# BGG sequences with weak regularity and applications

Andreas Čap and Kaibo Hu  Faculty of Mathematics, University of Vienna, Oskar-
Morgenstern-Platz 1, 1090 Wien, Austria<EMAIL_ADDRESS>Mathematical
Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory
Quarter, Oxford, OX2 6GG, UK<EMAIL_ADDRESS>
(Date: .)

###### Abstract.

We investigate some Bernstein-Gelfand-Gelfand (BGG) complexes on bounded
Lipschitz domains in $\mathbb{R}^{n}$ consisting of Sobolev spaces. In
particular, we compute the cohomology of the conformal deformation complex and
the conformal Hessian complex in the Sobolev setting. The machinery does not
require algebraic injectivity/surjectivity conditions between the input
spaces, and allows multiple input complexes. As applications, we establish a
conformal Korn inequality in two space dimensions with the Cauchy-Riemann
operator and an additional third order operator with a background in Möbius
geometry. We show that the linear Cosserat elasticity model is a Hodge-
Laplacian problem of a twisted de-Rham complex. From this cohomological
perspective, we propose potential generalizations of continuum models with
microstructures.

###### Contents

  1. 1 Abstract framework
    1. 1.1 The basic setup
    2. 1.2 Twisted complex
    3. 1.3 The BGG construction
    4. 1.4 Cohomology of the BGG sequence
  2. 2 Examples
    1. 2.1 The structure of the examples
    2. 2.2 Conformal Hessian complex
    3. 2.3 Conformal deformation complex
    4. 2.4 Higher order generalizations of the Hessian complex
  3. 3 Algebraic and geometric background
  4. 4 Conformal Korn inequalities in 2D
  5. 5 Continuum with microstructures
    1. 5.1 Linear Cosserat model as a Hodge-Laplacian
    2. 5.2 Potential generalizations
  6. 6 Conclusion

The origin of Bernstein-Gelfand-Gelfand (BGG) resolutions lies in
representation theory, where they are resolutions of finite dimensional
irreducible representations of semisimple Lie algebras by homomorphisms of
Verma modules. The original construction by I.N. Bernstein, I.M. Gelfand and
S.I. Gelfand was generalized by J. Lepowski to a resolution by homomorphisms
of so-called parabolic Verma modules. There is a duality between homomorphisms
between such modules and invariant differential operators acting on sections
of homogeneous vector bundles over generalized flag manifolds. This relation
was used in the book [14] to study the Penrose transform. A direct
construction of the resolutions in this dual picture, i.e. in terms of
differential operators was given in [21]. Representation theory still is an
important ingredient to this construction, but only finite dimensional
representations are needed in this case. This construction actually applies
not only to generalized flag manifolds but also to manifolds endowed with
curved geometries modelled on these homogeneous spaces, so-called parabolic
geometries, see [20]. In this more general setting, the construction does not
provide complexes, but it is important as a conceptual construction of
differential operators intrinsic to parabolic geometries, whose existence was
not established in general before.

For the purpose of the current article, the strong invariance properties of
the construction are not as important. In the setting of parabolic geometries,
the complexes we consider are related either to projective differential
geometry or to conformal differential geometry, but we only apply them to the
case of the geometries determined by the flat connection respectively the flat
Riemannian metric on open subsets of $\mathbb{R}^{n}$. What is important for
us is that one obtains an explicit construction of the relevant complexes from
the de-Rham complex of differential forms with values in an appropriate finite
dimensional vector space. Together with known results on the de-Rham complex,
this allows us to obtain analytical results which are important for the use of
BGG complexes in numerical analysis and applied mathematics.

The applications of the BGG construction in numerical analysis were motivated
by the construction of finite elements for linear elasticity (the Hellinger-
Reissner principle) [7, 12]. For elasticity problems, the stress tensor and
load fit in the so-called elasticity complex (Riemannian deformation complex),
which is a special case of the BGG complexes [7, 39, 40]. For numerical
stability (inf-sup conditions [13, 19]) and approximation properties, it is
desirable that the finite element spaces for the stress and load fit in a
discrete elasticity complex. This perspective inspired the construction of the
Arnold-Winther element [12], which solved a decades-long problem.

Mathematical and numerical analysis of PDE models often involves broader
classes of function spaces, e.g., Sobolev spaces. Analytic issues and
applications of the complexes, particularly the elasticity complex, were
discussed by several authors [1, 2, 33, 44, 55, 56, 62]. Inspired by the BGG
construction, Arnold and Hu [11] systematically derived a comprehensive list
of complexes and established their algebraic and analytic properties for a
large class of function spaces. Recently, there has been a surge of interests
in discretizing these BGG complexes [3, 22, 23, 24, 25, 26, 27, 29, 47, 48,
58]. Not only these BGG complexes, but also the “BGG diagrams” for deriving
these complexes are important. To discretize elasticity with standard “finite
element differential forms” [8, 10], Arnold, Falk and Winther [9] proposed a
scheme which imposes the symmetry of tensors weakly using Lagrange
multipliers. Diagram chasing on the BGG diagrams plays an important role in
this work. Babuška and Brezzi theories require that the discrete spaces in the
BGG diagram should not be arbitrary but rather satisfy some algebraic
conditions that are similar to the continuous level [11]. If finite element
spaces are perfectly matched in the BGG diagrams, the schemes with weakly
imposed symmetry imply strong symmetry [45]. Diagram chasing on the BGG
diagram also leads to null-homotopy operators for the elasticity complex [30]
and provides a constructive tool for deriving finite elements and complexes
[27, 28, 29].

However, the simplification in the geometric and algebraic setting in [11]
left several questions open. Firstly, examples in [11] include the conformal
deformation complex and the conformal Hessian complex which have important
applications in general relativity and continuum mechanics (c.f., Beig and
Chrusciel [15] on parameterizing the Einstein constraint equations using
differential complexes, where the “conformal complex” and the “scalar complex”
correspond to the conformal deformation complex and the conformal Hessian
complex, respectively). The cohomology of these complexes was left open in
[11]. Secondly, [11] shows that analytic results, e.g., various Poincaré or
Korn inequalities, are deeply rooted in the algebraic structures. However, the
conformal Korn inequality in two space dimensions remain open as the diagram
does not fulfill the assumptions in [11] (c.f., [37] for applications of
generalized Korn inequalities in general relativity). Last but not least, the
algebraic conditions (injectivity/surjectivity of the connecting maps) in [11]
raise challenges for constructing numerical methods. Only carefully designed
discrete spaces can fulfill these conditions.

In this paper, we provide a major generalization of [11] to solve the above
issues. We get to a construction that is closer to the developments in [21]
but takes into account the simplifications present in our situation. In
particular, the generalizations of the framework include:

  * •

allowing more than two rows in the diagram,

  * •

removing the injectivity/surjectivity conditions in the assumptions. The
resulting complexes will have a more complicated form, which can be simplified
to the cases treated in [11] when algebraic conditions hold.

As a result of this generalization, we compute the cohomology of important
complexes, e.g., the conformal deformation complex, the conformal Hessian
complex and higher order generalizations of the Hessian complex. We establish
a conformal Korn inequality in two space dimensions and generalizations of the
linear Cosserat elasticity model based on its connections to the twisted de-
Rham complexes. The above generalizations of the algebraic framework play a
critical role in these developments.

#### Conformal Korn inequalities.

A generalized version of the Korn inequality holds in $n$D for $n\geq 3$. In
the framework of [11], this inequality holds since the conformal deformation
complex has finite dimensional cohomology (thus the operators have closed
range). Nevertheless, this result does not hold in 2D [37]. This is consistent
with the fact that a 2D version of the conformal deformation BGG diagram does
not satisfy the injectivity/surjectivity conditions, thus not fitting in the
framework of [11]. In fact, in 2D the
$\mathrm{dev}\operatorname{sym}\operatorname{grad}$ operator involved in the
generalized Korn inequalities corresponds to the Cauchy-Riemann operator in
complex analysis. Based on the generalizations in this paper, we fix the 2D
conformal Korn inequality by adding a third order term, which has a geometric
interpretation via Möbius structures.

#### Linear Cosserat continuum.

In 1909, the Cosserat brothers introduced a new continuum model that
incorporated a microscopic rotational degree of freedom at each point of the
material [35]. This seminal work triggered a number of generalizations in
continuum mechanics, e.g., Eringen’s micropolor elasticity theory [41]. In
this paper, we observe that the linear Cosserat elasticity exactly corresponds
to the Hodge-Laplacian problem of the twisted de-Rham complex that leads to
the elasticity complex. A similar observation holds for the plate models. The
Kirchhoff and a modified version of the Reissner-Mindlin plate models are the
Hodge-Laplacian problems of the Hessian complex and the corresponding twisted
de-Rham complexes [11], respectively. This connection inspired the
construction of numerical methods for solving the biharmonic equation [5, 57],
although not formulated in terms of the twisted de-Rham complex. From this
perspective, the cohomology-preserving projection in [11] and this paper can
be understood as the process of eliminating the rotational degrees of freedom
from Cosserat models to get classical elasticity. The operators mapping the
BGG complex back to the twisted de-Rham complex can in turn be interpreted as
describing classical elasticity as the Cosserat model with the additional
rotational degree of freedom appropriately fixed. Similar conclusion holds for
the Kirchhoff and the modified Reissner-Mindlin plate models. Moreover,
classical elasticity and the Kirchhoff plate are $\Gamma$-limit of the
Cosserat elasticity and the Reissner-Mindlin plate, respectively [31, 52].
These connections are sketched in the following diagram:

(1) ${{\rm
Cosserat~{}elasticity}}$${{\rm{classical}~{}elasticity}}$${{\rm{(modified)}~{}Reissner-
Mindlin~{}plate}}$${{\rm
Kirchhoff~{}plate}}$$\scriptstyle{\mathrm{BGG~{}(elasticity)}}$$\scriptstyle{\Gamma-{\rm
convergence}}$$\scriptstyle{\Gamma-{\rm
convergence}}$$\scriptstyle{\mathrm{BGG~{}(hessian)}}$

This cohomological perspective of the linear Cosserat model has several
consequences, e.g., well-posed Hodge-Laplacian boundary value problems and
important tools for the analysis and computation of the Cosserat models.
Moreover, each of the BGG complexes leads to a Hodge-Laplacian problem (or,
equivalently, energy functional). Other complexes and diagrams may thus
provide generalizations of the Cosserat models to other types of
microstructures. In this paper, we show some examples in this direction
inspired by the Erlangen program, where different Lie symmetries correspond to
different geometries [50].

