# Flat hypercomplex nilmanifolds are ${\mathbb{H}}$-solvable

Yulia Gorginyan

###### Abstract

We say that a hypercomplex nilpotent Lie algebra is ${\mathbb{H}}$-solvable if
there exists a sequence of ${\mathbb{H}}$-invariant subalgebras

${\mathfrak{g}}_{1}^{{\mathbb{H}}}\supset{\mathfrak{g}}_{2}^{{\mathbb{H}}}\supset\cdots\supset{\mathfrak{g}}_{k-1}^{{\mathbb{H}}}\supset{\mathfrak{g}}_{k}^{{\mathbb{H}}}=0,$

such that
$[{\mathfrak{g}}_{i}^{{\mathbb{H}}},{\mathfrak{g}}_{i}^{{\mathbb{H}}}]\subset{\mathfrak{g}}^{{\mathbb{H}}}_{i+1}.$
Let $N=\Gamma\backslash G$ be a hypercomplex nilmanifold with flat Obata
connection and ${\mathfrak{g}}=\operatorname{Lie}(G)$. We prove that the Lie
algebra ${\mathfrak{g}}$ is ${\mathbb{H}}$-solvable.

###### Contents

  1. 1 Introduction
    1. 1.1 Affine manifolds
    2. 1.2 Hypercomplex affine nilmanifolds
    3. 1.3 ${\mathbb{H}}$-solvable Lie algebras and algebraic holonomy
  2. 2 Preliminaries: Nilpotent Lie groups and algebras
  3. 3 Algebraic holonomy group
    1. 3.1 Left equivariant vector bundles
    2. 3.2 Algebraic holonomy group
    3. 3.3 Maltsev completion
    4. 3.4 Unipotent holonomy group and ${\mathbb{H}}$-solvability

##  1 Introduction

###  1.1 Affine manifolds

A manifold $X$ together with a torsion-free flat connection $\nabla$ is called
an affine manifold. Equivalently, an affine manifold $X$ is a manifold with an
atlas such that all translation maps between charts are in
$\operatorname{Aff}({\mathbb{R}}^{n})$ ([FGH], [Sh]).

Recall that a group of affine transformations
$\operatorname{Aff}({\mathbb{R}}^{n})$ is a semidirect product
$\operatorname{GL}_{n}({\mathbb{R}})\ltimes{\mathbb{R}}^{n}$. If
$f\in\operatorname{Aff}({\mathbb{R}}^{n})$ then for any $x\in{\mathbb{R}}^{n}$
there exist a linear part $A\in\operatorname{GL}({\mathbb{R}}^{n})$ and a
translation part $t\in{\mathbb{R}}^{n}$ such that $f(x)=Ax+t$. The
linearization map is the natural homomorphism
$l:\operatorname{Aff}({\mathbb{R}}^{n}){\>\longrightarrow\>}\operatorname{GL}({\mathbb{R}}^{n})$
given by the formula $l(f)=A$.

An affine manifold $X$ is called complete if its universal cover is affine
equivalent to ${\mathbb{R}}^{n}$, i.e. $X={\mathbb{R}}^{n}/\Gamma$, where
$\Gamma\subset\operatorname{Aff}({\mathbb{R}}^{n})$ is a discrete subgroup of
the affine transformation group.

There are three famous open questions related to affine geometry. In the
historical context, the first was the conjecture of Chern:

Conjecture 1.1: (Chern) The Euler characteristic of a compact affine manifold
vanishes [Gol].

Kostant and Sullivan proved Chern’s conjecture for the compact complete affine
manifolds [KS]. Also, it was proven by Bruno Klingler [Klin] in the case when
a compact affine manifold admits a parallel volume form. As far as we know,
the general case of Chern’s conjecture remains open.

The conjecture of Markus links the existence of a parallel volume form with
the completeness of a compact affine manifold.

Conjecture 1.2: (Markus, 1962)[Mar] A compact affine manifold $X$ admits
parallel volume form if and only if the manifold $X$ is complete.

Recall that a subgroup $\Gamma$ of an affine group
$\operatorname{Aff}({\mathbb{R}}^{n})$ is called crystallographic [A] if its
action on ${\mathbb{R}}^{n}$ is properly discontinuous, free, and cocompact.

Recall that a group $G$ is said to have a certain property $P$ virtually if
$G$ contains a subgroup $H$ of finite index which has the property $P$.

Theorem 1.3: (Bieberbach, 1911) [Bieb] Every discrete subgroup $\Gamma$ of an
isometry group ${\rm Isom}({\mathbb{R}}^{n})$ is virtually abelian. Every
crystallographic subgroup of ${\rm Isom}({\mathbb{R}}^{n})$ is virtually a
translation group. For a given $n$ there exists only a finite number of such
groups $\Gamma\subset{\rm Isom}({\mathbb{R}}^{n})$ up to a conjugation.

One of the ways to generalize the theorem of Bieberbach is to consider an
affine transformation group $\operatorname{Aff}({\mathbb{R}}^{n})$ instead of
${\rm Isom}({\mathbb{R}}^{n})$.

Conjecture 1.4: (Auslander, 1964)[Aus] Every crystallographic subgroup of an
affine group is virtually solvable, i.e. contains a solvable subgroup of
finite index.

Auslander’s conjecture was proven by D. Fried and W. Goldman [FG] in the case
$n=3$ and the result was refined up to dimension $n=6$ by H. Abels, G.A.
Margulis, and G.A. Soifer [AMS]. Also, Auslander’s conjecture is true in a
case when the monodromy group preserves a metric of signature $(1,n)$ [GK] and
$(2,n)$ [AMS2].

###  1.2 Hypercomplex affine nilmanifolds

Recall that a nilmanifold $N$ is a compact quotient of a connected simply
connected nilpotent Lie group $G$ by a lattice subgroup $\Gamma$, which acts
on the group $G$ from the left. We denote the quotient as $N=\Gamma\backslash
G$. It is called an affine nilmanifold if $G$ has a left-invariant affine
structure, i.e. the flat torsion-free connection $\nabla$ such that
$L_{g}^{*}(\nabla)=\nabla$, where $L_{g}:G{\>\longrightarrow\>}G$ is the left-
translation.

Let $\pi:\widetilde{X}{\>\longrightarrow\>}X$ be the universal cover of a
manifold $X$ and $\pi_{1}(X)$ the fundamental group. An affine immersion
$D:\widetilde{X}{\>\longrightarrow\>}{\mathbb{R}}^{n}$ is called the
developing map. By a well-known theorem ([OV], [Gol1]) $D$ exists and is
uniquely defined up to an affine automorphism.

Definition 1.5:  The affine holonomy representation is a unique homomorphism
$h:\pi_{1}(X){\>\longrightarrow\>}\operatorname{Aff}({\mathbb{R}}^{m})$ which
satisfies $D\circ\gamma=h(\gamma)\circ D$ for every $\gamma\in\pi_{1}(X)$. The
affine holonomy group $\mathcal{H}:=\operatorname{Im}(h)$ is the image of the
homomorphism $h$ in $\operatorname{Aff}({\mathbb{R}}^{n})$. The linear
holonomy group is the image of the affine holonomy group under the
linearization map,
$\mathcal{L}:=l(\mathcal{H})\subset\operatorname{GL}({\mathbb{R}}^{n})$.

Definition 1.6:  A closed Lie subgroup $G\subset\operatorname{GL}(V)$ is
called a linear algebraic group if $G$ is given by a system of polynomial
equations.

Definition 1.7:  A representation of a linear algebraic group is called
unipotent if, in a certain basis, its image is the group of strictly upper-
triangular matrices.

Let $X$ be a compact affine manifold whose linear holonomy representation is
unipotent. Then $X$ admits a parallel volume form. The converse was proven by
Goldman, Fried, and Hirsch, [FGH, Theorem A]:

Theorem 1.8:  Let $X$ be a compact affine manifold with a parallel volume
form. Assume that its affine holonomy group is nilpotent. Then the linear
holonomy representation is unipotent.

We will use the following reformulation of 1.2:

Theorem 1.9:  Let $X$ be a compact affine manifold with a parallel volume
form. Assume that its fundamental group is nilpotent. Then the monodromy
representation is unipotent.

We are interested in affine nilmanifolds which also possess a hypercomplex
structure. First, recall a definition of a complex nilmanifold.

Definition 1.10:  Let $G$ be a Lie group equipped with a left-invariant
complex structure. A complex nilmanifold is a pair $(N=\Gamma\backslash G,I)$,
where $N$ is a nilmanifold and the complex structure $I$ is obtained from the
corresponding left-invariant complex structure on $G$.

An almost hypercomplex manifold $X$ is a smooth manifold equipped with three
endomorphisms $I,J$ and $K$ of the tangent bundle satisfying the quaternionic
relations $I^{2}=J^{2}=K^{2}=-\operatorname{Id}$ and $IJ=K$. When almost
complex structures are integrable, the quadruple $(X,I,J,K)$ is called a
hypercomplex manifold.

