^{*}\omega-d^{\mathfrak{g}}\eta=0\ \ \
\text{and}\ \ \ \eta\wedge\omega^{n}\neq 0.$ (5.1)

On the other side, if a collection
$(\mathbb{R}\ltimes_{D}\mathfrak{g},\Omega=\omega+\vartheta\wedge\eta)$
satisfies (5.1) then it defines an lcs algebra.

######  Definition 5.3.

Let $\mathfrak{g}$ be a $2n+1$-dimensional Lie algebra, $D$ a derivation of
$\mathfrak{g}$, $\omega$ a 2-form on $\mathfrak{g}$, and $\eta$ a 1-form on
$\mathfrak{g}$. If

$d\omega=0,\ \ \ \omega+D^{*}\omega-d\eta=0\ \ \ \text{and}\ \ \
\eta\wedge\omega^{n}\neq 0$

then we say that a collection $(\mathfrak{g},D,\omega,\eta)$ is a semi-contact
Lie algebra.

According to Definition 5.3 and Proposition 5.2,
$(\mathfrak{g},D,\omega,\eta)$ is a semi-contact Lie algebra if and if
$(\mathbb{R}\ltimes_{D}\mathfrak{g},\omega+\vartheta\wedge\eta)$ is lcs, where
$\vartheta$ is as above. Thus, there is a one-to-one correspondence between
lcs and Lie semicontact algebras.

######  Definition 5.4.

A semi-contact Lie group is a triple $(G,\theta,\hat{\Omega})$, where $G$ is a
Lie algebra, $\theta$ an action of $\mathbb{R}^{>0}$ on $G$, and
$\hat{\Omega}$ a symplectic form on $\mathbb{R}^{>0}\ltimes_{\theta}G$ such
that $\omega$ is invariant with respect to the left action of $G$ and
satisfying $\lambda_{q}^{*}\hat{\Omega}=q^{2}\hat{\Omega}$, where
$\lambda_{q}x=(q\ltimes 1)(x)$.

######  Proposition 5.5.

Let $G$ be a Lie group, $t$ a coordinate on $\mathbb{R}^{>0}$, $\theta$ an
automorphism of $G$, and $\Omega$ a 2-form on
$\mathbb{R}^{>0}\ltimes_{\theta}G$. Then the collection
$(G,\theta,\hat{\Omega})$ is a semi-contact Lie group if and only if
$(\mathbb{R}^{>0}\ltimes_{\theta}G,{\Omega}=t^{-2}\hat{\Omega},2tdt)$ is an
lcs Lie group.

###### Proof.

We have

$d\Omega=d\left(t^{-2}\hat{\Omega}\right)=2t^{-3}dt\wedge\hat{\Omega}+t^{-2}d\hat{\Omega}=\left(2t^{-1}dt\right)\wedge{\Omega}+t^{-2}d\hat{\Omega}.$

Thus, $d{\Omega}=2tdt\wedge{\Omega}$ if and only if $d\hat{\Omega}=0$.

Moreover $\hat{\Omega}=t^{2}\Omega$ satisfies
$\lambda_{q}^{*}\hat{\Omega}=q^{2}\hat{\Omega}$ if and only if
$\lambda_{q}\Omega={\Omega}$. ∎

######  Corollary 5.6.

Let $(G,\theta,\Omega)$ be a semi-contact Lie group. Then the form $\Omega$ on
$\mathbb{R}^{>0}\ltimes_{\theta}G$ can be written as
$\Omega=t^{2}\omega+tdt\wedge\eta$, where $\omega$ and $\eta$ are
$G$-invariant 2-form and 1-form.