The rest of this paper will be organized as follows. In Section 1, we
establish the abstract framework. In Section 2, we show examples that fit in
the general framework. In Section 3, we discuss the algebraic and geometric
background of our construction and its relation to BGG sequences for parabolic
geometries. In Section 4, we focus on the conformal Korn inequalities. In
Section 5, we show connections between the Cosserat model and the twisted de-
Rham complexes and discuss potential generalizations.

##  1\. Abstract framework

###  1.1. The basic setup

Let $Z^{i,j},~{}0\leq i\leq n,0\leq j\leq N$, be Hilbert spaces and
$d^{i,j}:Z^{i,j}\to Z^{i+1,j}$ bounded linear operators such that for each $j$
we obtain a compex
$(Z^{{\scriptscriptstyle\bullet},j},d^{{\scriptscriptstyle\bullet},j})$, i.e.
$d^{i+1,j}\circ d^{i,j}=0$ for all $i,j$. Let $S^{i,j}:Z^{i,j}\to Z^{i+1,j-1}$
and $K^{i,j}:Z^{i,j}\to Z^{i,j-1}$ be linear operators with properties
specified below. We collect these data into the following diagram called the
BGG diagram:

(2)
${0}$${Z^{0,0}}$${Z^{1,0}}$${\cdots}$${Z^{n,0}}$${0}$${0}$${Z^{0,1}}$${Z^{1,1}}$${\cdots}$${Z^{n,1}}$${0}$${\cdots}$${\cdots}$${\cdots}$${\cdots}$${0}$${Z^{0,N}}$${Z^{1,N}}$${\cdots}$${Z^{n,N}}$${0.}$$\scriptstyle{d^{0,0}}$$\scriptstyle{d^{1,0}}$$\scriptstyle{d^{n-1,0}}$$\scriptstyle{d^{0,1}}$$\scriptstyle{K^{0,1}}$$\scriptstyle{S^{0,1}}$$\scriptstyle{d^{1,1}}$$\scriptstyle{S^{1,1}}$$\scriptstyle{K^{1,1}}$$\scriptstyle{d^{n-1,1}}$$\scriptstyle{S^{n-1,1}}$$\scriptstyle{{K^{n,1}}}$$\scriptstyle{{K^{0,2}}}$$\scriptstyle{S^{0,2}}$$\scriptstyle{S^{1,2}}$$\scriptstyle{{K^{1,2}}}$$\scriptstyle{S^{n-1,2}}$$\scriptstyle{{K^{n,2}}}$$\scriptstyle{d^{0,N}}$$\scriptstyle{{K^{0,N}}}$$\scriptstyle{S^{0,N}}$$\scriptstyle{d^{1,N}}$$\scriptstyle{S^{1,N}}$$\scriptstyle{{K^{1,N}}}$$\scriptstyle{d^{n-1,N}}$$\scriptstyle{S^{n-1,N}}$$\scriptstyle{{K^{N,N}}}$

This is a generalization of the setup in [11] where only two rows were used.

The abstract results below are established in two steps that impose slightly
different assumptions. However, we collect the basic relations between the
operators here. We start with two basic assumptions:

(3) $S^{i,j-1}K^{i,j}=K^{i+1,j-1}S^{i,j},$ (4)
$S^{i,j}=d^{i,j-1}K^{i,j}-K^{i+1,j}d^{i,j}.$

Using the fact that the $d^{i,j}$ form a complex for each $j$, (4) readily
implies that

(5) $S^{i+1,j}d^{i,j}=-d^{i+1,j-1}S^{i,j},\quad\forall i,j\geq 0.$

Using (5) and (3), we get $S^{i+1,j-1}\circ S^{i,j}=d^{i+1,j-2}\circ
S^{i,j-1}\circ K^{i,j}+K^{i+1,j-1}\circ S^{i+1,j}\circ d^{i,j}$. Inserting for
both $S$ operators from (4) we conclude that

(6) $S^{i+1,j-1}\circ S^{i,j}=0,\quad\forall i,j\geq 0.$

In applications, each row of (2) will be an input complex which is well
understood, most often a de-Rham complex with values in a finite-dimensional
vector space. The $S$ operators that connect the basic input complexes are of
algebraic origin. However, we will use functions spaces of different
regularity, which leads to some subtleties.

Next, we denote by $Z^{i}$ the $i$-th column of (2), i.e.
$Z^{i}:=Z^{i,0}\oplus Z^{i,1}\oplus\dots\oplus Z^{i,N}$. Each element in
$Z^{i}$ can be viewed as a vector with the $j$-th component in $Z^{i,j}$. Then
any linear operator on $Z^{i}$ can be written in a matrix form. For example,
the operators $K^{i,j}$ can be collected into $K^{i}:Z^{i}\to Z^{i}$, which
has the matrix form

$K^{i}:=\left(\begin{array}[]{cccccc}0&K^{i,1}&0&0&\cdots&0\\\
0&0&K^{i,2}&0&\cdots&0\\\ &&\cdots&\cdots&&\\\ 0&0&0&0&\cdots&0\\\
\end{array}\right).$

Similarly, the operatos $d^{i,j}$ are collected into $d^{i}:Z^{i}\to Z^{i+1}$
given by

$d^{i}:=\left(\begin{array}[]{cccccc}d^{i,0}&0&0&0&\cdots&0\\\
0&d^{i,1}&0&0&\cdots&0\\\ &&\cdots&\cdots&&\\\ 0&0&0&0&\cdots&d^{i,N}\\\
\end{array}\right),$

and the $S^{i,j}$ are collected into $S^{i}:Z^{i}\to Z^{i+1}$ given by

$S^{i}:=\left(\begin{array}[]{cccccc}0&S^{{i},1}&0&0&\cdots&0\\\
0&0&S^{{i},2}&0&\cdots&0\\\ &&\cdots&\cdots&&\\\
0&0&0&0&\cdots&0\end{array}\right).$

Observe that the identities derived above have simple expressions in these
terms, e.g. $d^{i+1}\circ d^{i}=0$, $S^{i}=d^{i}\circ K^{i}-K^{i+1}\circ
d^{i}$, $S^{i+1}\circ S^{i}=0$, and so on. In particular,
$(Z^{\scriptscriptstyle\bullet},d^{\scriptscriptstyle\bullet})$ is a complex,
that can simply be viewed as the direct sum of the rows of (2). In particular,
the cohomology of this complex is just the direct sum of the cohomologies of
the individual rows
$(Z^{{\scriptscriptstyle\bullet},j},d^{{\scriptscriptstyle\bullet},j})$.

###  1.2. Twisted complex

In the first step, we show that the differential of the complex
$(Z^{\scriptscriptstyle\bullet},d^{\scriptscriptstyle\bullet})$ can be
modified without changing the cohomology. For this step, we assume (3) and (4)
as well as that all the operators $K^{i,j}$ and $S^{i,j}$ are bounded. This of
course implies that also $K^{i}:Z^{i}\to Z^{i}$ and $S^{i}:Z^{i}\to Z^{i+1}$
are bounded.

Now define the twisted operator $d_{V}^{i}:=d^{i}-S^{i}:Z^{i}\to Z^{i+1}$,
i.e.

$d_{V}^{i}:=\left(\begin{array}[]{cccccc}d^{i,0}&-S^{i,1}&0&0&\cdots&0\\\
0&d^{i,1}&-S^{i,2}&0&\cdots&0\\\ &&\cdots&\cdots&&\\\
0&0&0&0&\cdots&d^{i,N}\\\ \end{array}\right).$

From (5) and (6), we obtain $d_{V}^{i+1}\circ d_{V}^{i}=0$ for each $i$. The
complex
$(Z^{{\scriptscriptstyle\bullet}},d_{V}^{{\scriptscriptstyle\bullet}})$, i.e.,

(7)
${\cdots}$${Z^{i-1}}$${Z^{i}}$${Z^{i+1}}$${\cdots,}$$\scriptstyle{d_{V}^{i-1}}$$\scriptstyle{d_{V}^{i}}$

is referred to as the twisted complex, or the $d_{V}$-complex.

We can now prove that the twisted complex
$(Z^{{\scriptscriptstyle\bullet}},d_{V}^{{\scriptscriptstyle\bullet}})$ is
isomorphic
$(Z^{{\scriptscriptstyle\bullet}},d^{{\scriptscriptstyle\bullet}})$, so in
particular, their cohomologies are isomorphic, too. This means that we
construct isomorphisms $F^{i}:Z^{i}\to Z^{i},~{}i=0,1,\cdots,n$ such that the
following diagram commutes:

(8)
${Z^{i}}$${Z^{i+1}}$${Z^{i}}$${Z^{i+1}}$$\scriptstyle{d^{i}}$$\scriptstyle{F^{i}}$$\scriptstyle{F^{i+1}}$$\scriptstyle{d_{V}^{i}}$

In fact, define
$F^{i}:=\exp(K^{i})=I+K^{i}+\frac{1}{2}(K^{i})^{2}+\frac{1}{6}(K^{i})^{3}+\cdots$,
the matrix exponential of $K^{i}$, i.e.,

(9)
$F^{i}:=\left(\begin{array}[]{cccccc}I&K^{i,1}&\frac{1}{2}K^{i,1}K^{i,2}&\frac{1}{6}K^{i,1}K^{i,2}K^{i,3}&\cdots&\frac{1}{N!}K^{i,1}K^{i,2}\cdots
K^{i,N}\\\
0&I&K^{i,2}&\frac{1}{2}K^{i,2}K^{i,3}&\cdots&\frac{1}{(N-1)!}K^{i,2}K^{i,3}\cdots
K^{i,N}\\\ &&\cdots&\cdots&&\\\ 0&0&0&0&\cdots&I\end{array}\right).$

Observe that $(K^{i})^{N+1}=0$ by construction, so the exponential series
actually is a finite sum. Moreover each $F^{i}$ is obviously invertible, with
inverse given as $\exp(-K^{i})$. Now we are ready to formulate our first main
result.