Definition 1.11:  Let $G$ be a nilpotent Lie group with a left-invariant
hypercomplex structure $I,J,K$ and $\Gamma\subset G$ a cocompact lattice. Then
$(N=\Gamma\backslash G,I,J,K)$ is called a hypercomplex nilmanifold.

M. Obata showed [Ob] that on a hypercomplex manifold $M$, there exists a
unique torsion-free connection $\nabla$ preserving the complex structures:
$\nabla I=\nabla J=\nabla K=0$. It is called the Obata connection.

It could be written in the following form

$\nabla_{X}Y=\frac{1}{2}([X,Y]+I[IX,Y]-J[X,JY]+K[IX,JY]),$ (1.1)

for any $X,Y\in TM$ [Sol].

The existence of a parallel volume form on a hypercomplex nilmanifold is
guaranteed by the following theorem:

Theorem 1.12: [BDV, Theorem 3.2] Let $N=\Gamma\backslash G$ be a hypercomplex
nilmanifold, $n=\dim_{{\mathbb{C}}}G$. Then $G$ admits a left-invariant, non-
zero, holomorphic section $\Omega$ of the canonical bundle $\Lambda^{n,0}G$.
Moreover, $\nabla\Omega=0$, where $\nabla$ is the Obata connection.

###  1.3 ${\mathbb{H}}$-solvable Lie algebras and algebraic holonomy

An operator $I$ on a real Lie algebra ${\mathfrak{g}}$ is called a complex
structure operator if $I^{2}=-\operatorname{Id}$ and the
$\sqrt{-1}$-eigenspace ${\mathfrak{g}}^{1,0}$ is a Lie subalgebra in the
complexification
${\mathfrak{g}}_{{\mathbb{C}}}={\mathfrak{g}}\otimes{\mathbb{C}}$.

Remark 1.13: Take an endomorphism $I\in\operatorname{End}{\mathfrak{g}}$,
$I^{2}=-\operatorname{Id}$, and extend it to the left-invariant almost complex
structure on the Lie group $G$. Then $I$ is complex if and only if
$[{\mathfrak{g}}^{1,0},{\mathfrak{g}}^{1,0}]\subset{\mathfrak{g}}^{1,0}$. In
other words, this definition is compatible with 1.2.

A hypercomplex structure on a Lie algebra ${\mathfrak{g}}$ is a triple of
complex structure operators $I,J$ and $K$ on ${\mathfrak{g}}$ satisfying the
quaternionic relations.

Definition 1.14:  Let ${\mathfrak{g}}$ be a nilpotent hypercomplex Lie
algebra. Define inductively ${\mathbb{H}}$-invariant Lie subalgebras:

${\mathfrak{g}}_{i}^{{\mathbb{H}}}:={\mathbb{H}}[{\mathfrak{g}}_{i-1}^{{\mathbb{H}}},{\mathfrak{g}}_{i-1}^{{\mathbb{H}}}],$

where ${\mathfrak{g}}_{0}^{{\mathbb{H}}}={\mathfrak{g}}$ and

${\mathfrak{g}}_{1}^{{\mathbb{H}}}:={\mathbb{H}}[{\mathfrak{g}},{\mathfrak{g}}]=[{\mathfrak{g}},{\mathfrak{g}}]+I[{\mathfrak{g}},{\mathfrak{g}}]+J[{\mathfrak{g}},{\mathfrak{g}}]+K[{\mathfrak{g}},{\mathfrak{g}}].$

It is natural to study if ${\mathfrak{g}}_{1}^{{\mathbb{H}}}$ is a proper
subalgebra in ${\mathfrak{g}}$.

A hypercomplex nilpotent Lie algebra ${\mathfrak{g}}$ is called
${\mathbb{H}}$-solvable if the following sequence terminates for some
$k\in{\mathbb{Z}}_{>0}$:

${\mathfrak{g}}_{1}^{{\mathbb{H}}}\supset{\mathfrak{g}}_{2}^{{\mathbb{H}}}\supset\cdots\supset{\mathfrak{g}}_{k-1}^{{\mathbb{H}}}\supset{\mathfrak{g}}_{k}^{{\mathbb{H}}}=0.$
(1.2)

Conjecture 1.15:  Let $({\mathfrak{g}},I,J,K)$ be a nilpotent hypercomplex Lie
algebra. Then it is ${\mathbb{H}}$-solvable.

Let $(N=\Gamma\backslash G,I,J,K)$ be a hypercomplex nilmanifold. Consider a
complex nilmanifold $(N,L)$ with a general complex structure $L=aI+bJ+cK$,
where $(a,b,c)\in S^{2}$, obtained from the hypercomplex structure. It is
natural to try to describe complex submanifolds in $(N,L)$. We solved this
problem in the case of complex curves under the additional assumption on the
Lie algebra ${\mathfrak{g}}=\operatorname{Lie}(G)$ [Gor]. Precisely, if the
corresponding Lie algebra ${\mathfrak{g}}$ is ${\mathbb{H}}$-solvable, then
there are no complex curves in a complex nilmanifold $(N,L)$ for the general
complex structure $L$.

In this work, we show that in the case when a hypercomplex nilmanifold
$\Gamma\backslash G$ admits the flat Obata connection, then
${\mathfrak{g}}=\operatorname{Lie}(G)$ is ${\mathbb{H}}$-solvable. The main
argument we use relies on the notion of an algebraic holonomy group, which we
define below (3.3).

Example 1.16: It was shown in [DF] that all 8-dimensional hypercomplex
nilpotent Lie algebras are Obata-flat. However, in the same paper, I. Dotti
and A. Fino presented an example of the 3-step nilpotent hypercomplex Lie
algebra of dimension 12 which has non-zero curvature of the Obata connection.

Let ${\mathfrak{g}}$ be a nilpotent Lie algebra over the field ${\mathbb{R}}$.
A rational structure in ${\mathfrak{g}}$ is a subalgebra
${\mathfrak{g}}_{{\mathbb{Q}}}\subset{\mathfrak{g}}$ defined over the rational
numbers such that
${\mathfrak{g}}_{{\mathbb{Q}}}\otimes_{{\mathbb{Q}}}{\mathbb{R}}={\mathfrak{g}}$.
Assume that a Lie algebra ${\mathfrak{g}}$ admits a rational structure
${\mathfrak{g}}_{{\mathbb{Q}}}$. By [CG] it happens if and only if there
exists a nilmanifold $\Gamma\backslash G$ such that
${\mathfrak{g}}_{{\mathbb{Q}}}:=\rm{span}_{{\mathbb{Q}}}\langle\log\Gamma\rangle$.

In the case when a Lie group $G$ admits a left-invariant hypercomplex
structure with the flat Obata connection, we prove that
${\mathfrak{g}}_{i}^{{\mathbb{H}}}$ is a proper subalgebra of
${\mathfrak{g}}_{i-1}$ using the following approach.

A connection on a Lie algebra ${\mathfrak{g}}$ is an ${\mathbb{R}}$-linear map

$\nabla:{\mathfrak{g}}{\>\longrightarrow\>}{\mathfrak{g}}^{*}\otimes{\mathfrak{g}}$

such that $X\mapsto\nabla_{X}\in\operatorname{End}({\mathfrak{g}})$. Note that
a connection on a Lie algebra ${\mathfrak{g}}$ is the same as a left-invariant
connection on a Lie group $G$ (3.1). This notion can be generalized to
arbitrary $G$-equivariant vector bundle on a Lie group $G$, giving the
algebraic version of an equivariant connection (see (3.1)).

The curvature tensor $R$ of the connection $\nabla$ defined in the following
way:

$R(X,Y)=[\nabla_{X},\nabla_{Y}]-\nabla_{[X,Y]},$

where $X,Y\in{\mathfrak{g}}$. The connection $\nabla$ is called flat if $R=0$.
Notice that left invariant flat connections on $G$ are equivalent to the
representations of a Lie algebra on itself considered as a vector space (3.1).

In the end, we prove the following theorem:

Theorem 1.17:  Let $(N,I,J,K)$ be a hypercomplex nilmanifold with the flat
Obata connection. Then the corresponding Lie algebra is
${\mathbb{H}}$-solvable.

##  2 Preliminaries: Nilpotent Lie groups and algebras

Let $G$ be a Lie group. Define inductively the descending chain of normal
subgroups of $G$

$G=G_{0}\supset G_{1}\supset G_{2}\supset\cdots\supset G_{k}\supset\cdots,$
(2.1)

where $G_{j}:=[G_{j-1},G]$ is the subgroup generated by the elements of the
form $xyx^{-1}y^{-1},\,x\in G_{j-1},\,y\in G$.

Definition 2.1:  A Lie group $G$ is called a nilpotent Lie group if (2.1)
terminates for some $k\in{\mathbb{Z}}_{>0}$, i.e. $G_{k}=\\{e\\}$.