###### Proof.

By Proposition 5.5,
$\left(\mathbb{R}^{>0}\ltimes_{\theta}G,\hat{\Omega}=t^{-2}\Omega\right)$ is
lcs. Let $D$ be a derivation of $\mathfrak{g}$ such that $\exp D=\theta$.
Consider an identification of $\mathbb{R}\ltimes_{D}\mathfrak{g}$ with the
left invariant fields on $\mathbb{R}^{>0}\ltimes_{\theta}G$ such that

$1\ltimes_{D}0=t\frac{\partial}{\partial t}.$

Let $\vartheta$ be as in Proposition 5.2. Then

$\vartheta=t^{-1}dt.$

By Proposition 5.2, we have

$\hat{\Omega}=\omega+t^{-1}dt\wedge\eta.$

Thus, we have

$\Omega=t^{2}\omega+tdt\wedge\eta.$

∎

######  Theorem 5.7.

Let $\mathfrak{g}$ be a Lie algebra of left invariant vector fields on a Lie
group $G$, $D$ a derivation of $\mathfrak{g}$, $\theta=\exp D$ the
corresponding automorphism of $G$, $\omega$ and $\eta$ left invariant $2$-form
and $1$-form correspondingly. Then the following conditions are equivalent:

  * (i)

$(\mathfrak{g},D,\omega,\eta)$ is a semicontact Lie algebra.

  * (ii)

$\left(\mathbb{R}^{>0}\ltimes_{\theta}G,\Omega=\omega+t^{-1}\eta\wedge
dt,2t^{-1}dt\right)$ is an lcs Lie group.

  * (iii)

$(G,\theta,\hat{\Omega}=t^{2}\omega+tdt\wedge\eta)$ is a semi-contact Lie
group.

###### Proof.

$\text{(ii)}\Leftrightarrow\text{(iii)}$ It follows from Proposition 5.5.

$\text{(i)}\Leftrightarrow\text{(ii)}$ Consider an identification of
$\mathbb{R}\ltimes_{D}\mathfrak{g}$ with the left invariant fields on
$\mathbb{R}^{>0}\ltimes_{\theta}G$ such that

$1\ltimes_{D}0=\frac{1}{2}t\frac{\partial}{\partial t}.$

Let $\vartheta$ be as in Proposition 5.2. Then

$\vartheta=2t^{-1}dt.$

By Proposition 5.2, $(\mathfrak{g},D,\omega,\eta)$ is a semicontact Lie
algebra if and only if
$\left(\mathbb{R}\ltimes_{D}\mathfrak{g},\Omega,\vartheta=2t^{-1}dt\right)$ is
an lcs Lie algebra. By the definition, $(G,\theta,\omega,\eta)$ is a semi-
contact Lie group if and only if
$\left(\mathbb{R}\ltimes_{D}\mathfrak{g},\Omega,\vartheta\right)$ is lcs Lie
algebra. ∎

######  Definition 5.8.

A locally conformally Kähler (shortly, lck) Lie algebra is an lcs
$(\mathfrak{h},D,\Omega,\vartheta)$ Lie algebra endowed with a complex
structure $I$ such that

$g:=\Omega(*,I*)$

is a positive definite symmetric bilinear form (see [HK]). The Lie group
corresponding to an lck Lie algebra is called an lck Lie group.

######  Definition 5.9.

A semi-Sasakian Lie algebra is a semi-contact Lie algebra
$(\mathfrak{g},D,\omega,\eta)$ with a left-invariant integrable almost complex
structure $I$ on $(\mathfrak{g}\ltimes_{D}\mathbb{R})$ such that

$g:=\hat{\Omega}(*,I*)$

is a positive definite symmetric bilinear form, where
$\hat{\Omega}=\omega+\vartheta\wedge\eta$ and $\vartheta$ is as in Proposition
5.2.

######  Definition 5.10.

A semi-Sasakian Lie group is a semi-contact Lie group
$(G,\theta,\hat{\Omega})$ with a left invariant complex structure $I$ on
$\mathbb{R}^{>0}\ltimes_{\theta}G$ such that
$(\mathbb{R}^{>0}\ltimes_{\theta}G,\hat{\Omega},I)$ is a Kähler manifold.