######  Theorem 1.

Assume (3) and (4) and that all the operators $K^{i,j}$ and $S^{i,j}$ are
bounded. Then the operators $F^{i}$ defined above are bounded and induce an
isomorphism between the twisted complex (7) and the complex
$(Z^{\scriptscriptstyle\bullet},d^{\scriptscriptstyle\bullet})$. Hence for the
cohomology of (7), we get
$\mathcal{H}^{k}(Z^{\scriptscriptstyle\bullet},d_{V}^{\scriptscriptstyle\bullet})\cong\oplus_{j=0}^{N}\mathcal{H}^{k}(Z^{{\scriptscriptstyle\bullet},j},{d^{{\scriptscriptstyle\bullet},j}})$.
Here for a complex
$(X^{{\scriptscriptstyle\bullet}},d^{{\scriptscriptstyle\bullet}})$, we use
$\mathcal{H}^{k}(X^{{\scriptscriptstyle\bullet}},d^{{\scriptscriptstyle\bullet}})$
to denote the cohomology at index $k$.

###### Proof.

Since we have verified that each $F^{i}$ is an isomorphism, it remains to
prove that the diagram (8) commutes for each $i$. We first use induction to
prove that

(10) $d^{i}(K^{i})^{m}-(K^{i+1})^{m}d^{i}=mS^{i}(K^{i})^{m-1},\quad m\geq 1.$

For $m=1$, this is (4). Assume that (10) holds for $m=\ell$ and let us leave
out the index $i$, i.e., $dK^{\ell}-K^{\ell}d=\ell SK^{\ell-1}$. Then it
suffices to verify (10) for $m=\ell+1$. In fact,

$dK^{\ell+1}-K^{\ell+1}d=dK^{\ell}K-K^{\ell+1}d=K^{\ell}dK+\ell
SK^{\ell}-K^{\ell+1}d=K^{\ell}S+\ell SK^{\ell}=(\ell+1)SK^{\ell},$

where the last equality is by (3). Now writing (10) as
$K^{m}d=dK^{m}-SmK^{m-1}$ and summing appropriately, we readily get
$\exp(K)\circ d=(d-S)\circ\exp(K)$ as claimed. ∎

###  1.3. The BGG construction

The second step of the construction starts from the twisted complex and
constructs new differentials on certain subspaces of the spaces $Z^{i}$. We
will then prove that the cohomology of the resulting complex is isomorphic to
the cohomology of the twisted complex. In this second step, we will not need
the operators $K^{i,j}$ any more, so we will only need to assume that we have
bounded operators $S^{i,j}$ that satisfy (5) and (6). In addition, we need
that for each $i,j$, the range $\mathcal{R}(S^{i,j})$ is closed in
$Z^{i+1,j-1}$, which of course implies that $\mathcal{R}(S^{i})$ is closed in
$Z^{i+1}$.

The starting point for this is that (6) implies that also
$S^{\scriptscriptstyle\bullet}$ defines a differential on
$Z^{\scriptscriptstyle\bullet}$, so the “diagonals” in (2) are complexes, too.
(In the geometric BGG construction, these correspond to point-wise standard
complexes for Lie algebra cohomology.) In particular, by the assumption on
closed ranges, we obtain a decomposition

(11)
$Z^{i,j}=\mathcal{R}(S^{i-1,j+1})\oplus\mathcal{R}(S^{i-1,j+1})^{\perp}=\mathcal{R}(S^{i-1,j+1})\oplus\mathcal{N}(S^{i,j})^{\perp}\oplus\Upsilon^{i,j},$

where
$\Upsilon^{i,j}:=\mathcal{R}(S^{i-1,j+1})^{\perp}\cap\mathcal{N}(S^{i,j})$
plays the role of cohomology or of harmonic forms in a “Hodge decomposition”.
In the geometric BGG theory, the decomposition (11) is obtained without
involving an inner product, here we have used the Hilbert space structures
(reflected in $\mathcal{R}(S^{i-1,j+1})^{\perp}$) to simplify the
presentation. In the sequel, we will use $P_{\mathcal{R}(S^{i,j})}$,
$P_{\mathcal{N}(S^{i,j})}$ and $P_{\Upsilon^{i.j}}$ to denote the projections
to $\mathcal{R}(S^{i,j})$, $\mathcal{N}(S^{i,j})$ and $\Upsilon^{i,j}$,
respectively. We also write $P_{\mathcal{R}(S)}$, $P_{\mathcal{N}(S)}$ and
$P_{\Upsilon}$ for brevity.

Another important ingredient in the Hodge theory is the adjoint of the $S$
operators. With respect to the decomposition (11), we define bounded operators
$T^{i,j}:Z^{i,j}\to Z^{i-1,j+1}$ by

$T^{i,j}:=(S^{i-1,j+1})^{-1}\circ P_{\mathcal{R}(S^{i-1,j+1})},$

where
$(S^{i-1,j+1})^{-1}:\mathcal{R}(S^{i-1,j+1})\to\mathcal{N}(S^{i-1,j+1})^{\perp}$
is the inverse of the isomorphism
$\mathcal{N}(S^{i-1,j+1})^{\perp}\to\mathcal{R}(S^{i-1,j+1})$ defined by
$S^{i-1,j+1}$. By definition,
$\mathcal{R}(T^{i,j})=\mathcal{N}(S^{i-1,j+1})^{\perp}$ is also closed. Of
course, the operators $T^{i,j}$ can be collected into a bounded operator
$T^{i}:Z^{i}\to Z^{i-1}$ with closed range.

######  Lemma 1.

The operators defined above satisfy the following properties for any $i,j$:

(12) $T^{i-1,j+1}\circ T^{i,j}=0,$ (13) $T^{i,j}\circ S^{i-1,j+1}\circ
T^{i,j}=T^{i,j},$ (14) $S^{i,j}\circ T^{i+1,j-1}\circ S^{i,j}=S^{i,j}.$ (15)
$\mathcal{R}(S^{i,j})=\mathcal{N}(T^{i+1,j-1})^{\perp},\quad\mathcal{R}(T^{i,j})=\mathcal{N}(S^{i-1,j+1})^{\perp}.$

###### Proof.

To prove (12), note that

$T^{i-1,j+1}\circ T^{i,j}=(S^{i-2,j+2})^{-1}\circ
P_{\mathcal{R}(S^{i-2,j+2})}\circ(S^{i-1,j+1})^{-1}\circ
P_{\mathcal{R}(S^{i-1,j+1})}=0,$

since $(S^{i-1,j+1})^{-1}\circ
P_{\mathcal{R}(S^{i-1,j+1})}u\in\mathcal{N}(S^{i-1,j+1})^{\perp}\subset\mathcal{R}(S^{i-2,j+2})^{\perp}$.

For (13), we have

$T^{i,j}\circ S^{i-1,j+1}\circ T^{i,j}=T^{i,j}\circ
S^{i-1,j+1}\circ(S^{i-1,j+1})^{-1}\circ
P_{\mathcal{R}(S^{i-1,j+1})}=T^{i,j}\circ
P_{\mathcal{R}(S^{i-1,j+1})}=T^{i,j}.$

Similarly we can prove (14). ∎

######  Remark 1.

By definition, if $S^{i,j}$ is bijective, then $T^{i+1,j-1}$ is just the
inverse of $S^{i,j}$ and hence is bijective, too. Conversely, if $T^{i,j}$ is
bijective, then from (13) we have that $S^{i-1,j+1}$ is its inverse and thus
also bijective.

We next sketch the BGG construction, for details some technical results will
be needed. By construction, $T^{i+1}d^{i}$ maps $Z^{i}$ to $Z^{i}$ but it
“moves each component down by a row”, i.e. it maps $Z^{i,j}$ to $Z^{i,j+1}$
and therefore is nilpotent. Thus we can define a bounded operator
$G^{i}:Z^{i}\to Z^{i-1}$ by

(16) $G^{i}=-\sum_{k=0}^{\infty}(T^{i}d^{i-1})^{k}T^{i},$

where the sum in (16) is finite since $T^{i}d^{i-1}$ is nilpotent. Now define

$\Upsilon^{i}:=\mathcal{R}(S^{i-1})^{\perp}\cap\mathcal{N}(S^{i}),$

so this is the sum of the spaces $\Upsilon^{i,j}$ from (11). Here we use the
convention that $\mathcal{R}(S^{-1})^{\perp}=Z^{0}$ and
$\mathcal{N}(S^{N})=Z^{N}$. Further, define $A^{i}:\Upsilon^{i}\to Z^{i}$ by
$A^{i}:=I-G^{i+1}d_{V}^{i}$. As we shall verify below, $d_{V}^{i}A^{i}$ has
values in $\mathcal{R}(S^{i-1})^{\perp}$, so using the projection from (11) to
define
$D^{i}:=P_{\mathcal{N}(S^{i+1})}d_{V}^{i}A^{i}=P_{\Upsilon^{i+1}}d_{V}^{i}A^{i}$,
we get an opertor $D^{i}:\Upsilon^{i}\to\Upsilon^{i+1}$. These operators form
the BGG sequence

(17)
${\cdots}$${\Upsilon^{i-1}}$${\Upsilon^{i}}$${\Upsilon^{i+1}}$${\cdots,}$$\scriptstyle{D^{i-1}}$$\scriptstyle{D^{i}}$

We will see that $A^{{\scriptscriptstyle\bullet}}$ defines a complex map from
the BGG complexes to the twisted complexes which induces an isomorphism in
cohomology. The explicit matrix form of some operators in this section will be
given in an Appendix.