The descending series of a real Lie algebra ${\mathfrak{g}}$ is the chain of
ideals defined as follows:

${\mathfrak{g}}_{0}\supset{\mathfrak{g}}_{1}\supset\dots\supset[{\mathfrak{g}}_{k},{\mathfrak{g}}]\supset\dots,$

${\mathfrak{g}}_{0}={\mathfrak{g}}$ and
${\mathfrak{g}}_{k}=[{\mathfrak{g}}_{k-1},{\mathfrak{g}}]$.

Definition 2.2:  A Lie algebra ${\mathfrak{g}}$ is called nilpotent if
${\mathfrak{g}}_{s}=0$ for some $s\in{\mathbb{Z}}_{>0}$.

Let ${\mathfrak{g}}$ be a real Lie algebra and ${\mathfrak{g}}^{*}$ its dual
space. For any $\alpha\in{\mathfrak{g}}^{*}$ and $\xi,\theta\in{\mathfrak{g}}$
the Chevalley–Eilenberg differential
$d:{\mathfrak{g}}^{*}{\>\longrightarrow\>}\Lambda^{2}{\mathfrak{g}}^{*}$ is
defined as follows

$d\alpha(\xi,\theta)=-\alpha([\xi,\theta]).$ (2.2)

It extends to a finite-dimensional complex

$0{\>\longrightarrow\>}{\mathfrak{g}}^{*}{\>\longrightarrow\>}\Lambda^{2}{\mathfrak{g}}^{*}{\>\longrightarrow\>}\cdots{\>\longrightarrow\>}\Lambda^{2n}{\mathfrak{g}}^{*}{\>\longrightarrow\>}0$
(2.3)

by the Leibniz rule:
$d(\alpha\wedge\beta)=d\alpha\wedge\beta+(-1)^{\widetilde{\alpha}}\alpha\wedge
d\beta$, where $\alpha,\beta\in{\mathfrak{g}}^{*}$. The identity $d^{2}=0$
follows from the Jacobi identity [CE].

Notice that the kernel of a closed $1$-form is an ideal in the Lie algebra
${\mathfrak{g}}$. According to the definition of the Chevalley–Eilenberg
differential, any closed 1-form $\alpha\in{\mathfrak{g}}^{*}$ vanishes on the
commutator ideal $[{\mathfrak{g}},{\mathfrak{g}}]\subseteq\ker\alpha$.

Remark 2.3:  The intersection of all kernels of closed 1-forms
$\alpha\in{\mathfrak{g}}^{*}$

$\Sigma\,\,:=\bigcap_{\alpha\in{\mathfrak{g}}^{*},d\alpha=0}{\ker\alpha}$

also forms an ideal in the Lie algebra ${\mathfrak{g}}$ which coincides with
the commutator ideal $[{\mathfrak{g}},{\mathfrak{g}}]$.

Recall that a distribution on a smooth manifold $N$ is a sub-bundle
$\Sigma\subset TN$. The distribution is called involutive if it is closed
under the Lie bracket. A leaf of the distribution $\Sigma$ is a maximal
connected, immersed submanifold $L\subset N$ such that $L$ tangent to $\Sigma$
at each point. If $\Sigma$ is involutive, then the set of all its leaves is
called a foliation.

##  3 Algebraic holonomy group

###  3.1 Left equivariant vector bundles

Recall that a map $\varphi:X{\>\longrightarrow\>}Y$ of two manifolds $X$ and
$Y$ with an action of a group $G$ is called $G$-equivariant if $\varphi(g\cdot
x)=g\cdot\varphi(x)$ for any $g\in G$, $x\in X$. A vector bundle
$\pi:\mathbb{B}{\>\longrightarrow\>}X$ on a manifold $X$ is called a
$G$-equivariant vector bundle if its total space is equipped with a $G$-action
such that the projection $\pi$ is a $G$-equivariant map and the action of $G$
is linear on the fibers, i.e.
$\mathbb{B}_{x}{\>\longrightarrow\>}\mathbb{B}_{gx}$ is a linear map for all
$g\in G$.

Let $G$ be a Lie group, $L_{g}:G{\>\longrightarrow\>}G$ the left translation
($h\mapsto g\cdot h$), and $B$ a finite-dimensional vector space with a basis
$\\{b_{1},\dots,b_{n}\\}$.

Definition 3.1:  Let ${\mathfrak{g}}=\operatorname{Lie}(G)$ be a Lie algebra
and ${\mathfrak{g}}^{*}=\operatorname{Hom}({\mathfrak{g}},{\mathbb{R}})$ its
dual. An algebraic connection is an ${\mathbb{R}}$-linear map

$\nabla:B{\>\longrightarrow\>}{\mathfrak{g}}^{*}\otimes B.$

We use the following notation

$\nabla(b)=\sum_{i}\theta_{i}\otimes A_{i}(b)$ (3.1)

where $\theta_{i}\in{\mathfrak{g}}^{*}$ and $A_{i}\in\operatorname{End}(B)$,
$b\in B$.

Let $d:{\mathfrak{g}}^{*}{\>\longrightarrow\>}\Lambda^{2}{\mathfrak{g}}^{*}$
be the Chevalley–Eilenberg differential. We define the curvature of the
connection $\nabla$ by the following formula

$\Theta_{\nabla}:=d\omega+\omega\wedge\omega,$ (3.2)

where $\omega:=\sum_{i}\theta_{i}\otimes A_{i}$,
$d\omega=\sum_{i}d\theta_{i}\otimes A_{i}$ and
$\omega\wedge\omega=\sum_{i<j}\theta_{i}\wedge\theta_{j}\otimes[A_{i},A_{j}]$.

Consider a vector bundle $\pi:\mathbb{B}{\>\longrightarrow\>}G$ on a Lie group
$G$. From now we denote the fiber at the identity $e\in G$ by
$\mathbb{B}_{e}=B$.

Definition 3.2:  A left equivariant vector bundle
$\pi:\mathbb{B}{\>\longrightarrow\>}G$ over a Lie group $G$ is a
$G$-equivariant vector bundle $\pi$ on a Lie group $G$ with an action given by
the left translations. A left invariant section $s$ is determined by its value
$s_{e}$ at $e\in G$. This gives a map

$\mathbb{L}:\mathbb{B}_{e}{\>\longrightarrow\>}H^{0}(G,\mathbb{B}),\quad
s=\mathbb{L}(s_{e}),$ (3.3) $\mathbb{L}(s_{e})(g)=s(g):=(L_{g})_{*}(s_{e}).$

identified the fiber $\mathbb{B}_{e}$ with the space $H^{0}(G,\mathbb{B})$ of
all $G$-invariant sections of $\mathbb{B}$.

Remark 3.3: All tensor powers of a $G$-equivariant vector bundle are
$G$-equivariant.

Claim 3.4: The category of left equivariant vector bundles on a Lie group $G$
is equivalent to the category of vector spaces.

Proof: The group $G$ acts on itself by the left multiplication freely and
transitively. Any left equivariant bundle $\mathbb{B}$ is trivial as a vector
bundle and there is a basis of left invariant sections
$\\{s_{1},\dots,s_{n}\\}$, such that $s_{i}=\mathbb{L}(b_{i})$. For each left
equivariant vector bundle the vector space of its left invariant sections
gives a functor to the category of vector spaces. Conversely, given a vector
space, consider it as a space of sections evaluated at the identity $e\in G$,
and then extend them via all left translations (3.3) to obtain a left
equivariant vector bundle.

Now we are able to give a definition of an invariant connection on the bundle
$\mathbb{B}$. We use an algebraic connection (3.1) on the fiber
$B=\mathbb{B}_{e}$.

Definition 3.5:  Let $\pi:\mathbb{B}{\>\longrightarrow\>}X$ be a
$G$-equivariant vector bundle over a manifold $X$. A connection
$\nabla\\!\\!\\!\\!\nabla$ on $\mathbb{B}$ is called equivariant if it defines
a left invariant differential operator

$\nabla\\!\\!\\!\\!\nabla:\mathbb{B}{\>\longrightarrow\>}\Lambda^{1}(X)\otimes\mathbb{B}.$

Claim 3.6:  A connection
$\nabla\\!\\!\\!\\!\nabla:\mathbb{B}{\>\longrightarrow\>}\Lambda^{1}(G)\otimes\mathbb{B}$
on a left equivariant vector bundle $\mathbb{B}$ over a Lie group $G$ is
equivariant if it satisfies

$\nabla\\!\\!\\!\\!\nabla(\mathbb{L}(s_{e}))=\mathbb{L}((\nabla\\!\\!\\!\\!\nabla
s)_{e}),$ (3.4)

where $\nabla\\!\\!\\!\\!\nabla s$ is understood as a section of the left-
equivariant bundle $\Lambda^{1}(G)\otimes\mathbb{B}$, $\mathbb{L}$ is the
left-translation map defined in (3.3), and $(\nabla\\!\\!\\!\\!\nabla s)_{e}$
means the value of the section in $e\in G$.