Equivalently, a semi-Sasakian Lie is a Lie group $G$ equipped with an
automorphism $\theta$ and a Kähler structure $(g,I)$ on
$\mathbb{R}^{>0}\ltimes_{\theta}G$ such that $I$ is
$\mathbb{R}^{>0}\ltimes_{\theta}G$-invariant, $g$ is $G$-invariant and
satisfies $\lambda_{q}^{*}g=q^{2}g$, where $\lambda_{q}$ is as above.

######  Corollary 5.11.

Let $\mathfrak{g}$ be a Lie algebra of the invariant vector fields on a Lie
group $G$, $D$ the derivation of $\mathfrak{g}$, $\theta=\exp D$ a
corresponding automorphism of $G$, $I$ a complex structure on
$\mathbb{R}\ltimes\mathfrak{g}$, $\omega$ and $\eta$ left invariant $2$-form
and $1$-form correspondingly. Then the following conditions are equivalent:

  * (i)

$(\mathfrak{g},D,\omega,\eta,I)$ is a semi-Sasakian Lie algebra.

  * (ii)

$\left(\mathbb{R}^{>0}\ltimes_{\theta}G,\Omega=\omega+t^{-1}\alpha\wedge
dt,2t^{-1}dt,I\right)$ is an lck Lie group.

  * (iii)

$(G,\theta,\hat{\Omega}=t^{2}\omega+tdt\wedge\eta,I)$ is a semi-Sasakian Lie
group.

######  Remark 5.1.

Let $(G,\theta,g,I)$ be a semi-Sasakian Lie group. Let $\rho$ be a vector
field corresponding to the left action of $\mathbb{R}^{>0}$ on
$\mathbb{R}\ltimes_{\theta}G$. Define
$s:\mathbb{R}^{>0}\ltimes_{\theta}G\to\mathbb{R}^{>0}\ltimes_{\theta}G$ by the
rule

$s(x)=g(\rho,\rho).$

Then $M=\\{x\in\mathbb{R}\ltimes_{\theta}G|\ g(\rho,\rho|_{M})=1\\}$ is a
Sasakian manifold. Moreover, we have an isomorphism of manifolds

$\alpha:\mathbb{R}^{>0}\ltimes_{\theta}G\to\mathbb{R}^{>0}\times G\ \ \
\alpha(p)=s(p)\times\left(\gamma_{p}\cap M\right),$

where $\gamma_{p}$ be an integral curve of $\rho$ containing $p$. Then, by
Proposition 3.5 we have

$(\mathbb{R}^{>0}\ltimes_{\theta}G,g)\simeq(\mathbb{R}^{>0}\times
M,g=s^{2}g_{M}+ds^{2})$

where $g_{M}=g|_{M}$. Note that $M\subset G\times_{\theta}\mathbb{R}^{>0}$ is
not necessary a subgroup.

##  6 Projective Hessian Lie groups

######  Definition 6.1.

A Hessian Lie group $(G,\nabla,g)$ is a Lie group $G$ endowed with a left
invariant affine structure $\nabla$ and a left invariant Hessian metric $g$.

######  Theorem 6.2.

Let $(G,\nabla,g)$ be an $n$-dimensional simply connected Hessian Lie group
and $\theta$ the linear part of the corresponding affine action of $G$. Then
there exists a left invariant Kähler metric on
$G_{\nabla}=G\ltimes_{\theta}\mathbb{R}^{n}=TG.$