Before presenting the proof, we consider the example of a diagram with two
rows. This is the setting of [11], but we do not assume any
injectivity/surjectivity conditions. For $N=2$, we have $G^{i}=-T^{i}$.
Therefore

$Gd_{V}=\left(\begin{array}[]{cc}0&0\\\
-T&0\end{array}\right)\left(\begin{array}[]{cc}d&-S\\\
0&d\end{array}\right)=\left(\begin{array}[]{cc}0&0\\\
-Td&TS\end{array}\right)=\left(\begin{array}[]{cc}0&0\\\
-Td&P_{\mathcal{N}(S)^{\perp}}\end{array}\right),$

and

$D=P_{\mathcal{N}(S)}d_{V}A=P_{\mathcal{N}(S)}\left(\begin{array}[]{cc}d&-S\\\
0&d\end{array}\right)\left(\begin{array}[]{cc}I&0\\\
Td&P_{\mathcal{N}(S)}\end{array}\right)=\left(\begin{array}[]{cc}P_{\mathcal{R}(S)^{\perp}}d&0\\\
P_{\mathcal{N}(S)}dTd&dP_{\mathcal{N}(S)}\end{array}\right).$

The BGG complex (17) is simplified with
$\Upsilon^{i}=\mathcal{R}(S^{i-1})^{\perp}\cap\mathcal{N}(S^{i})$, and for
$(\alpha_{0},\alpha_{1})\in\Upsilon^{i}$,

(18) ${D}^{i}\left(\begin{array}[]{c}\alpha_{0}\\\
\alpha_{1}\end{array}\right)=\left(\begin{array}[]{c}P_{\mathcal{R}(S)^{\perp}}d\alpha_{0}\\\
d\alpha_{1}+P_{\mathcal{N}(S)}dTd\alpha_{0}\end{array}\right).$

We have simplified the operator using the fact that
$\alpha_{0}\in\mathcal{R}(S^{i-1})^{\perp}$ and
$\alpha_{1}\in\mathcal{N}(S^{i})$.

Next we show how this formula simplifies in the presence of
injectivity/surjectivity conditions. The discussions will involve three $S$
operators indicated in the following diagram

(19)
${\cdots}$${Z^{i,0}}$${Z^{i+1,0}}$${\cdots}$${\cdots}$${Z^{i,1}}$${Z^{i+1,1}}$${\cdots,}$$\scriptstyle{d^{i,0}}$$\scriptstyle{d^{i-1,1}}$$\scriptstyle{S^{i-1,1}}$$\scriptstyle{d^{i,1}}$$\scriptstyle{S^{i,1}}$$\scriptstyle{S^{i+1}}$

and there are three cases.

  1. (1)

Assume that $S^{i-1,1}$, $S^{i,1}$ and $S^{i+1,1}$ are injective. In this case
$\mathcal{N}(S^{i,1})=0$, so $\alpha_{1}$ is eliminated from the system. The
operator ${D}^{i}$ is reduced to

${D}^{i}\left(\begin{array}[]{c}\alpha_{0}\\\
0\end{array}\right):=\left(\begin{array}[]{c}P_{\mathcal{R}(S)^{\perp}}d\alpha_{0}\\\
0\end{array}\right).$

  2. (2)

Assume that $S^{i-1}$ is injective, $S^{i}$ is bijective and $S^{i+1}$ is
surjective. In this case $\mathcal{N}(S^{i,1})=0$, so $\alpha_{1}$ is
eliminated from the system. The operator ${D}^{i}$ is reduced to

${D}^{i}\left(\begin{array}[]{c}\alpha_{0}\\\
0\end{array}\right):=\left(\begin{array}[]{c}0\\\
dTd\alpha_{0}\end{array}\right).$

Note that we have removed $P_{\mathcal{N}(S)}$ in front of $dTd\alpha_{0}$.
From Remark 1, $T=S^{-1}$ is bijective. Therefore $S(dTd)=-dSTd=0$. This
implies that $dTd\alpha$ is automatically in $\mathcal{N}(S)$.

  3. (3)

Assume that $S^{i-1}$, $S^{i}$ and $S^{i+1}$ are surjective. In this case
$\mathcal{R}(S)^{\perp}=0$, so $\alpha_{0}$ is eliminated from the system. The
operator ${D}^{i}$ is reduced to

${D}^{i}\left(\begin{array}[]{c}0\\\
\alpha_{1}\end{array}\right):=\left(\begin{array}[]{c}0\\\
d\alpha_{1}\end{array}\right).$

The three differential operators in the elasticity complex in three dimensions
[11] correspond to these three cases, respectively.

###  1.4. Cohomology of the BGG sequence

We next add the details for the construction of the sequence (17), prove that
it is a complex and show that it computes the same cohomology as the twisted
complex (7). We prove this fact by exploiting the properties of $G^{i}$ and
$A^{i}$. The explicit cohomology-preserving cochain maps will be given in the
proof.

First we can verify the following properties of $G^{i}$:

(20) $G^{i}|_{\mathcal{N}(T^{i})}=0,$ (21) $\phi-
d_{V}^{i-1}G^{i}\phi\in\mathcal{N}(T^{i}),$ (22)
$G^{i}\phi\in\mathcal{R}(T^{i}).$

In fact, (20) and (22) are by the definition (16). It suffices to verify (21).
To this end, we note that

$T^{i}d_{V}^{i-1}T^{i}=T^{i}(d^{i-1}-S^{i-1})T^{i}=T^{i}d^{i-1}T^{i}-T^{i}S^{i-1}T^{i}=(-I+T^{i}d^{i-1})T^{i}.$

This implies that on $\mathcal{R}(T^{i})$, we get
$T^{i}d_{V}^{i-1}=-I+T^{i}d^{i-1}$. Since
$\mathcal{R}(G)\subset\mathcal{R}(T)$, we have
$T^{i}d_{V}^{i-1}{G}^{i}=(-I+T^{i}d^{i-1})G^{i}=T^{i}$.

Indeed, property (21) provides the motivation for the construction of $G^{i}$:
As we have noted above, restricting $T^{i}d_{V}^{i-1}$ to
$\mathcal{R}(T^{i})$, one obtains a map
$\mathcal{R}(T^{i})\to\mathcal{R}(T^{i})$ that is nilpotent and hence
$I-T^{i}d_{V}^{i-1}$ is invertible on that subspace. Composing the inverse
with $-T$, one obtains a map $Z^{i}\to Z^{i-1}$ that satisfies (21), and
$G^{i}$ is simply obtained from writing the inverse as a Neumann series.

From (20)-(22), we have the following lemma.

######  Lemma 2.

  1. (1)

For any $\alpha\in\Upsilon^{i}$, we get
$A^{i}\alpha\in\mathcal{R}(S^{i})^{\perp}$ and
$P_{\mathcal{N}(S^{i})}A^{i}\alpha=P_{\Upsilon^{i}}A^{i}\alpha=\alpha$.
Moreover, we get $d_{V}^{i}A^{i}(\alpha)\in\mathcal{R}(S^{i})^{\perp}$.

  2. (2)

Conversely, suppose that $\psi\in\mathcal{R}(S^{i-1})^{\perp}$ has the
property that $d_{V}^{i}\psi\in\mathcal{R}(S^{i-1})^{\perp}$. Then
$\psi=A^{i}(\beta)$, where
$\beta=P_{\mathcal{N}(S^{i})}\psi=P_{\Upsilon^{i}}\psi$.

###### Proof.

  1. (1)

Since $G^{i}$ has value in $\mathcal{N}(S^{i})^{\perp}$, the first two
properties follow readily. Moreover, $d_{V}^{i}A^{i}(\alpha)=d_{V}^{i}\alpha-
d_{V}^{i}G^{i+1}d_{V}^{i}\alpha$, so applying (21) to $d_{V}^{i}\alpha$, the
last property follows.

  2. (2)

From (1) we conclude that $\psi-A^{i}(\beta)$ lies in
$\mathcal{N}(S^{i})^{\perp}$ and satisfies
$d_{V}^{i}(\psi-A^{i}(\beta))\in\mathcal{R}(S^{i})^{\perp}$. Thus it suffices
to prove that $\psi\in\mathcal{N}(S^{i})^{\perp}$ and
$d_{V}^{i}\psi\in\mathcal{R}(S^{i})^{\perp}$ imply $\psi=0$. But the latter
condition says that $d^{i}\psi-S^{i}\psi\in\mathcal{R}(S^{i})^{\perp}$. Thus
the component of $d^{i}\psi$ in ${\mathcal{R}(S^{i})}$ is ${S^{i}(\psi)}$.
Writing $\psi=(\psi^{0},\psi^{1},\cdots,\psi^{N})$,
$\psi\in\mathcal{N}(S^{i})^{\perp}$ implies $\psi^{0}=0$. Then
$d^{i}\psi=(0,d^{i,1}\psi^{1},\cdots,d^{i,N}\psi^{N})$ while
$S^{i}\psi=(S^{i,1}\psi^{1},\cdots,S^{i,N-1}\psi^{N-1},0)$. Comparing the
first component, we get $S^{i,1}\psi^{1}=0$ and thus $\psi^{1}=0$ since
$\psi^{1}\in\mathcal{N}(S^{i,1})^{\perp}$. Similarly, we get
$S^{i,2}\psi^{2}=0$ and thus $\psi^{2}=0$. Repeating this argument, we see
that $\psi=0$.

∎

Using this, we can formulate our second main result.

######  Theorem 2.

Assume that we start from the twisted complex (7) associated to (2) with
bounded operators $S^{i,j}$ such that (5) and (6) are satisfied. Then the BGG
sequence (17) is a complex, whose cohomology is isomorphic to the cohomology
of the twisted complex (7) (and thus isomorphic to the cohomology of the input
by Theorem 1). Explicitly, the map $A^{i}$ and the map $B^{i}$ defined by (24)
in the proof are complex maps that induce inverse isomorphisms in cohomology.