Definition 3.7:  Let $\theta\in{\mathfrak{g}}^{*}$ and $b\in B$. Define a map

$d_{\nabla}:{\mathfrak{g}}^{*}\otimes_{{\mathbb{R}}}B{\>\longrightarrow\>}\Lambda^{2}{\mathfrak{g}}^{*}\otimes_{{\mathbb{R}}}B$

by the following formula:

$d_{\nabla}(\theta\otimes b)=d\theta\otimes b-\theta\wedge\nabla b.$ (3.5)

This defines the curvature of an algebraic connection

$\Theta=d^{2}_{\nabla}:B{\>\longrightarrow\>}\Lambda^{2}{\mathfrak{g}}^{*}\otimes
B.$

As in (3.2) one can also write $\Theta=d\omega+\omega\wedge\omega$.

Claim 3.8:  The curvature of a left equivariant connection
$\nabla\\!\\!\\!\\!\nabla$ on a G-equivariant bundle
$\pi:\mathbb{B}{\>\longrightarrow\>}X$ is an ${\mathbb{R}}$-linear map
$\Theta_{\nabla\\!\\!\\!\\!\nabla}:\mathbb{B}{\>\longrightarrow\>}\Lambda^{2}(X)\otimes\mathbb{B}$
given by
$\Theta_{\nabla\\!\\!\\!\\!\nabla}(s):=\mathbb{L}(d^{2}_{\nabla}(s_{e}))$.

We say that bundle $\mathbb{B}$ is flat if
$\Theta_{{\nabla\\!\\!\\!\\!\nabla}}=0$.

Claim 3.9:  Let $\pi:\mathbb{B}{\>\longrightarrow\>}G$ be a left equivariant
vector bundle over a Lie group. Then the bundle is flat if and only if
$\nabla_{X}:{\mathfrak{g}}{\>\longrightarrow\>}\operatorname{End}(B)$ is a Lie
algebra representation.

Proof: Assume that $\nabla:{\mathfrak{g}}{\>\longrightarrow\>}B^{*}\otimes
B\cong\operatorname{End}(B)$ is a representation and the map is given as
follows $X\mapsto\nabla_{X}$. Then for any $X,Y\in{\mathfrak{g}}$ we
have$[\nabla_{X},\nabla_{Y}]-\nabla_{[X,Y]}=0$, hence by 3.1 the curvature
vanishes. Conversely, if $\Theta_{\nabla\\!\\!\\!\\!\nabla}=0$, then
$\Theta_{\nabla}=[\nabla_{X},\nabla_{Y}]-\nabla_{[X,Y]}=0$, hence $\nabla_{X}$
is a representation.

Consider a left-equivariant vector bundle
$\pi:\mathbb{B}{\>\longrightarrow\>}G$ over a Lie group $G$. Let $\Gamma$ be a
discrete subgroup of a Lie group $G$ which acts on $G$ from the left and
$q:G{\>\longrightarrow\>}\Gamma\backslash G$ the quotient map. Denote by
$\mathbb{B}_{\Gamma}$ the induced vector bundle
$\pi_{\Gamma}:\mathbb{B}_{\Gamma}{\>\longrightarrow\>}\Gamma\backslash G$ on a
manifold $\Gamma\backslash G$ obtained by taking the fiberwise quotients by
$\Gamma$ such that the following diagram is commutative

${\mathbb{B}}$${\mathbb{B}_{\Gamma}}$${G}$${{\Gamma\backslash
G}.}$$\scriptstyle{q_{\Gamma}}$$\scriptstyle{\pi}$$\scriptstyle{\pi_{\Gamma}}$$\scriptstyle{q}$

Given a vector bundle $(\mathbb{B},\nabla\\!\\!\\!\\!\nabla)$ with an
equivariant connection $\nabla\\!\\!\\!\\!\nabla$, we denote by
$\nabla\\!\\!\\!\\!\nabla^{\Gamma}$ the induced connection on a bundle
$\mathbb{B}_{\Gamma}$.

###  3.2 Algebraic holonomy group

Recall that there is a bijection between the set of isomorphism classes of
flat vector bundles over $X$ and the set of conjugacy classes of homomorphisms
$\varphi:\pi_{1}(X){\>\longrightarrow\>}\operatorname{GL}({\mathbb{R}}^{n})$
(It is called “the Riemann-Hilbert correspondence”, see e.g. [OV]).

Let $p:E{\>\longrightarrow\>}M$ be a vector bundle over a manifold $M$ and
$\delta:[0,1]{\>\longrightarrow\>}M$ a smooth path in $M$. An equation
$\nabla_{\dot{\delta}(t)}v=0$ defines a parallel transport of a section $v$ of
$E_{|_{\delta(t)}}$ along the curve $\delta$. The parallel transport along a
loop $\delta:[0,1]{\>\longrightarrow\>}M$ such that $x=\delta(0)=\delta(1)$
gives an endomorphism $p_{\delta}\in\operatorname{End}(E_{x})$ of the fiber
$E_{x}$.

Definition 3.10:  Let $\nabla:E{\>\longrightarrow\>}\Lambda^{1}(M)\otimes E$
be a connection on a vector bundle $\pi:E{\>\longrightarrow\>}G$. The holonomy
group of the connection $\nabla$ is the group of linear transformations of a
fiber $E_{x}$ given by all parallel translations along all smooth loops based
at $x$:

$\mathcal{H\\!ol}_{\nabla}=\\{p_{\delta}\,|\,p_{\delta}\in\operatorname{End}(E_{x})\\}.$

Definition 3.11:  The holonomy group of a flat connection $\nabla$ is called
the monodromy group of $\nabla$.

When $G$ is a Lie subgroup $\operatorname{GL}(n,K),K={\mathbb{R}}$ or
${\mathbb{C}}$, its Lie algebra is canonically identified with the
corresponding Lie subalgebra in $\operatorname{Mat}(n,K)$. In this case,
recall that the exponential map $\exp:{\mathfrak{g}}{\>\longrightarrow\>}G$ is
defined as follows

$X\mapsto
e^{tX}=\operatorname{Id}+\displaystyle\frac{t}{1!}X+\frac{t^{2}}{2!}X^{2}+\dots+\frac{t^{k}}{k!}X^{k}+\cdots,\,t\in{\mathbb{R}},X\in{\mathfrak{g}},$
(3.6)

and its local inverse is given by a logarithm

$1+\gamma\mapsto\log(1+\gamma)=\gamma-\frac{\gamma^{2}}{2}+\frac{\gamma^{3}}{3}-\dots+(-1)^{k+1}\frac{\gamma^{k}}{k}+\dots,$
(3.7)

for all $\gamma\in U\subset G$ in a sufficiently small open neighborhood $U$
of the identity $e\in G$.

Let $G$ be a simply connected nilpotent Lie group (it is diffeomorphic to a
Euclidean space ${\mathbb{R}}^{n}$ [VGO, Corollary 2, p. 53]) and $\Gamma$ a
cocompact lattice in $G$, i.e. the left quotient $\Gamma\backslash G$ is a
nilmanifold.

Theorem 3.12:  Let $(\mathbb{B},\nabla\\!\\!\\!\\!\nabla)$ be a flat left
equivariant vector bundle over a nilpotent Lie group $G$, $\Gamma$ a cocompact
lattice and $(\mathbb{B}_{\Gamma},\nabla\\!\\!\\!\\!\nabla^{\Gamma})$ the
corresponding flat bundle on the nilmanifold $\Gamma\backslash G$. Then the
monodromy group $\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}$ of the
connection $\nabla\\!\\!\\!\\!\nabla^{\Gamma}$ is generated by
$e^{\log\gamma}$, where $\gamma\in\Gamma$.

Proof: Let $\delta:[0,1]{\>\longrightarrow\>}G$ be a $1$-parametric subgroup
in a Lie group $G$, such that $\delta(0)=e$, $\delta(1)\in\Gamma$ and tangent
to a vector $X=\log\gamma$. Then the solution of a linear differential
equation $\nabla_{X}b=0$ along the curve $\gamma$ is given by the exponents
$e^{tX}=e^{t\log\gamma}$.

###  3.3 Maltsev completion

In this section, we assume that a group $G$ is a finitely generated torsion-
free nilpotent group.

Definition 3.13: [Mal] A nilpotent group $G$ is called Maltsev complete if for
each $g\in G$ and for all $n\in{\mathbb{Z}}_{>0}$ the equation $x^{n}=g$ has
solutions in $G$.

Definition 3.14: [Mal] Let $\Gamma$ be a subgroup of a Maltsev complete
nilpotent group ${G}$. Then the set $\hat{\Gamma}=\\{g\in
G\,|\,g^{n}\in\Gamma,\,n\in{\mathbb{Z}}\\}\subset G$ is called the Maltsev
completion of a group $\Gamma$. In [Mal], the definition of the Maltsev
completion was given over the field ${\mathbb{Q}}$ but it could be done over
any field $k$ of characteristic zero.