###### Proof.

The group $G\ltimes_{\theta}\mathbb{R}^{n}=TG$ is locally biholomorphic to
$\mathfrak{g}\oplus I\mathfrak{g}$. Hence, we have the local coordinates
$x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}$ on
$G\ltimes_{\theta}\mathbb{R}^{n}=TG$ such that $I\frac{\partial}{\partial
x_{i}}=\frac{\partial}{\partial y_{i}}$ and $x_{1},\ldots,x_{n}$ are constant
along any fiber of $\pi:TG\to G$. The Hessian metric $g$ is locally equivalent
to $\text{Hess}\varphi$. Define the associated Hermitian metric $g^{T}$ as in
Section 2. By Proposition 2.3, $g^{T}$ is a Hessian metric which is locally
expressed by $\text{Hess}_{\mathbb{C}}(4\pi^{*}\varphi)$. The subgroup
$\text{id}\ltimes\mathbb{R}^{n}\subset G\ltimes_{\theta}\mathbb{R}^{n}$ acts
on fibers of $TG\to G$. The function $\pi^{*}\varphi$ is constant along the
fibers hence $g^{T}=\text{Hess}_{\mathbb{C}}(4\pi^{*}\varphi)$ is invariant
under the action of $\text{Id}\ltimes_{\theta}\mathbb{R}^{n}\subset
G\ltimes_{\theta}\mathbb{R}^{n}$. By Proposition 2.5,

$g^{T}(X,Y)=\pi^{*}g(X,Y)+\pi^{*}g(IX,IY)+\sqrt{-1}\pi^{*}g(IX,Y)-\sqrt{-1}\pi^{*}g(X,IY).$

Moreover, $g$ is invariant under the action of $G\ltimes 0\subset
G\ltimes_{\theta}\mathbb{R}^{n}$. Thus, $g^{T}$ is invariant under the action
of $G\ltimes 0\subset G\ltimes_{\theta}\mathbb{R}^{n}$, also. Therefore,
$g^{T}$ is invariant under the action of the group
$G\ltimes_{\theta}\mathbb{R}^{n}$. ∎

######  Definition 6.3.

A Lie algebra $\mathfrak{g}$ is called projective if there is an invariant
affine structure $\nabla$ on $\mathfrak{g}\times\mathbb{R}$ such that

$\nabla_{X}E=\nabla_{E}X=X,$

where $X\in\mathfrak{g}$ and $E\in\mathbb{R}$. A Lie group is called
projective if the corresponding Lie algebra is projective.

Note that if $G$ is a projective Lie group then there is an invariant affine
structure on $G\times\mathbb{R}^{>0}$.

######  Definition 6.4.

A projective Hessian Lie group $(G,g_{G})$ is a projective Lie group $G$
endowed with a left-invariant Riemannian metric $g_{G}$ such that
$(G\times\mathbb{R}^{>0},\nabla,g=t^{2}g_{G}+dt^{2})$ is a Hessian manifold,
where $t$ is a coordinate on $\mathbb{R}^{>0}$ and $\nabla$ be the affine
connection on $G\times\mathbb{R}^{>0}$ corresponding to the projective
structure on $G$. A Lie algebra corresponding to projective Hessian Lie group
is called a projective Hessian Lie algebra.

######  Theorem 6.5.

Let $(G,g_{G}$ be an $n$-dimensional simply connected projective Hessian Lie
group and $\theta$ the linear part of the corresponding affine representation
of $G\times\mathbb{R}^{>0}$. Then there exists a structure of a semi-Sasakian
Lie group on $G\ltimes_{\theta}\mathbb{R}^{n+1}$. Moreover,
$G\ltimes_{\theta}\mathbb{R}^{n+1}\simeq TG\times\mathbb{R}$.