###### Proof.

Part (1) of Lemma 2 shows that $D^{i}\alpha$ is the component of
$d_{V}^{i}A^{i}\alpha$ in $\Upsilon^{i+1}$. Since
$d_{V}^{i}A^{i}\alpha\in\mathcal{R}(S^{i})^{\perp}$ and
$d_{V}^{i+1}d_{V}^{i}A^{i}\alpha=0$, part (2) of Lemma 2 shows that

(23) $d_{V}^{i}A^{i}\alpha=A^{i+1}D^{i}\alpha.$

This in turn readily implies $d_{V}^{i+1}A^{i+1}D^{i}\alpha=0$ and hence
$D^{i+1}D^{i}\alpha=0$. Hence the BGG sequence is a complex and (23) shows
that $A^{\scriptscriptstyle\bullet}$ defines a complex map from the BGG
complex to the twisted complex.

Now for a form $\phi$ consider $\tilde{\phi}:=\phi-d_{V}^{i-1}G^{i}\phi$. By
(21), this liees in $\mathcal{R}(S^{i-1})^{\perp}$ and we define $B^{i}(\phi)$
to be its component in $\Upsilon^{i}$, i.e.,

(24) $B^{i}\phi:=P_{\Upsilon^{i}}(I-d_{V}^{i-1}G^{i})\phi,$

where $P_{\Upsilon^{i}}$ is the projection to
$\mathcal{R}(S^{i-1})^{\perp}\cap\mathcal{N}(S^{i})$. Now consider

$\tilde{\phi}-G^{i+1}d_{V}^{i}\tilde{\phi}=\phi-
d_{V}^{i-1}G^{i}\phi-{G}^{i+1}d_{V}^{i}\phi.$

This lies in $\mathcal{R}(S^{i-1})^{\perp}$ and applying $d_{V}^{i}$, it gives
$d_{V}^{i}\phi-d_{V}^{i}{G}^{i+1}d_{V}^{i}\phi$, which as above, lies in
$\mathcal{R}(S^{i})^{\perp}$ by (21). Moreover, its component in
$\Upsilon^{i}$ still equals $B^{i}\phi$, so by part (2) of Lemma 2, we get

(25) $A^{i}B^{i}(\phi)=\phi-d_{V}^{i-1}{G}^{i}\phi-{G}^{i+1}d_{V}^{i}\phi.$

Now $B^{i+1}d_{V}^{i}\phi$ is the component in $\Upsilon^{i+1}$ of
$d_{V}^{i}\phi-d_{V}^{i}{G}^{i+1}d_{V}^{i}\phi=d_{V}^{i}A^{i}B^{i}\phi$. This
shows that $B^{i+1}d_{V}^{i}\phi=D^{i}B^{i}\phi$, so $B$ is a complex map,
too. Equation (25) then shows that $AB$ is chain homotopic to the identity. On
the other hand, since
$A^{i}\alpha=\alpha-{G}^{i+1}d_{V}^{i}\alpha\in\mathcal{R}(S^{i-1})^{\perp}$,
we get ${G}^{i}A^{i}\alpha=0$. Hence $B^{i}A^{i}\alpha$ simply is the
component in $\Upsilon^{i}$ of $A^{i}\alpha$ which equals $\alpha$. Thus
$B^{i}A^{i}=I$ and we see that $B$ and $A$ induce inverse isomorphism in
cohomology. ∎

##  2\. Examples

We demonstrate several examples of the abstract framework. In particular,
these include some complexes in [15] with applications in general relativity.
Following the general framework, for each example we should specify the
diagram (2), i.e., the spaces $Z^{i,j}$, the operators $d^{i,j}$, $K^{i,j}$
(and thus $S^{i,j}$ defined by (4)), such that the assumptions (3) and (4)
hold. We’ll say more about how these operators are obtained and the relation
to the constructions in [21] (that apply in a smooth setting) in Section 3
below. As we shall see below, to derive complexes with more general function
spaces (e.g., Sobolev spaces), these algebraic structures are not enough due
to regularity issues. With a similar idea as [11], this subtlety caused by the
regularity of functions can be fixed by requiring in addition that the input
complexes have a uniform representation of cohomology, not depending on the
regularity. For de-Rham sequences, this boils down to the results by Costabel
and McIntosh [36]. However, the argument given here is different from the
proof in [11] based on Poincaré operators.

In Section 1.3, we have seen that for diagrams with two rows and assuming
injective/surjectivity conditions on the $S$-operators, we recover the
examples from [11], so we focus on diagrams with at least three rows.

Throughout the rest of this paper, we assume that $U$ is a bounded Lipschitz
domain in $\mathbb{R}^{n}$. Thus the results in [36] apply and in particular
there exists a uniform representation of cohomology for the de-Rham complex,
c.f., [36, Theorem 1.1].

###  2.1. The structure of the examples

To set up one of our examples, we fix finite dimensional vector spaces
$\mathbb{V}_{j}$ and real numbers $q_{j}$ for $j=0,\dots,N$, and the examples
apply to bounded Lipschitz domains $U\subset\mathbb{R}^{n}$, mostly with
$n=3$. The spaces $Z^{i,j}$ can be interpreted as differential $i$-forms on
$U$ with values in $\mathbb{V}_{j}$ or equivalently as functions to
$\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V}_{j}$ of appropriate regularity.
The operators we construct all make sense in the smooth setting and the
identities can usually be obtained respectively checked there, but the main
case to consider is that the functions in $Z^{i,j}$ lie in the Sobolev space
$H^{q_{j}-i}$. The operators $d^{i,j}:Z^{i,j}\to Z^{i+1,j}$ are all induced by
the exterior derivative, so since the regularity in the target is always one
lower than in the source, they are bounded. The regularities $q_{j}$ depend on
some fixed number $q\in\mathbb{R}$, but we’ll need different choices for the
two steps and a key lemma to connect the two steps. In the first step we will
take $q_{j}=q$ for all $j$, while in the second step $q_{j}=q-j$ for each $j$.

To start with the first step, we take $q_{j}=q$ for all $j$, and hence
$Z^{i,j}=H^{q-i}\otimes\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V}_{j}$. The
operators $K^{i,j}$ will always be built up from the operators of
mulitplication by bounded smooth functions, so since the regularity in
$Z^{i,j}$ and $Z^{i,j-1}$ are the same, they define bounded operators
$Z^{i,j}\to Z^{i,j-1}$ for each $i$ and $j$. One next verifies directly that
$S^{i.j}:=d^{i,j-1}K^{i,j}-K^{i+1,j}d^{i,j}$ in the smooth case satisfies (3),
so this extends to the Sobolev setting, and all the assumptions for Theorem 1
are satisfied. Thus we conclude that the complex
$(Z^{\scriptscriptstyle\bullet},d_{V}^{\scriptscriptstyle\bullet})$ is
isomorphic to the sum of the Sobolev de-Rham complexes with values in
$\mathbb{V}_{j}$, and hence its cohomology is well understood.

To pass to the second step, we need an additional observation: In each of the
examples, one verifies directly that the operator $S^{i,j}$ defined by (4)
maps a smooth function $\phi$ to $\partial^{i,j}\circ\phi$, where
$\partial^{i,j}:\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V}_{j}\to\Lambda^{i+1}\mathbb{R}^{n*}\otimes\mathbb{V}_{j-1}$
is a linear map. Hence also in the Sobolev setting $S^{i,j}$ is given by
composition with $\partial^{i,j}$. This readily shows that $S^{i,j}$ also
extends as a bounded linear operator $\tilde{Z}^{i,j}\to\tilde{Z}^{i+1,j-1}$
where
$\tilde{Z}^{i,j}:=H^{q-i-j}\otimes\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V}_{j}$.
The identities (5) and (6) of course extend to this regularity since they hold
on smooth forms. Moreover, $\mathcal{R}(S^{i,j})\subset\tilde{Z}^{i+1,j-1}$ is
the subspace $H^{q-i-j}\otimes\mathcal{R}(\partial^{i,j})$, which is evidently
closed in $\tilde{Z}^{i+1,j-1}$ (and the change of regularity is needed to get
this). Hence all conditions needed to apply Theorem 2 are satisfied, and it
remains to understand the cohomology of
$(\tilde{Z}^{\scriptscriptstyle\bullet},d_{V}^{\scriptscriptstyle\bullet})$.
This is described by the following key lemma, which crucially depends on the
existence of uniform smooth representatives for the Sobolev-de-Rham
cohomology. To formulate this, let us write
$Z_{q}^{\scriptscriptstyle\bullet}$ for the complex in regularity $q$, i.e.,
$Z_{q}^{i,j}:=H^{q-i}\otimes\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V}_{j}$.
Then by definition $Z_{q}^{i,j}\subset\tilde{Z}^{i,j}\subset Z_{q-N}^{i,j}$,
so we have inclusions
$\mu:Z_{q}^{\scriptscriptstyle\bullet}\to\tilde{Z}^{\scriptscriptstyle\bullet}$
and $\nu:\tilde{Z}^{\scriptscriptstyle\bullet}\to
Z_{q-N}^{\scriptscriptstyle\bullet}$.

######  Lemma 3.

Assume that $U$ is a bounded Lipschitz domain. Then in the notation introduced
above, both
$\mu:(Z_{q}^{\scriptscriptstyle\bullet},d_{V}^{\scriptscriptstyle\bullet})\to(\tilde{Z}^{\scriptscriptstyle\bullet},d_{V}^{\scriptscriptstyle\bullet})$
and
$\nu:(\tilde{Z}^{\scriptscriptstyle\bullet},d_{V}^{\scriptscriptstyle\bullet})\to(Z_{q-N}^{\scriptscriptstyle\bullet},d_{V}^{\scriptscriptstyle\bullet})$
are inclusions of subcomplexes that induce isomorphisms in cohomology.