Definition 3.15: [GH] Let $k$ be a field of characteristic zero. The Maltsev
completion functor $\mathscr{M}_{k}$ is a functor from the category of
finitely generated torsion-free nilpotent groups to the category of unipotent
algebraic $k$-groups. If $G$ is a finitely generated nilpotent group, then
$\mathscr{M}_{k}(G):=\overline{\Phi(G)}$, where

$\Phi:G{\>\longrightarrow\>}\operatorname{End}(B)$

is a faithful unipotent representation and $\overline{\Phi(G)}$ is the Zariski
closure of the image, which lies in the subgroup of upper-triangular matrices.

Remark 3.16: We will also denote the rational Maltsev completion of a group
$G$ by $\hat{G}$, as it was introduced in 3.3.

Below we are listing some properties of the Maltsev completion [GH]:

Properties 3.17:

  1. 0.

A connected nilpotent Lie group is Maltsev complete;

  2. 2.

Let $\Gamma$ be a subgroup of a nilpotent torsion-free Lie group $G$. By
definition, $\mathscr{M}_{k}(\Gamma)\subset G$ is a Maltsev complete group.
Its isomorphism class does not depend on $G$ and the embedding of $\Gamma$ to
$G$;

  3. 3.

The Maltsev completion $\mathscr{M}_{k}(\Gamma)$ is a minimal complete
subgroup which contains $\Gamma$;

  4. 4.

The Maltsev completion $\mathscr{M}_{k}(\Gamma)$ is equipped with a natural
structure of an algebraic group over $k$.

Claim 3.18:  The functor $\mathscr{M}_{{\mathbb{Q}}}$ provides a bijection
between the finite-dimensional unipotent representations of $\Gamma$ over
${\mathbb{Q}}$ and the finite-dimensional ${\mathbb{Q}}$-representations of
$\hat{\Gamma}$, considered as an algebraic group:

${\Gamma}$${\hat{\Gamma}}$${\operatorname{End}(B)}$$\scriptstyle{\mathscr{M}_{{\mathbb{Q}}}}$$\scriptstyle{\Phi}$$\scriptstyle{\hat{\Phi}}$

Moreover, the image $\Phi(\Gamma)$ is Zariski dense in
$\hat{\Phi}(\hat{\Gamma})$.

Proof: [GH].

Note that in the context of 3.2, the holonomy group
$\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}$ is isomorphic to the
image of the representation of the lattice subgroup $\Gamma$:

$\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}\cong\Phi(\Gamma).$ (3.8)

The notion of a holonomy group has a geometric nature. We define an algebraic
version of it as follows.

Definition 3.19:  Let $B$ be a vector space, ${\mathfrak{g}}$ a Lie algebra,
and $\nabla:B{\>\longrightarrow\>}{\mathfrak{g}}^{*}\otimes B$ an algebraic
connection (3.1). An algebraic holonomy group $\mathcal{H\\!ol}^{a}_{\nabla}$
is a subgroup of $\operatorname{GL}(B)$ generated by the matrix exponents:

$\mathcal{H\\!ol}^{a}_{\nabla}:=\langle
e^{t\nabla_{X}}\,|\,t\in{\mathbb{R}},\,\text{for
all}\,X\in{\mathfrak{g}}\,\rangle.$

Remark 3.20: We assume that an equivariant connection
$\nabla\\!\\!\\!\\!\nabla$ on a $G$-equivariant vector bundle $B$ is flat. In
this case, by 3.1 the associated algebraic connection defines a representation
of a Lie algebra $X\mapsto\nabla_{X}\in\operatorname{End}(B)$.

Let $G$ be a nilpotent group and $\hat{G}$ its rational Maltsev completion,
which is a rational algebraic group, i.e. it can be written as
$\hat{G}=\operatorname{Spec}(A)$, where $A$ is its ring of regular functions.

Remark 3.21:  In 3.3 we defined the functor of Maltsev completion
$\mathscr{M}_{k}$ with coefficients in a field $k$. The real Maltsev
completion $\mathscr{M}_{{\mathbb{R}}}$ can be equivalently defined as
follows:

$\hat{G}\otimes_{{\mathbb{Q}}}{\mathbb{R}}:=\operatorname{Spec}(A\otimes_{{\mathbb{Q}}}{\mathbb{R}})=\mathscr{M}_{{\mathbb{R}}}(G).$

Theorem 3.22:  Let $(\mathbb{B},\nabla\\!\\!\\!\\!\nabla)$ be a flat left
equivariant vector bundle over a nilpotent Lie group $G$, $\Gamma\subset G$ a
cocompact lattice, and
$(\mathbb{B}_{\Gamma},\nabla\\!\\!\\!\\!\nabla^{\Gamma})$ the induced bundle
over the nilmanifold $\Gamma\backslash G$. Suppose that
$\nabla:B{\>\longrightarrow\>}{\mathfrak{g}}^{*}\otimes B$ is the algebraic
connection associated with $\nabla\\!\\!\\!\\!\nabla$. Assume that the
monodromy representation
$\Phi:\Gamma{\>\longrightarrow\>}\operatorname{End}(B)$ is unipotent.

Then

$\overline{\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}}={\mathcal{H\\!ol}}^{a}_{\nabla},$
(3.9)

where
$\overline{\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}}\subseteq\operatorname{GL}(B)$
denotes the Zariski closure of the monodromy group of
$(\mathbb{B}_{\Gamma},\nabla\\!\\!\\!\\!\nabla^{\Gamma})$.

Proof: Consider the chain of embeddings

$\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}\subseteq\widehat{\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}}\subseteq\widehat{\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}}\otimes_{{\mathbb{Q}}}{\mathbb{R}},$
(3.10)

where $\widehat{\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}}$ is the
rational Maltsev completion of the group
$\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}$.

Notice that
$\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}\cong\Phi(\Gamma)$ is
Zariski dense in
$\widehat{\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}}$ by [BrVer,
Lemma 3.11] and
$\widehat{\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}}$ is Zariski
dense in
$\widehat{\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}}\otimes_{{\mathbb{Q}}}{\mathbb{R}}$.
Hence,

$\overline{\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}}=\widehat{\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}}\otimes_{{\mathbb{Q}}}{\mathbb{R}}.$
(3.11)

By 3.3, the algebraic holonomy group
${\mathcal{H\\!ol}}^{a}_{\nabla}\subseteq\widehat{\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}}$
is a real group, which also contains
$\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}$. Therefore, it has to
be isomorphic to
$\widehat{\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}}\otimes_{{\mathbb{Q}}}{\mathbb{R}}$,
because of the minimality of the real Maltsev completion.

Corollary 3.23:  Let $\Gamma\backslash G$ be a hypercomplex nilmanifold with
the flat Obata conection $\nabla^{Ob}$ (1.1) on $TG$. Then the action of the
algebraic holonomy on ${\mathfrak{g}}=\operatorname{Lie}(G)$ is unipotent.

Proof: Let $\nabla$ be a flat algebraic connection on the Lie algebra
${\mathfrak{g}}=T_{e}G$, associated with $\nabla^{Ob}$. Define the action
$\mathcal{H\\!ol}^{a}_{\nabla}\times{\mathfrak{g}}{\>\longrightarrow\>}{\mathfrak{g}}$
on ${\mathfrak{g}}$ as follows:

$(e^{\nabla_{X}},Y)\mapsto e^{\nabla_{X}}\cdot Y,$ (3.12)

where $X,Y\in{\mathfrak{g}}$. By 1.2, the monodromy representation
$\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}$ is unipotent. From 3.3
it follows that $\mathcal{H\\!ol}_{\nabla\\!\\!\\!\\!\nabla^{\Gamma}}$ is
Zariski dense in ${\mathcal{H\\!ol}}^{a}_{\nabla}$. Hence
${\mathcal{H\\!ol}}^{a}_{\nabla}$ also unipotent.

###  3.4 Unipotent holonomy group and ${\mathbb{H}}$-solvability

Let $(G,I,J,K)$ be a nilpotent Lie group with a left-invariant hypercomplex
structure. In what follows we denote by $\nabla$ the Obata connection (1.1),
which we assumed to be flat, by $\mathcal{H\\!ol}_{\nabla}$ and
$\mathcal{H\\!ol}^{a}_{\nabla}$ its holonomy (monodromy) and algebraic
holonomy groups respectively.

Let ${\mathfrak{g}}$ be a nilpotent hypercomplex Lie algebra. Consider the
smallest $\mathbb{H}$-invariant subspace of ${\mathfrak{g}}$ containing the
commutator ideal $[{\mathfrak{g}},{\mathfrak{g}}]$:

${\mathfrak{g}}_{1}^{\mathbb{H}}:={\mathbb{H}}[{\mathfrak{g}},{\mathfrak{g}}]=[{\mathfrak{g}},{\mathfrak{g}}]+I[{\mathfrak{g}},{\mathfrak{g}}]+J[{\mathfrak{g}},{\mathfrak{g}}]+K[{\mathfrak{g}},{\mathfrak{g}}].$

Since ${\mathbb{H}}[{\mathfrak{g}},{\mathfrak{g}}]$ contains the commutator
ideal, it is an ideal, hence a subalgebra of ${\mathfrak{g}}$.