######  Lemma 6.6.

Let $F,G,H$ be groups, $\alpha$ and $\beta$ actions of $F$ and $G$ on $H$
respectively such that for any $f\in F,g\in G$, and $h\in H$ we have

$\alpha(f)\beta(g)h=\beta(g)\alpha(f)h.$

Then

$(F\times
G)\ltimes_{\alpha\times\beta}H=F\ltimes_{\text{id}\times\alpha}(G\ltimes_{\beta}H).$

###### Proof.

Both semidirect products are equal to $F\times G\times H$ as sets. Thus, it is
enough to check that two multiplications coincide. Write multiplication on the
group $(F\times G)\ltimes_{\alpha\times\beta}H$

$((f_{1}\times g_{1})\ltimes_{\alpha\times\beta}h_{1})((f_{2}\times
g_{2})\ltimes_{\alpha\times\beta}h_{2})=(f_{1}f_{2}\times
g_{1}g_{2})\ltimes_{\alpha\times\beta}h_{1}\alpha(f_{1})\beta(g_{1})h_{2}.$

Write multiplication on the group
$F\ltimes_{\text{id}\times\alpha}(G\ltimes_{\beta}H)$

$(f_{1}\ltimes_{\text{id}\times\alpha}(g_{1}\ltimes_{\beta}h_{1}))(f_{2}\ltimes_{\text{id}\times\alpha}(g_{2}\ltimes_{\beta}h_{2}))=f_{1}f_{2}\ltimes_{\text{id}\times\alpha}(g_{1}\ltimes_{\beta}h_{1})(g_{2}\ltimes_{\beta}\alpha(f_{1})h_{2})=$
$=f_{1}f_{2}\ltimes_{\text{id}\times\alpha}(g_{1}g_{2}\ltimes_{\beta}h_{1}\beta(g_{1})\alpha(f_{1})h_{2})=f_{1}f_{2}\ltimes_{\text{id}\times\alpha}(g_{1}g_{2}\ltimes_{\eta}h_{1}\alpha(f_{1})\beta(g_{1})h_{2}).$

Two multiplications coincide.

∎

###### Proof of Theorem 6.5.

There exist an invariant torsion free flat connection $\nabla$ on
$G\times\mathbb{R}^{>0}$ and a Hessian metric $g$ invariant under the left
action of $G$. Define a Kähler metric $g^{T}$ on
$(G\times\mathbb{R}^{>0})\ltimes_{\theta}\mathbb{R}^{n+1}$ as in the proof of
Theorem 6.2. This metric is invariant under the left action of
$(G\times\text{Id})\ltimes_{\theta}\mathbb{R}^{n+1}\subset(G\times\mathbb{R}^{>0})\ltimes_{\theta}\mathbb{R}^{n+1}$
by the same argument as in the proof of Theorem 6.2. Moreover,

$g^{T}(X,Y)=\pi^{*}g(X,Y)+\pi^{*}g(IX,IY)+\sqrt{-1}\pi^{*}g(IX,Y)-\sqrt{-1}\pi^{*}g(X,IY),$

where $\pi:G\times\mathbb{R}^{>0}\ltimes_{\theta}\mathbb{R}^{n+1}\simeq
T(G\times\mathbb{R}^{>0})\to G\times\mathbb{R}^{>0}$ is a projection. Let

$\lambda_{q}:G\times\mathbb{R}^{>0}\to G\times\mathbb{R}^{>0},\ \
\lambda_{q}(p\times r)=p\times qr$

and

$\mu_{q}:\left(G\ltimes_{\theta}\mathbb{R}^{>0}\right)\ltimes_{\theta}\mathbb{R}^{n+1}\to\left(G\ltimes_{\theta}\mathbb{R}^{>0}\right)\ltimes_{\theta}\mathbb{R}^{n+1},\
\ \lambda_{q}((p\times r)\ltimes_{\theta}v)=(p\times
qr)\ltimes_{\theta}\theta(q)v.$

Then we have the commutative diagram

$\begin{CD}\left(G\ltimes_{\theta}\mathbb{R}^{>0}\right)\ltimes_{\theta}\mathbb{R}^{n+1}@>{\mu_{q}}>{}>\left(G\ltimes_{\theta}\mathbb{R}^{>0}\right)\ltimes_{\theta}\mathbb{R}^{n+1}\\\
@V{}V{\pi}V@V{}V{\pi}V\\\
G\times\mathbb{R}^{>0}@>{\lambda_{q}}>{}>G\times\mathbb{R}^{>0}\end{CD},$