###### Proof.

By construction, both $\mu$ and $\nu$ are compatible with the differentials
$d_{V}$ and hence induce maps in cohomology. The composition $\nu\circ\mu$ is
just the inclusion
$(Z_{q}^{\scriptscriptstyle\bullet},d^{\scriptscriptstyle\bullet}_{v})\to(Z_{q-N}^{\scriptscriptstyle\bullet},d_{V}^{\scriptscriptstyle\bullet})$.
Via the isomorphisms $F$ from Theorem 1, this is conjugate to the inclusion
$(Z_{q}^{\scriptscriptstyle\bullet},d^{\scriptscriptstyle\bullet})\to(Z_{q-N}^{\scriptscriptstyle\bullet},d^{\scriptscriptstyle\bullet})$
of a sum of Sobolev-de-Rham complexes. The latter induces an isomorphisms in
cohomology by [36], and this carries over to the $d_{V}$ complexes via $F$.
This implies that $\mu$ induces an injection in cohomology, while $\nu$
induces a surjection in cohomology and to complete the proof, it suffices to
show that the map in cohomology induced by $\nu$ is injective.

To prove this, we have to take forms $\psi_{j}\in\tilde{Z}^{i,j}$ for some $i$
and $j=0,\dots N$ such that there are forms $\phi_{j}\in Z_{q-N}^{i-1,j}$ such
that $d_{V}(\phi_{0},\dots,\phi_{N})=(\psi_{0},\dots,\psi_{N})$ (which of
course implies that $d_{V}(\psi_{0},\dots,\psi_{N})=0$). We then have to show
that we can also find forms $\tilde{\phi}_{j}\in\tilde{Z}^{i-1,j}$ such that
$d_{V}(\tilde{\phi}_{0},\dots,\tilde{\phi}_{N})=(\psi_{0},\dots,\psi_{N})$,
i.e. we have to improve the regularity of the $j$th component from $H^{q-N+1}$
to $H^{q-j+1}$. In particular, there is nothing to do for $j=N$ and we use
this as the starting point of a backwards induction. So take $j<N$ and assume
that $\phi_{k}\in\tilde{Z}^{i,k}$ for $k=j+1,\dots,N$. Then by assumption, the
$j$th component of $d_{V}(\phi_{0},\dots,\phi_{N})=(\psi_{0},\dots,\psi_{N})$
reads as $\psi_{j}=d\phi_{j}-S\phi_{j+1}$. By the inductive hypothesis,
$\phi_{j+1}\in\tilde{Z}^{i-1,j+1}$ so $S\phi_{j+1}\in\tilde{Z}^{i,j}$. Hence
$\psi_{j}+S\phi_{j+1}$ lies in $H^{q-j}$ and by assumption
$\psi_{j}+S\phi_{j+1}=d\phi_{j}$. But this means that $\psi_{j}+S\phi_{j+1}$
is exact in the sum of Sobolev-de-Rahme complexes in regularity $q-N$, so it
is also exact in regularity $q-j$ by [36]. Hence there is a form
$\hat{\phi}_{j}\in\tilde{Z}^{i-1,j}$ such that
$\psi_{j}+S\phi_{j+1}=d\hat{\phi}_{j}$. Now $d(\hat{\phi}_{j}-\phi_{j})=0$, so
by [36], we can write $\hat{\phi}_{j}=\phi_{j}+\alpha+d\beta$, where $\alpha$
is a smooth representative for the cohomology class of
$\hat{\phi}_{j}-\phi_{j}$ and $\beta\in Z_{q-N}^{i-2,j}$. Now defining
$\tilde{\phi}_{j}:=\hat{\phi}_{j}-\alpha\in\tilde{Z}^{i-1,j}$, we get
$d\tilde{\phi}_{j}=d\hat{\phi}_{j}=\psi_{j}+S\phi_{j+1}$ and
$S\tilde{\phi}_{j}=S\phi_{j}-dS\beta$. Hence if we in addition replace
$\phi_{j-1}$ by $\tilde{\phi}_{j-1}=\phi_{j-1}+S\beta$ and put
$\tilde{\phi}_{\ell}=\phi_{\ell}$ for $\ell\neq j,j-1$, we get
$d_{V}(\tilde{\phi}_{0},\dots,\tilde{\phi}_{N})=(\psi_{0},\dots,\psi_{N})$.
This completes the inductive step and hence the proof. ∎

We will usually work in dimension $n=3$ and to be better compatible with the
literature, we often replace differential forms by their vector proxies and
the exterior derivative by gradient, curl and divergence. Following the
notation in [11], we introduce notation for the algebraic operations we need
in this picture. Put $\mathbb{V}:=\mathbb{R}^{n}$, let $\mathbb{M}$ be the
space of all $n\times n$-matrices and, $\mathbb{S}$ and $\mathbb{K}$ and
$\mathbb{T}$ be the subspaces of matrices that are symmetric, skew-symmetric
and trace-free, respectively. Now we consider the following some basic linear
algebraic operations: $\operatorname{skw}:\mathbb{M}\to\mathbb{K}$ and
$\operatorname{sym}:\mathbb{M}\to{\mathbb{S}}$ are the skew and symmetric part
operators; $\operatorname{tr}:\mathbb{M}\to\mathbb{R}$ is the matrix trace;
$\iota:\mathbb{R}\to\mathbb{M}$ is the map $\iota u:=uI$ identifying a scalar
with a scalar matrix; $\mathrm{dev}:\mathbb{M}\to\mathbb{T}$ given by
$\mathrm{dev}u:=u-1/n\operatorname{tr}(u)I$ is the deviator, or trace-free
part. In three space dimensions, we can identify a skew symmetric matrix with
a vector,

$\operatorname{mskw}\left(\begin{array}[]{c}v_{1}\\\ v_{2}\\\
v_{3}\end{array}\right):=\left(\begin{array}[]{ccc}0&-v_{3}&v_{2}\\\
v_{3}&0&-v_{1}\\\ -v_{2}&v_{1}&0\end{array}\right).$

Consequently, we have $\operatorname{mskw}(v)w=v\times w$ for
$v,w\in{\mathbb{V}}$. We also define
$\operatorname{vskw}=\operatorname{mskw}^{-1}\circ\operatorname{skw}:\mathbb{M}\to{\mathbb{V}}$.
We further define the Hessian operator
$\operatorname{hess}:=\operatorname{grad}\circ\operatorname{grad}$ and for any
matrix valued function $u$, $\mathcal{S}u:=u^{T}-\operatorname{tr}(u)I$.

###  2.2. Conformal Hessian complex

In the language of Section 2.1, we put $n=3$, $N=2$,
$\mathbb{V}_{0}=\mathbb{V}_{2}=\mathbb{R}$ and
$\mathbb{V}_{1}=\mathbb{R}^{3}$. So elements in $Z^{i,0}$ and $Z^{i,2}$ are
just differential forms of degree $i$, which we denote by $\phi$ and $\omega$,
respectively. An element of $Z^{i,1}$ can be written as a triple
$(\psi_{1},\psi_{2},\psi_{3})$ of $i$-forms. In this notation, we define
$K^{i,1}(\psi_{1},\psi_{2},\psi_{3}):=\sum_{\ell}x^{\ell}\psi_{\ell}$, where
$x^{1},x^{2},x^{3}$ are the coordinate functions on $\mathbb{R}^{3}$.
Likewise, we define
$K^{i,2}(\omega):=({x^{1}\omega,x^{2}\omega,x^{3}\omega})$. Defining $S^{i,j}$
by (4), we obtain

$\displaystyle
S^{i,1}(\psi_{1},\psi_{2},\psi_{3})=\textstyle\sum_{\ell}(d(x^{\ell}\psi_{\ell})-x^{\ell}d\psi_{\ell})=\sum_{\ell}dx^{\ell}\wedge\psi_{\ell}$
$\displaystyle
S^{i,2}\omega=(dx^{1}\wedge\omega,dx^{2}\wedge\omega,dx^{3}\wedge\omega).$

In particular, in the picture of functions, the $S^{i,j}$ can be written as
compositions with linear maps $\partial^{i,j}$ as discussed in Section 2.1.

There is only one relevant instance of equation (3), namely
$K^{i+1,1}S^{i,2}=S^{i,1}K^{i,2}$, and both sides evidently map $\omega$ to
$\sum_{\ell}x^{\ell}dx^{\ell}\wedge\omega$. Hence we have made all the
verifications needed to apply our machinery. Converting to vector proxies, the
BGG diagram has the form

(26)
${0}$${H^{q_{0}}\otimes\mathbb{R}}$${H^{q_{0}-1}\otimes\mathbb{V}}$${H^{q_{0}-2}\otimes\mathbb{V}}$${H^{q_{0}-3}\otimes\mathbb{R}}$${0}$${0}$${H^{q_{1}}\otimes\mathbb{V}}$${H^{q_{1}-1}\otimes\mathbb{M}}$${H^{q_{1}-2}\otimes\mathbb{M}}$${H^{q_{1}-3}\otimes\mathbb{V}}$${0}$${0}$${H^{q_{2}}\otimes\mathbb{R}}$${H^{q_{2}-1}\otimes\mathbb{V}}$${H^{q_{2}-2}\otimes\mathbb{V}}$${H^{q_{2}-3}\otimes\mathbb{R}}$${0.}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{I}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{\operatorname{mskw}}$$\scriptstyle{1/3\iota}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{x\cdot}$$\scriptstyle{I}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{2\operatorname{vskw}}$$\scriptstyle{x\cdot}$$\scriptstyle{1/3\operatorname{tr}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{\operatorname{tr}}$$\scriptstyle{x\cdot}$$\scriptstyle{-2\operatorname{vskw}}$$\scriptstyle{x\cdot}$$\scriptstyle{I}$$\scriptstyle{x\otimes}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\iota}$$\scriptstyle{x\otimes}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{-\operatorname{mskw}}$$\scriptstyle{x\otimes}$$\scriptstyle{x\otimes}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{I}$$\scriptstyle{x\otimes}$