Define inductively the ${\mathbb{H}}$-invariant Lie subalgebras:

${\mathfrak{g}}_{i}^{{\mathbb{H}}}:={\mathbb{H}}[{\mathfrak{g}}_{i-1}^{{\mathbb{H}}},{\mathfrak{g}}_{i-1}^{{\mathbb{H}}}],i\in{\mathbb{Z}}_{\geqslant
2}.$

Claim 3.24:  Let $({\mathfrak{g}},I,J,K)$ be a hypercomplex nilpotent Lie
algebra with the flat Obata connection. Then
$\nabla_{X}Y\in{\mathfrak{g}}^{{\mathbb{H}}}_{i+1}$ for any
$X,Y\in{\mathfrak{g}}^{{\mathbb{H}}}_{i}$.

Proof: Suppose that $X,Y\in{\mathfrak{g}}^{{\mathbb{H}}}_{i}$. Then by (1.1)
$\nabla_{X}Y\in{\mathfrak{g}}^{{\mathbb{H}}}_{i+1}$.

For each $i\in{\mathbb{Z}}_{\geqslant 0}$,
$X\in{\mathfrak{g}}^{{\mathbb{H}}}_{i}$, the elements
$e^{\nabla_{X}}\in\operatorname{End}({{\mathfrak{g}}^{{\mathbb{H}}}_{i}})$
generate the algebraic holonomy group, associated to the Lie subalgebra
${\mathfrak{g}}^{{\mathbb{H}}}_{i}$:

$\mathcal{H\\!ol}^{a}_{\nabla^{i}}=\langle
e^{t\nabla_{X}}\,|\,t\in{\mathbb{R}},\,X\in{\mathfrak{g}}^{{\mathbb{H}}}_{i}\rangle.$

Remark 3.25:  Notice that $\mathcal{H\\!ol}^{a}_{\nabla^{i}}$ acts on
${\mathfrak{g}}^{{\mathbb{H}}}_{i}$ as in (3.12) and it could happen that it
is not a subgroup or even a subset of the algebraic holonomy group
$\mathcal{H\\!ol}^{a}_{\nabla}$. Indeed, the action of $\nabla_{X}$ on
${\mathfrak{g}}_{i}^{{\mathbb{H}}}$ could be trivial, but act non-trivially on
${\mathfrak{g}}$ for some $X\in{\mathfrak{g}}^{{\mathbb{H}}}_{i}$. However,
the action of $\mathcal{H\\!ol}^{a}_{\nabla^{i}}$ on
${\mathfrak{g}}^{{\mathbb{H}}}_{i}$ coincides with the action of the
corresponding subgroup of $\mathcal{H\\!ol}^{a}_{\nabla}$. Therefore, it is
unipotent on ${\mathfrak{g}}^{{\mathbb{H}}}_{i}$ if
$\mathcal{H\\!ol}^{a}_{\nabla}$ is unipotent.

The following theorem is the main result of this paper.

Theorem 3.26: Let $({\mathfrak{g}},I,J,K)$ be a hypercomplex nilpotent Lie
algebra with the flat Obata connecton. Assume that the algebraic monodromy
representation is unipotent. Then ${\mathfrak{g}}$ is ${\mathbb{H}}$-solvable.

Proof: The action of $\mathcal{H\\!ol}^{a}_{\nabla^{i}}$ on the Lie subalgebra
${\mathfrak{g}}^{{\mathbb{H}}}_{i}$ defined as in (3.12). By the assumption,
the representation of the algebraic holonomy on ${\mathfrak{g}}$ is unipotent,
hence, by 3.4, the action of $\mathcal{H\\!ol}^{a}_{\nabla^{i}}$ is unipotent
on the ${\mathfrak{g}}^{{\mathbb{H}}}_{i}$.

Let
$\varphi:\mathcal{H\\!ol}^{a}_{\nabla^{i}}{\>\longrightarrow\>}\operatorname{End}(({\mathfrak{g}}^{{\mathbb{H}}}_{i})^{*})$
be the action of the algebraic holonomy group on the dual Lie algebra
$({\mathfrak{g}}_{i}^{{\mathbb{H}}})^{*}$. Every unipotent representation of a
Lie algebra has an invariant vector. This implies the existence of a non-zero
Obata-parallel 1-form for
$\alpha\in\Lambda^{1}({\mathfrak{g}}_{i}^{{\mathbb{H}}})^{*}$. Again, we
consider the holonomy action on $({\mathfrak{g}}_{i}^{{\mathbb{H}}})^{*}$ and
for each $i$ we obtain a non-zero parallel (hence, closed111For any torsion-
free connection $\nabla$ and any 1-form $\beta$ we have the equality
$d\beta=\rm{Alt}(\nabla\beta)$, where $\rm{Alt}$ is the skew-symmetrization
map.) $1$-form
$\alpha_{i}\in\Lambda^{1}({\mathfrak{g}}_{i}^{{\mathbb{H}}})^{*}$.

The intersection of kernels $\Sigma_{\alpha_{i}}=\ker\alpha_{i}\cap\ker
I\alpha_{i}\cap\ker J\alpha_{i}\cap\ker K\alpha_{i}$ gives an
${\mathbb{H}}$-invariant foliation, which contains
${\mathfrak{g}}^{{\mathbb{H}}}_{i}$ as a proper subspace (2). This implies
${\mathfrak{g}}_{i+1}^{{\mathbb{H}}}\subsetneq{\mathfrak{g}}^{{\mathbb{H}}}_{i}$.
The sequence
${\mathfrak{g}}\subsetneq{\mathfrak{g}}_{1}^{{\mathbb{H}}}\subsetneq\dots$
terminates in finitely many steps since ${\mathfrak{g}}$ is finite-
dimensional.

Corollary 3.27: Let $(N,I,J,K)$ be a hypercomplex nilmanifold with flat Obata
connection $\nabla$. Then the Lie algebra
${\mathfrak{g}}=\operatorname{Lie}(G)$ is ${\mathbb{H}}$-solvable.

Proof: By 3.3, the representation of the algebraic holonomy on
${\mathfrak{g}}$ is unipotent.

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Yulia Gorginyan
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil
also:
Laboratory of Algebraic Geometry,
National Research University (HSE),
Department of Mathematics, 6 Usacheva Str.
Moscow, Russia
<EMAIL_ADDRESS>


# Learning from 2D: machine learning of 3D effective properties of
heterogeneous materials based on 2D microstructure sections

Guangyu Hu  Marat I. Latypov<EMAIL_ADDRESS>
###### Abstract

Microstructure–property relationships are key to effective design of
structural materials for advanced applications. Advances in computational
methods enabled modeling microstructure-sensitive properties using 3D models
(e.g., finite elements) based on microstructure representative volumes. 3D
microstructure data required as input to these models are typically obtained
from either 3D characterization experiments or digital reconstruction based on
statistics from 2D microstructure images. In this work, we present machine
learning (ML) approaches to modeling effective properties of heterogeneous
materials directly from 2D microstructure sections. To this end, we consider
statistical learning models based on spatial correlations and convolutional
neural networks as two distinct ML strategies. In both strategies, models are
trained on a dataset of synthetically generated 3D microstructures and their
properties obtained from micromechanical 3D simulations. Upon training, the
models predict properties from 2D microstructure sections. The advantage of
the presented models is that they only need 2D sections, whose experimental
acquisition is more accessible compared to 3D characterization. Furthermore,
the present models do not require digital reconstruction of 3D
microstructures.

###### keywords:

Homogenization theories , Two-phase composites , Effective properties ,
Machine learning

††journal: Frontiers in Metals and Alloys

This is a postprint of the article: G. Hu & M.I. Latypov, Learning from 2D:
Machine learning of 3D effective properties of heterogeneous materials based
on 2D microstructure sections, Frontiers in Metals and Alloys (2022).
DOI:10.3389/ftmal.2022.1100571

##  1 Introduction

Multiphase alloys and metal-matrix composites constitute an important class of
structural materials [1, 2, 3]. The presence of two or more phases allows
unique combinations of properties inaccessible to single-phase materials.
Accelerated design of multiphase materials and their process optimization
could benefit from efficient computational tools that can accurately capture
the relationships between mulitphase microstructure and overall engineering
properties.

In this context, a number of modeling approaches and strategies have been
developed to date. Historically earliest, the Voigt and Reuss models [4, 5]
estimated the overall or _effective_ properties of composites from the volume
fractions of their constituents. These models are based on the assumptions of
either uniform stress or uniform strain throughout the composite and serve as
the lower and upper bounds of the effective property of interest (e.g.,
stiffness or strength). While straightforward, these bounds are often widely
separated, especially in _high-contrast_ composites with a large difference in
the properties between the constituents [6]. Tighter bounds for effective
elastic properties were obtained by Hashin and Shtrikman for composites with
moderate elastic contrasts [7]. Self-consistent models also offer better
estimates of effective properties compared to bounds. Most of these
approaches, however, are based on volume fractions of the constituents, which
limits their use for _microstructure-sensitive_ design of composite materials
and structural components [8].