By the same argument as in Proposition 3.6, we get

$\mu_{q}^{*}g^{T}=q^{2}g^{T}.$ (6.1)

We constructed the Kähler metric $g^{T}$ on
$(G\times\mathbb{R}^{>0})\ltimes_{\theta}\mathbb{R}^{n+1}$ which is invariant
under the action of $G\ltimes_{\theta}\mathbb{R}^{n+1}$ and satisfies (6.1).
Also, by Lemma 6.6, we have

$(G\times\mathbb{R}^{>0})\ltimes_{\theta}\mathbb{R}^{n+1}=\mathbb{R}^{>0}\ltimes(G\ltimes_{\theta}\mathbb{R}^{n+1}).$

Thus, $G\ltimes_{\theta}\mathbb{R}^{n+1}$ is a semi-Sasakian Lie group. ∎

Examples of projective Hessian Lie groups are described in the next two
sections.

##  7 Regular convex cones

######  Definition 7.1.

A subset $V\subset\mathbb{R}^{n}$ is called regular if $V$ does not contain
any straight line.

Let $V\subset\mathbb{R}^{n}$ be a convex regular domain. We denote the maximal
subgroup of $\text{GL}(\mathbb{R}^{n})$ preserving $V$ by $\text{Aut}(V)$.
Note that if $V$ is a regular convex cone then

$\text{Aut}(V)=(\text{Aut}(V)\cap\text{SL}(\mathbb{R}^{n}))\times\mathbb{R}^{>0}.$

The following theorem summarized known results.

######  Theorem 7.2 ([V], [VGP], [Sh]).

Let $V\subset\mathbb{R}^{n}$ be a convex homogeneous regular domain,

${U=\mathbb{R}^{n}\oplus\sqrt{-1}V\subset\mathbb{C}^{n}},$

and

$\pi:U=\mathbb{R}^{n}\oplus\sqrt{-1}V\to\sqrt{-1}V\simeq V$

be a projection. Then there exist a function $\varphi$ on $V$ and a subgroup
$T\subset\text{Aut}(V)$ such that the following conditions are satisfied:

  * (i)

$g_{can}=\text{Hess}\varphi$ is $\text{Aut}(V)$-invariant Hessian metric on
$V$.

  * (ii)

$T$ acts on $V$ simply transitively.

  * (iii)

A group $\text{Aut}(V)\ltimes\mathbb{R}^{n}$ acts on $U$ by holomorphic
automorphisms. Moreover, the subgroup
$T\ltimes\mathbb{R}^{n}\subset\text{Aut}(V)\ltimes\mathbb{R}^{n}$ acts on $U$
simply transitively.

  * (iv)

The bilinear form $\text{Hess}_{\mathbb{C}}(4\pi^{*}\varphi)$ is a
$\text{Aut}(V)\ltimes\mathbb{R}^{n}$-invariant Kähler metric on $U$.

See [V] for (i) and (ii), [VGP] for (iii), [Sh] for (iv).

######  Theorem 7.3.

Let $V\subset\mathbb{R}^{n}$ be a convex homogeneous regular cone and $U$,
$\pi$, $T$, $\varphi$ as in Theorem 7.2. Denote
${T_{\text{SL}}=T\cap\text{SL}(\mathbb{R}^{n})}$. Then following conditions
are satisfied:

  * (i)

The exists a $\text{Aut}_{\text{SL}}(V)$-invariant conical Hessian metric
$g_{con}=\text{Hess}\varphi$ on $V$. The dilation subgroup
$\mathbb{R}^{>0}\subset\text{Aut}(V)$ acts on $g_{con}$ by the rule
$\lambda_{q}^{*}g_{con}=q^{2}g_{con}$, for any $q\in\mathbb{R}^{>0}$.