With the emergence of high-performance computing, computational homogenization
has become a viable and powerful approach to calculating effective properties
of heterogeneous materials [9]. The finite element (FE) methods [10, 11, 12,
6] and fast Fourier solvers [13, 14, 15, 16] both allow calculating effective
elastic and inelastic properties with the direct account for three-dimensional
(3D) microstructure. The microstructure effects are explicitly incorporated by
performing calculations on 3D representative volume elements (RVEs) or
microstructure volume elements (MVEs) of the multiphase microstructure of the
material. The account for microstructure is associated with a significant
increase in the computational cost compared to analytical calculations of
bounds and faster numerical self-consistent methods discussed above. The
computational cost of RVE-based methods is still prohibitively high for
routine use in multiscale modeling of commercial metal forming processes.

Surrogate or reduced-order models have been proposed as a strategy to address
the trade-off between the computational cost and incorporation of 3D
microstructure effects. Materials Knowledge Systems (MKS) is an example
computational approach of surrogate model development [17, 18]. In MKS, 3D
microstructures are quantified using the $n$-point spatial statistics with
subsequent establishment of quantitative microstructure–property relationships
in the form of polynomial functions fitted to data from numerical (e.g., FE)
simulations. Other forms of microstructure–property relationships besides
polynomial functions are also seen in literature, including statistical
learning models (e.g., Gaussian process regression [19]), or neural networks,
including convolutional neural networks, CNN [20, 21, 22, 23] and graph neural
networks [24, 25]. Owing to the computational efficiency and account for 3D
microstructure, surrogate models offer a promising pathway towards practical
implementation and industrial adoption of microstructure-sensitive multiscale
models of full-scale metal forming processes.

Both the computational homogenization methods and their data-driven surrogates
require three-dimensional (3D) microstructure data in the form of an RVE as
input for property calculations. The RVE needs to have sufficiently high
resolution and sufficiently large size to capture both the 3D morphology and
spatial distribution of the microstructure constituents (continuous phases,
particles, or grains). Advances in experimental characterization have made it
possible to obtain 3D microstructure datasets for a wide variety of materials.
Experimental 3D datasets are typically obtained from serial sectioning [26,
27] or directly using non-desctructive 3D characterization techniques [28,
29]. Most of the experimental techniques that yield high-quality 3D
microstrcuture data require access to high-cost and unique experimental
facilities. Consequently high-quality 3D experimental data are currently
scarce in the materials research community, especially in industry. An
alternative to 3D characterization is digital generation of 3D microstructure
RVEs or MVEs based on 2D microstructure data [30]. For example, Dream.3D is an
open-source software commonly used for 3D microstructure generation based on
statistical distributions of sizes, shapes, and volume fractions of the
microstructure constituents [31]. Dream.3D has been widely used for building
RVEs as input for microstructure-based simulations [32, 33]. Recently, new
approaches have been emerging for 3D RVE generation from 2D microstructure
data, including a 3D generation algorithm inspired by solid texture synthesis
in computer graphics [34] or transfer learning [35].

In this work, we propose a machine learning framework that relates the
microstructure from 2D sections directly to effective 3D properties of
heterogeneous materials and specifically two-phase composites. The framework
is based on a hypothesis that 2D microstructure data encapsulates sufficient
statistical information to capture the effective properties of heterogeneous
media to a satisfactory extent. We explore the application of this framework
to modeling effective mechanical properties of two-phase materials using two
approaches: (i) statistical learning with spatial correlations as
microstructure descriptors and (ii) deep learning with CNNs as microstructure
feature extractors. Section 2 describes these two approaches followed by their
application on stiffness of two-phase microstructures presented and discussed
in Sections 3–5.

Figure 1: Network architecture proposed in the multisection CNN approach.

##  2 New Multisection modeling framework

The key distinguishing feature of the computational framework developed in
this study is the extraction of quantitative microstructure features from 2D
orthogonal microstructure sections, their aggregation and use for machine
learning models for effective properties. To this end, we develop and
critically compare two approaches. In the first approach, which we refer to as
the _Multisection MKS approach_ , we use $n$-point correlation functions and
principal component analysis (PCA) to engineer microstructure features from 2D
orthogonal sections. In the second, _Multisection CNN approach_ , we train
CNNs as feature extractors for microstructure images. In both cases, features
extracted from individual 2D sections are combined into a single feature
vector for establishing quantitative relationships between microstructures and
effective properties of interest.

###  2.1 Multisection MKS approach

As in the original developments of the MKS framework [36], our Multisection
MKS approach relies on the $n$-point correlation functions as a rigorous
statistical description of heterogeneous microstructures. The theoretical
basis for the use of $n$-point statistics as microstructure description in
homogenization models is the classical statistical continuum theories, where
an effective property is expressed as a volume-averaged quantity with
corrections to higher-order microstructure effects represented by $n$-point
correlation functions of increasing order with corresponding influence
coefficients [17, 37, 38, 39, 40]. Most of the prior MKS models were
successfully developed based on two-point ($n=2$) correlation functions [17,
33, 36, 41] together with PCA as a dimensionality reduction technique utilized
to facilitate regression-type model calibration. In our framework, two-point
correlation functions are computed individually from 2D orthogonal sections
and then combined into a single vector prior to the application of PCA for
dimensionality reduction. PCA is a linear transformation of multidimensional
data to a new basis represented by orthogonal vectors corresponding to
sequentially decreasing variance in the dataset [42]. When PCA is applied to
two-point statistics, principal component basis vectors can be interpreted as
patterns in two-point correlation maps, in respect to which the spatial
correlations vary the most in a set of microstructures [33, 43]. Once a new
basis is established with patterns as basis vectors, the weights of these
patterns are obtained as principal component scores for two-point correlations
of microstructures for which prediction of the effective properties are
sought. Prior MKS studies focused on 3D microstructures involved calculations
of directionally-resolved 3D two-point statistics so that PCA provided 3D
patterns in two-point statistics [33, 36, 41]. In the present computational
framework, PCA yields sets of three 2D patterns in two-point statistics that
correspond to the three orthogonal sections of the microstructures. Principal
component scores, which represent weights of these patterns are used as
features describing microstructure for machine learning of effective
properties. Since PCA automatically ranks patterns according to the variance
in two-point statistics, usually only a few first principal components are
used for statistical learning.

###  2.2 Multisection CNN approach

Deep learning has been shown promising for predicting effective properties of
heterogeneous microstructures. For modeling relationships between 3D
microstructures and properties, 3D CNN architectures have been used [21]. 3D
CNNs serve as extractors of microstructure features relevant for the property
of interest. In the present Multisection CNN approach, we use 2D CNNs for
microstructure feature extraction from three orthogonal 2D sections. The
features obtained from the individual sections using separate CNNs are then
combined into a single microstructure feature vector for establishing a
quantitative linkage with the property of interest. Drawing inspiration from
multi-view CNN methods for classification of 3D objects from 2D images taken
at different angles [44], we employ view pooling for feature aggregation from
the 2D sections. An example architecture of the Multisection CNN is shown in
Figure 1 with two 2D convolutional layers followed by concatenation and
average pooling of the features. A linear regression layer concludes the
architecture for relating features from the Multisection CNN to a property of
interest. While aggregation of features from individual 2D sections of the
microstructure is key to our Multisection CNN approach, the other parameters
of the architecture (e.g., number of convolutional layers) can be selected and
tuned for each specific problem and dataset at hand.

##  3 Case study: elastic properties of two-phase microstructures

We demonstrate and evaluate the new framework in a case study of effective
elastic properties in high-contrast two-phase microstructures. To this end, we
leverage a dataset previously published by Cecen et al. [21]. The dataset
includes 5900 two-phase microstructures and their corresponding effective
Young’s modulus values obtained using FE simulations. Periodic 3D MVEs of the
microstructures measured $51\times 51\times 51$ voxels and had a volume
fraction of the hard phase varied from 25% to 75%. The contrast of Young’s
modulus between hard and soft phases was 50, i.e., the hard phase was 50 times
stiffer than the soft phase. Figure 2 shows microstructures typical of the
dataset. Using this dataset, we trained two new Multisection models (MKS and
CNN) as well as a 3D MKS model used as a benchmark for evaluation of the new
Multisection models. For all three models, the dataset of 5900 MVEs and their
properties was split into a training subset (90%) and a testing subset. The
testing subset was held out during fitting of the MKS models and training of
the CNN model.

Figure 2: Typical 3D MVEs of two-phase microstructures contained in the
dataset used for training and testing the multisection MKS and CNN models in
the present study. Three orthogonal sections of the MVEs and the corresponding
two-point autocorrelation maps of the stiff phase are also shown.