  * (ii)

The bilinear form $\text{Hess}_{\mathbb{C}}(4\pi^{*}\varphi)$ is a
$T_{\text{SL}}$-invariant Kähler metric. Moreover, $\lambda_{q}^{*}g=q^{2}g$.

###### Proof.

This theorem is similar to the previous one. The metric $g_{con}$ on a convex
regular cone $V$ is defined as $\text{Hess}(\varphi)$, where $\varphi$ is a
characteristic function of the cone (see [V]). The item (ii) is analogous to
the item (iv) from Theorem 7.2. ∎

By the item (ii) from Theorem 7.2, $T$ is a Hessian Lie group. By the item
(iv) from Theorem 7.2, $T\ltimes\mathbb{R}^{n}$ is a Kähler Lie group. On the
other hand, we can get the same result using Theorem 6.2. Thus, Theorem 6.2
generalizes the item (iv) from Theorem 7.2.

Analogically, by Theorem 7.3, $T_{\text{SL}}$ is a projective Hessian Lie
group and $T_{\text{SL}}\ltimes\mathbb{R}^{n}$ is a Sasakian Lie group. We can
get the same result using Theorem 6.5. Thus, Theorem 6.5 generalizes known
constructions for homogeneous regular cones.

######  Definition 7.4.

The group $T$ from the Theorem 7.2 is called a Lie group associated with $V$.
The corresponding Lie algebra $\mathfrak{t}$ is a Lie algebra associated with
$V$.

######  Proposition 7.5 ([V]).

Let $T_{\text{SL}}$ be a Lie group from Theorem 6.3 then there exist a convex
regular domain $W$ such that $T_{\text{SL}}$ is an associated with $W$ Lie
group. Conversely, if $S$ is a Lie group associated with a regular convex
domain $W$ then there exist regular convex $V$ such that $S=T_{\text{SL}}$,
where $T_{\text{SL}}$ is as in Theorem 7.3.

######  Corollary 7.6.

Let $T$ be a Lie group associated with a homogeneous regular domain. Then $T$
is both a Hessian Lie group and a projective Hessian Lie group.

######  Example 7.7.

Let $V$ be the vector space of all real symmetric matrices of rank $n$ and
$\Omega$ the set of all positive definite symmetric matrices in $V$. Then
$\Omega$ is a regular convex cone and the group of upper triangular matrix
$\text{T}(\mathbb{R}^{n})$ acts simply transitively on $\Omega$ by
$s(x)=sxs^{T}$, where $x\in\Omega$ and $s\in\text{T}(n)$.The characteristic
function is equal to

$\varphi(x)=(\det x)^{-\frac{n+1}{2}}\varphi(e),$

where $e$ is the unit matrix (see [Sh]).

##  8 Projective Hessian structure on $\text{SO}(2)$ and $\text{SU}(2)$

######  Example 8.1.

Consider the group $\mathbb{R}$ as the universal covering of
$\text{U}(1)=\text{SO(2)}$. The identification
$\text{SO}(2)\times{\mathbb{R}^{>0}}\simeq\mathbb{R}^{2}\backslash\\{0\\}$
sets a projective Hessian structure on $\text{SO}(2)$ and on the universal
covering $\mathbb{R}$. The corresponding to $\mathbb{R}$ semi-Sasakian group
$\mathbb{R}\ltimes\mathbb{R}^{2}$ is the universal covering to the group of
Euclidean motions $\text{E}(2)=\text{SO}(2)\ltimes\mathbb{R}^{2}$. Hence, the
Lie algebra of Euclidean motions $\mathfrak{e}(2)$ is semi-Sasakian. Let
$t,x,y,r$ be standard coordinates on the group
$\mathbb{R}\ltimes\mathbb{R}^{2}\times\mathbb{R}^{>0}$. The Kähler metric is
given by $\text{Hess}_{\mathbb{C}}(r^{2})$. The Kähler structure on
$\mathbb{R}\ltimes\mathbb{R}^{2}\times\mathbb{R}^{>0}$ induces the Kähler
structure on $\text{SO}(2)\ltimes\mathbb{R}^{2}\times\mathbb{R}^{>0}$. Thus,
the group of Euclidean motions $\text{E}(2)$ is semi-Sasakian, too.