###  3.1 Multisection MKS

The application of the Multisection MKS approach in this case study included
computation of two-point correlation functions on 2D microstructure sections
and PCA of the two-point statistics vectors aggregated from the three 2D
sections. For two-phase microstructures, a single two-point correlation
function is sufficient to fully describe the microstructure [45] so that we
only calculated autocorrelation for the stiff phase using an efficient fast
Fourier transform-based algorithm [46]. Figure 2 shows two-point correlation
maps for 2D sections of MVEs representative of the dataset. The three
concatenated 2D autocorrelations for each MVE results in a high dimensional
feature vector with $3\times 50\times 50=7500$ elements. To facilitate machine
learning, we reduced the dimensionality of the aggregated autocorrelation
vector using the PCA. The selected principal component scores were then used
as the final features for building a statistical learning model. In this case
study, we included the first 13 principal components in a third-order
polynomial function because this combination of principal components and
polynomial order showed good performance for modeling effective stiffness in
high-contrast two-phase microstructures [21].

###  3.2 Multisection CNN

Our Multisection CNN model relies on CNNs for extracting microstructure
features from 2D sections. Our network architecture includes three parallel
branches that take 2D image as input and then pass it through two 2D
convolutional layers followed by a rectified linear layer and average pooling
layer. The features produced by these three parallel channels are then further
averaged to generate a single 6400-dimensional vector of features representing
three orthogonal sections of the microsructure. We obtained this specific
architecture by experimentation with different network configurations and by
hyperparameter optimization. Bayesian optimization (on SigOpt platform [47])
was used to identify the optimal number of convolutional layers, the number of
filters in the convolutional layer(s), the kernel size of the convolutional
layer(s), the pooling size, batch size, learning rate, number of for the final
dense layer, the final pooling method (mean or max), optimizer (Adam or
stochastic gradient descent). Figure 1 shows the architecture obtained by
hyperparameter optimization, which includes two convolutional layers with 16
filters of $10\times 10$ kernel size in the first layer and 256 filters with
$4\times 4$ kernel size in the second layer, followed by an average pooling
layer of $8\times 8$ size. We employed mean absolute error (MAE) as the loss
function during training with a learning rate of $10^{-4}$. Early stopping was
introduced and controlled by decrease of less than $1\text{\,}\mathrm{\char
37\relax}$ in the MAE on portion of the training data. The learning curves for
the model is shown in Figure 3. The Multisection CNN models were trained and
tuned on the Ocelote cluster of the UArizona High Performance Computing system
with the following specifications: two Xeon E5 14-core processors (28 CPUs
total) with $2.3\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ and
$6\text{\,}\mathrm{G}\mathrm{B}$ RAM per CPU.

Figure 3: Learning curves corresponding to training of the Multisection CNN
model adopted in this study in terms of mean absolute percentage error for
training and validation sets.

###  3.3 Benchmark 3D MKS

We additionally built a 3D MKS model which served as a benchmark for critical
evaluation of the new Multisection models. We developed the benchmark 3D MKS
model using a procedure similar to the Multisection MKS model. The key
difference of the 3D MKS model from the Multisection MKS was that we
calculated full 3D two-point correlation functions for entire volumes of the
3D MVEs. In this case, we obtained the microstructure features for machine
learning by PCA of 3D two-point correlations vectors. For the benchmark 3D MKS
model, we also fitted a third-order polynomial including 13 principal
components of the 3D two-point correlations.

##  4 Results

The key results of the case study with the two new models – Multisection MKS
and Multisection CNN – are shown in the form of parity plots in Figure 4b–c.
The performance of the two models is compared to the benchmark 3D MKS model
(Figure 4a). The parity plot depicts predictions of the developed machine
learning models against the ground-truth FE simulation results. Each point
represents a prediction and ground truth result for one MVE; a perfect
agreement would be seen as points lying on the parity line. All the models
produce a reasonable agreement between the trained machine learning models and
the FE simulations. Indeed, the predictions of all three models show a
reasonable homoscedastic distribution along the parity line. The Multisection
MKS model predictions are characterized by a wider scatter compared to the 3D
MKS model, which is the “price” of relying on 2D microstructure information
rather than full 3D data. The Multisection CNN model also shows a wider
scatter compared to the benchmark model but to a smaller extent than the
Multisection MKS model. The quantitative comparison of the MAE normalized by
ground truth values supports these observations: the Multisection MKS model
has the highest MAE of $26.6\text{\,}\mathrm{\char 37\relax}$ vs.
$13.1\text{\,}\mathrm{\char 37\relax}$ of the benchmark 3D MKS model, while
the MAE of the Multisection CNN is $19.2\text{\,}\mathrm{\char 37\relax}$ (all
for the testing set).

Figure 4: Parity plots showing the fitting results (for training set, dark
grey) and the prediction results (for testing set, blue) for (a) benchmark 3D
MKS model based on 3D microstructures; (b) Multisection MKS and (c)
Multisection CNN models based on three 2D microstructure sections. Mean
absolute errors (MAE) for the testing set normalized by ground trugh values
are also shown.

##  5 Discussion

The results of the case study presented above show the feasibility of machine
learning models that relate effective properties of two-phase microstructures
directly to 2D microstructure sections. The value of the developed
Multisection models arises from the prevalence of 2D microstructure data in
the materials community, especially in industry, compared to scarce 3D data.
The transition from full 3D microstructure data to 2D sections used for the
MKS model development comes at the price of about $13\text{\,}\mathrm{\char
37\relax}$ loss in accuracy. The trade-off between the completeness of the
microstructure information and the accuracy can be addressed by more advanced
models such as CNNs: we report only a $6\text{\,}\mathrm{\char 37\relax}$
increase in MAE with the Multisection CNN compared to the MKS model based on
full 3D microstructure information. This sacrifice in accuracy can be
reasonable in many use case scenarios given the significant savings in time,
labor, and resources needed to collect 2D section data compared to 3D
characterization. An additional advantage of the best performing Multisection
CNN is manifested in the computational cost of training: it takes only about
$12\text{\,}\min$ to train or Multisection CNN with CPUs (hardware specified
in Section 3.1.2), while previous studies reported hours of training for 3D
CNNs on the same dataset of high-contrast composites [21]. The Multisection
CNN model developed here is thus a light-weight data-driven model which
strikes a balance between minimal requirements in terms of microstructure
input (2D sections), decent accuracy, and modest computational resources and
time needed for training.

The Multisection models presented here show a viable strategy of modeling
effective properties of heterogeneous materials without the need of
experimental 3D microstructure data or an extra computational step of
reconstructing 3D micrsotructures from 2D data. The models trained on 2D
microstructure sections from 3D synthetic microstructures used in the
simulations can be leveraged for predicting properties of materials based on
experimental microstructure images. While we demonstrated the predictive
capabilities of the Multisection framework on effective elastic properties, it
can be readily extended to effective inelastic properties of multiphase
materials [18] as well as elastic and inelastic properties of polycrystalline
materials [41]. Indeed, CNNs can be configured to take crystal orientations as
input (e.g., [48]), while calculation of two-point correlations for
polycrystalline MVEs within the MKS framework is also available using
generalized spherical harmonics [41] as the microstructure description basis.
Finally, in this study, we trained a relatively simple architecture from
scratch to test the concept, however, use and fine-tuning of existing state-
of-the-art models (such as ResNet) as the base CNN architecture for each 2D
section can futher improve the results and accuracy of the Multisection CNN
approach.

##  6 Summary

We presented a new Multisection machine learning framework for predicting
effective properties of heterogeneous materials. The framework is based on the
hypothesis that the microstructure data from 2D sections suffice for
establishing a quantitative relationship between the microstructure and the
effective property of interest. We tested the hypothesis in a case study of
predicting effective stiffness in high-contrast two-phase composite
microstructures. To this end, we tested two computational approaches:
Mutisection MKS and Multisection CNN. Both approaches rely on aggregating
microstructure features extracted from three 2D sections. In the case of
Multisection MKS, we obtain microstructure features by principal component
analysis of two-point correlation functions calculated for 2D sections. In the
case of Multisection CNN, we utilize CNN filters as feature extractors with
subsequent pooling of features from individual 2D sections. We show that both
the Multisection MKS and Multisection CNN models show reasonable accuracy with
$13\text{\,}\mathrm{\char 37\relax}$ and $6\text{\,}\mathrm{\char 37\relax}$
increase in the mean average error compared to the previously published MKS
model relying on full 3D microstructure data. Our results confirm the
hypothesis and demonstrate the Multisection CNN approach as an optimal model
for predicting effective properties of heterogeneous media without the need to
collect or reconstruct 3D microstructure volumes. The Multisection CNN is
therefore a powerful addition to the computational materials science toolkit
for modeling microstructure–property relationships.

## Acknowledgments

Our machine learning models were trained and tuned using High Performance
Computing (HPC) resources supported by the University of Arizona TRIF, UITS,
and the Office of Research, Innovation, and Impact, and maintained by the
UArizona Research Technologies Department.

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