All $3$-dimensional Sasakian Lie algebras are classified.

######  Proposition 8.2 ([AFV]).

Any 3-dimensional Sasakian Lie algebra is isomorphic to one of the following:
$\mathfrak{su}(2),\mathfrak{sl}(2,\mathbb{R}),\mathfrak{aff}(\mathbb{R})\times\mathbb{R}$,
and the Heisenberg algebra $\mathfrak{h}_{3}$.

The algebras $\mathfrak{su}(2)$ and $\mathfrak{sl}(2,\mathbb{R})$ are
semisimple, and the algebras $\mathfrak{aff}(\mathbb{R})\times\mathbb{R}$ and
$\mathfrak{h}_{3}$ are nilpotent. The algebra $\mathfrak{e}(2)$ is solvable
but not nilpotent. Therefore, the algebra $\mathfrak{e}(2)$ is semi-Sasaian
but not Sasakian.

######  Example 8.3.

There is an identifications $\text{SU}(2)\simeq S^{3}$ and
$\text{SU}(2)\times\mathbb{R}^{>0}=\mathbb{R}^{4}\backslash\\{0\\}$. The group
structure on $S^{3}$ equals to the restriction on $S^{3}$ of the standard
$\text{SU}(2)$-action on $\mathbb{C}^{2}\simeq\mathbb{R}^{4}$. The
corresponding semi-Sasakian Lie group is equal to
$\text{SU}(2)\ltimes\mathbb{C}^{2}$.

Notice that the group $\text{SU}(2)$ is not Hessian just because the sphere
$S^{3}$ does not admit an affine structure. However, there exists an invariant
affine structures on
$\text{SU}(2)\times\mathbb{R}^{>0}\simeq\mathbb{R}^{4}\backslash\\{0\\}$.
Moreover, if the group $G$ is one of the previous examples of projective
Hessian Lie groups (a clan or $\text{U}(1)$) then the group
$G\times\mathbb{R}^{>0}$ is Hessian. We prove that the group
$\text{SU}(2)\times\mathbb{R}^{>0}$ is not Hessian.

######  Proposition 8.4.

The manifold $S^{3}\times\mathbb{R}^{>0}$ does not admit an
$\mathbb{R}^{>0}$-invariant Hessian structure.

###### Proof.

If $S^{3}\times\mathbb{R}^{>0}$ admits an $\mathbb{R}^{>0}$-invariant Hessian
structure then the Hopf manifold

$S^{3}\times\mathbb{R}^{>0}/_{(x\times t)\sim(x\times 2t)}\simeq S^{3}\times
S^{1}$

admits a Hessian structure. In [Sh], Shima proved that if $M$ is compact
Hessian manifold then the universal covering $\tilde{M}$ is convex domain in
$\mathbb{R}^{n}$. The universal covering of the Hopf manifold is diffeomorphic
to $S^{3}\times\mathbb{R}^{>0}$ that cannot be diffeomorphic to any domain in
$\mathbb{R}^{4}$. Therefore, $S^{3}\times\mathbb{R}^{>0}$ does not admit an
$\mathbb{R}^{>0}$-invariant Hessian structure. ∎

######  Corollary 8.5.

The group $\text{SU}(2)\times\mathbb{R}^{>0}$ is not Hessian.

### Acknowledgements.

I would like to thank M. Verbitsky for fruitful discussions, and D.V.
Alekseevsky, for his useful comments and help with preparation of the paper.

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