Here the maps $x\cdot$ send an $\mathbb{R}^{3}$-valued function $f$ to the
function $x\mapsto{x\cdot f(x)}$ while $x\otimes$ maps a real valued function
$g$ to the function $x\mapsto g(x)x$. The operators in the twisted complex in
this picture then read as

$d_{V}^{0}=\left(\begin{array}[]{ccc}\operatorname{grad}&-I&0\\\
0&\operatorname{grad}&-\iota\\\
0&0&\operatorname{grad}\end{array}\right),\quad
d_{V}^{1}=\left(\begin{array}[]{ccc}\operatorname{curl}&-2\operatorname{vskw}&0\\\
0&\operatorname{curl}&\operatorname{mskw}\\\
0&0&\operatorname{curl}\end{array}\right),\quad
d_{V}^{2}=\left(\begin{array}[]{ccc}\operatorname{div}&-\operatorname{tr}&0\\\
0&\operatorname{div}&-I\\\ 0&0&\operatorname{div}\end{array}\right).$

To pass to the BGG complex, we first move to $\tilde{Z}^{i,j}$ as discussed in
Section 2.1. Then we just have to understand the complements to the image of
each $S$ operators in the kernel of the next $S$ operators, and this just
happens point-wise. From (26) we can immediately read off that all operators
$S^{i,1}$ are surjective and $S^{0,1}$ is bijective, while all $S^{i,2}$ are
injective and $S^{2,2}$ is bijective. This immediately leads to the complex

(27)
${0}$${H^{q}\otimes\mathbb{R}}$${H^{q-2}\otimes(\mathbb{S}\cap\mathbb{T})}$${H^{q-3}\otimes(\mathbb{S}\cap\mathbb{T})}$${H^{q-5}\otimes\mathbb{R}}$${0,}$$\scriptstyle{\mathrm{dev}\operatorname{hess}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}\operatorname{div}}$

which is known as the conformal Hessian complex.

In the language of differential forms, we can also show that our construction
is compatible with Euclidean motions in an appropriate sense. Let $f$ be a
Euclidean motion, so $f(x)=Ax+b$ for an orthgonal matrix $A=(a^{i}_{j})$ and a
vector $b\in\mathbb{R}^{3}$. Then for a bounded Lipschitz domain $U$, also
$f(U)\subset\mathbb{R}^{3}$ is a bounded Lipschitz domain, and we have the
standard pullback operator on differential forms (of any regularity), which we
denote by $f^{*}$. Now in the notation above, for forms $\phi\in
Z^{i,0}(f(U))$ and $\omega\in Z^{i,2}(f(U))$ we use this standard pullback
$f^{*}\phi\in Z^{i,0}(U)$ and $f^{*}\omega\in Z^{i,2}(U)$. On $Z^{i,1}$, we
need a different version of the pullback. Namely, representing an element of
$Z^{i,1}(f(U))$ as a triple $(\psi_{1},\psi_{2},\psi_{3})$ we define

$f^{*}(\psi_{1},\psi_{2},\psi_{3}):=(\textstyle{\sum}_{j}a_{1}^{j}f^{*}\psi_{j},\textstyle{\sum}_{j}a_{2}^{j}f^{*}\psi_{j},\textstyle{\sum}_{j}a_{3}^{j}f^{*}\psi_{j}).$

This may look arbitrary, the conceptual explanation is that we should actually
view elements of $Z^{i,1}$ as $i$-forms with values in the tangent bundle and
this is the natural action of a diffeomorphism on such forms. See Section 3
for more details.

The standard compatibility of a pullback of differential forms with the
exterior derivative together with the fact that the $a^{i}_{j}$ are constants,
implies that the pullback we have defined commutes with the exterior
derivatives $d^{i}$ on all spaces in question. Moreover, we can verify by a
direct computation that it is also compatible with the operators $S^{i,j}$. By
definition, $f(x)^{i}=\sum_{j}a^{i}_{j}x^{j}+b^{i}$, where the $b^{j}$ are the
components of $b$, which implies that $f^{*}dx^{i}=\sum_{j}a^{i}_{j}dx^{j}$
for any $i$. Inserting this, one immediately computes that for
$(\psi_{1},\psi_{2},\psi_{3})\in Z^{i,1}(f(U))$ both
$S^{i,1}(f^{*}(\psi_{1},\psi_{2},\psi_{3}))$ and
$f^{*}(S^{i,1}(\psi_{1},\psi_{2},\psi_{3}))$ are given by
$\sum_{j,\ell}a^{j}_{\ell}dx^{j}\wedge f^{*}\psi_{\ell}$. On the other hand,
for $\omega\in Z^{i,2}(f(U))$ the $k$th component of $S^{i,2}f^{*}\omega$ is
given by $dx^{k}\wedge f^{*}\omega$, while for the $k$th component of
$f^{*}(S^{i,2}\omega)$, we obtain

$\textstyle\sum_{j}a^{j}_{k}f^{*}(dx^{j}\wedge\omega)=\textstyle\sum_{j,\ell}a^{j}_{k}a^{j}_{\ell}dx^{\ell}\wedge
f^{*}\omega.$

But since $A^{t}=A^{-1}$ we get
$\sum_{j}a^{j}_{k}a^{j}_{\ell}=\delta_{k,\ell}$, the Kronecker delta, so we
again get $dx^{k}\wedge f^{*}\omega$.

This readily implies that the pullback is compatble with $d^{i}_{V}$ for each
$i$, so we obtain a pullback for the twisted complexes. Further the pullback
preseves each of the spaces $\mathcal{N}(S^{i,j})$ and $\mathcal{R}(S^{i,j})$
as well as their orthcomplements. Hence the pullback is also compatible with
the operators $T^{i,j}$ and there is an induced pullback on the spaces
$\Upsilon^{i}$. This in turn implies that all the further operations we
construct are compatible with the pullback and hence so are the BGG operators
$D^{i}$. The pullbacks on the spaces $\Upsilon^{i}$ can be made explicit, they
correspond to the natural pullback on tensor fields of appropriate type. In
particular, if some Euclidean motion maps a bounded Lipschitz domain to itself
than all our constructions are invariant under that motion.

For more general maps, the compatibility need not hold. The corresponding
phenomenon for BGG complexes related to projective differential geometry
involves affine transformations instead of Euclidean motions. The failure of
compatibility with more general maps was encountered in the context of the
isogeometric analysis [3] for the Hessian complex.

###  2.3. Conformal deformation complex

In the language of Section 2.1, we put $n=3$, $N=2$,
$\mathbb{V}_{0}=\mathbb{R}^{3}$,
$\mathbb{V}_{1}:=\mathfrak{o}(3)\oplus\mathbb{R}$, and
$\mathbb{V}_{2}=\mathbb{R}^{3*}$, so in the language of Section 3 we use
$\mathbb{V}=\mathfrak{g}$ (with a shift of the grading by $1$ for
consistency). The conceptual notation to use here is to write elements of
$Z^{i,0}$ as column vectors of $i$-forms $\phi_{\ell}$, and elements of
$Z^{i,2}$ as row vectors of $i$-forms $\omega_{\ell}$ for $\ell=1,2,3$.
Elements of $Z^{i,1}$ in this convention are viewed as $3\times 3$-matrices of
$i$-forms of the form
$(\psi_{\ell},\psi):=\begin{pmatrix}\psi&\psi_{3}&-\psi_{2}\\\
-\psi_{3}&\psi&\psi_{1}\\\ \psi_{2}&-\psi_{1}&\psi\end{pmatrix}$. Next,
$K^{i,1}$ is defined by acting with a matrix on the positions vector, i.e.

$K^{i,1}\begin{pmatrix}\psi&\psi_{3}&-\psi_{2}\\\ -\psi_{3}&\psi&\psi_{1}\\\
\psi_{2}&-\psi_{1}&\psi\end{pmatrix}=\begin{pmatrix}x^{1}\psi+x^{2}\psi_{3}-x^{3}\psi_{2}\\\
-x^{1}\psi_{3}+x^{2}\psi+x^{3}\psi_{1}\\\
x^{1}\psi_{2}-x^{2}\psi_{1}+x^{3}\psi\end{pmatrix}.$

As for the conformal Hessian complex, this readily leads to

$S^{i,1}\begin{pmatrix}\psi&\psi_{3}&-\psi_{2}\\\ -\psi_{3}&\psi&\psi_{1}\\\
\psi_{2}&-\psi_{1}&\psi\end{pmatrix}=\begin{pmatrix}dx^{1}\wedge\psi+dx^{2}\wedge\psi_{3}-dx^{3}\wedge\psi_{2}\\\
-dx^{1}\wedge\psi_{3}+dx^{2}\wedge\psi+dx^{3}\wedge\psi_{1}\\\
dx^{1}\wedge\psi_{2}-dx^{2}\wedge\psi_{1}+dx^{3}\wedge\psi\end{pmatrix}.$

The operator $K^{i,2}$ can be obtained from a Lie bracket with a position
vector, see Section 3, but one can simply start with an explicit formula:

$K^{i,2}(\omega_{1},\omega_{2},\omega_{3}):=\begin{pmatrix}-\textstyle\sum_{\ell}x^{\ell}\omega_{\ell}&x^{2}\omega_{1}-x^{1}\omega_{2}&x^{3}\omega_{1}-x^{1}\omega_{3}\\\
x^{1}\omega_{2}-x^{2}\omega_{1}&-\textstyle\sum_{\ell}x^{\ell}\omega_{\ell}&x^{3}\omega_{2}-x^{2}\omega_{3}\\\
x^{1}\omega_{3}-x^{3}\omega_{1}&x^{2}\omega_{3}-x^{3}\omega_{2}&-\textstyle\sum_{\ell}x^{\ell}\omega_{\ell}\end{pmatrix},$

and this readily gives

$S^{i,2}(\omega_{1},\omega_{2},\omega_{3}):=\begin{pmatrix}-\textstyle\sum_{\ell}dx^{\ell}\wedge\omega_{\ell}&dx^{2}\wedge\omega_{1}-dx^{1}\wedge\omega_{2}&dx^{3}\wedge\omega_{1}-dx^{1}\wedge\omega_{3}\\\
dx^{1}\wedge\omega_{2}-dx^{2}\wedge\omega_{1}&-\textstyle\sum_{\ell}dx^{\ell}\wedge\omega_{\ell}&dx^{3}\wedge\omega_{2}-dx^{2}\wedge\omega_{3}\\\
dx^{1}\wedge\omega_{3}-dx^{3}\wedge\omega_{1}&dx^{2}\wedge\omega_{3}-dx^{3}\wedge\omega_{2}&-\textstyle\sum_{\ell}dx^{\ell}\wedge\omega_{\ell}\end{pmatrix}.$

Now as for the conformal Hessian complex, there is just one instance of (3)
that needs to be checked, namely $K^{i+1,1}\circ S^{i,2}=S^{i,1}\circ
K^{i,2}$. This is verified by a simple direct computation: The first rows of
$K^{i+1,1}(S^{i,2}(\omega_{1},\omega_{2},\omega_{3}))$ and of
$S^{i,1}(K^{i,2}(\omega_{1},\omega_{2},\omega_{3}))$ are, respectively, given
by

$\displaystyle-\textstyle\sum_{\ell}x^{1}dx^{\ell}\wedge\omega_{\ell}+x^{2}dx^{2}\wedge\omega_{1}-x^{2}dx^{1}\wedge\omega_{2}+x^{3}dx^{3}\wedge\omega_{1}-x^{3}dx^{1}\wedge\omega_{3}$
$\displaystyle-\textstyle\sum_{\ell}x^{\ell}dx^{1}\wedge\omega_{\ell}+x^{2}dx^{2}\wedge\omega_{1}-x^{1}dx^{2}\wedge\omega_{2}+x^{3}dx^{3}\wedge\omega_{1}-x^{1}dx^{3}\wedge\omega_{3},$

so these agree. Similarly, one verifies that the other rows agree. Converting
to vector proxies, we obtain the following BGG diagram

(28)
${0}$${H^{q_{0}}\otimes\mathbb{V}}$${H^{q_{0}-1}\otimes\mathbb{M}}$${H^{q_{0}-2}\otimes\mathbb{M}}$${H^{q_{0}-3}\otimes\mathbb{V}}$${0}$${0}$${H^{q_{1}}\otimes(\mathbb{R}\oplus\mathbb{V})}$${H^{q_{1}-1}\otimes(\mathbb{V}\oplus\mathbb{M})}$${H^{q_{1}-2}\otimes(\mathbb{V}\oplus\mathbb{M})}$${H^{q_{1}-3}\otimes(\mathbb{R}\oplus\mathbb{V})}$${0}$${0}$${H^{q_{2}}\otimes\mathbb{V}}$${H^{q_{2}-1}\otimes\mathbb{M}}$${H^{q_{2}-2}\otimes\mathbb{M}}$${H^{q_{2}-3}\otimes\mathbb{V}}$${0.}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{K^{0,1}}$$\scriptstyle{S^{0,1}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{K^{1,1}}$$\scriptstyle{S^{1,1}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{K^{2,1}}$$\scriptstyle{S^{2,1}}$$\scriptstyle{K^{3,1}}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{K^{0,2}}$$\scriptstyle{S^{0,2}}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{K^{1,2}}$$\scriptstyle{S^{1,2}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{K^{2,2}}$$\scriptstyle{S^{2,2}}$$\scriptstyle{K^{3,2}}$

The $S$ and $K$ operators can alternatively be obtained from the diagram [11,
(32)] where we have maps between scalar and vector valued de-Rham complexes.
Then the corresponding operators in (28) is a combination of them.
Specifically, $S^{0,1}=(\iota,-\operatorname{mskw})^{T}$, i.e.,
$S^{0,1}(w,v)=wI-\operatorname{mskw}v$. Similarly,
$S^{1,1}=(-\operatorname{mskw},\mathcal{S})^{T}$,
$S^{2,1}=(I,2\operatorname{vskw})^{T}$,
$S^{0,2}=(I,-\operatorname{mskw})^{T}$,
$S^{1,2}=(2\operatorname{vskw},\mathcal{S})^{T}$,
$S^{2,2}=(\operatorname{tr},2\operatorname{vskw})^{T}$. The $K$ operators are
defined by $K^{1,i}=(x\otimes,x\wedge)^{T}$, $K^{2,i}=(x\cdot,x\wedge)$.

To understand the form of the resulting BGG complex, we observe that the
definitions easily imply that the linear maps $\partial^{i,j}$ that underlie
$S^{0,1}$, $S^{0,2}$ and $S^{1,2}$ are injective while those for $S^{1,1}$,
$S^{1,2}$ and $S^{2,2}$ are surjective. Since
$\partial^{1,1}\circ\partial^{0,2}=0$, the dimensions of the spaces in
question imply that $\mathcal{N}(\partial^{1,1})=\mathcal{R}(\partial^{0,2})$
so the corresponding statement for the $S$-operators follows. In the same way,
we conclude that $\mathcal{N}(S^{2,1})=\mathcal{R}(S^{1,2})$. This implies
that non-zero harmonic parts $\Upsilon^{i,j}$ in (11) only occur for
$(i,j)=(0,0)$, $(1,0)$, $(2,2)$, and $(3,2)$ and we conclude that the
resulting BGG complex is

(29)
${0}$${H^{q}\otimes\mathbb{V}}$${H^{q-1}\otimes(\mathbb{S}\cap\mathbb{T})}$${H^{q-4}\otimes(\mathbb{S}\cap\mathbb{T})}$${H^{q-5}\otimes\mathbb{V}}$${0.}$$\scriptstyle{\mathrm{dev}\operatorname{def}}$$\scriptstyle{\cot}$$\scriptstyle{\operatorname{div}}$

Here
$\mathrm{dev}\operatorname{def}=\mathrm{dev}\operatorname{sym}\operatorname{grad}$
is the symmetric trace-free part of the gradient, and
$\cot:={\operatorname{curl}\mathcal{S}^{-1}\operatorname{curl}\mathcal{S}^{-1}\operatorname{curl}}$
leads to the linearized Cotton-York tensor with modified trace. Compatibility
of the construction with Euclidean motions can be obtained similarly as for
the conformal Hessian complex. However, this time the action of a motion mixes
components in all three rows. We do not go into details of this here.

###  2.4. Higher order generalizations of the Hessian complex

The next example deals with an arbitrary number of rows. It should be
mentioned that the setup here is unusally simple, in general it is very hard
to treat complexes in higher orders simultaneously. Indeed, the Hessian
complex is the simplest example of a BGG complex (beyond the de-Rham complex).
For simplicity, we again restrict to the case $n=3$, but we take an arbitrary
number $N\in\mathbb{N}$ and define $\mathbb{V}_{j}$ to be the symmetric power
$\odot^{j}\mathbb{R}^{3*}$ for $j=0,\dots,N$, so in particular
$\mathbb{V}_{0}=\mathbb{R}$ and $\mathbb{V}_{1}=\mathbb{R}^{3*}$ (the
definition for symmetric powers can be found in, e.g., [43]). Elements of
$Z^{i,0}$ will be viewed as $i$-forms, while for $i>0$, we view an element of
$Z^{i,j}$ as a family $\phi_{k_{1}\dots k_{j}}$ of differential forms of
degree $i$ indexed by $j$-numbers $k_{1},\dots,k_{j}\in\\{1,2,3\\}$ which are
completely symmetric, i.e. $\phi_{k_{1}\dots k_{j}}=\phi_{k_{\sigma_{1}}\dots
k_{\sigma_{j}}}$ for any permutation $\sigma$ of $j$ elements.

In this language, we define $K^{i,j}$ for $j\geq 1$ by

$K^{i,j}(\phi)_{k_{1},\dots,k_{j-1}}:=\textstyle\sum_{\ell}x^{\ell}\phi_{\ell
k_{1}\dots k_{j-1}},$

so symmetry of the family $\phi_{k_{1}\dots k_{j}}$ implies symmetry of
$K^{i,j}(\phi)_{k_{1}\dots k_{j-1}}$. As before, this immediately implies that

$S^{i,j}(\phi)_{k_{1},\dots,k_{j-1}}:=\textstyle\sum_{\ell}dx^{\ell}\wedge\phi_{\ell
k_{1}\dots k_{j-1}}.$

Hence we conclude that $S^{i,j}$ is indeed induced by composition with a
linear map
$\partial^{i,j}:\Lambda^{i}\mathbb{R}^{3*}\otimes\odot^{j}\mathbb{R}^{3*}\to\Lambda^{i+1}\mathbb{R}^{3*}\otimes\odot^{j-1}\mathbb{R}^{3*}$.
To verify (3) we observe that for $j\geq 2$, we get

$S^{i,j-1}(K^{i,j}(\phi))_{k_{1}\dots
k_{j-2}}=\textstyle\sum_{\ell,m}x^{\ell}dx^{m}\wedge\phi_{m\ell k_{1}\dots
k_{j-2}},$

while $K^{i+1,j-1}(S^{i,j}(\phi))_{k_{1}\dots k_{j-2}}$ is given by the same
expression with $\phi_{\ell mk_{1}\dots k_{j-2}}$ instead. Hence (3) holds by
symmetry of the indices of $\phi$. Thus we have completed all verifications
needed to get the machinery going and it remains to understand what the
resulting BGG complex looks like.

We first observe that $\partial^{0,1}:\mathbb{R}^{3*}\to\mathbb{R}^{3*}$ is
the identity, so in the case $q_{j}=q-j$, which is relevant for step 2,
$S^{0,1}$ is the identity, too. In the next “diagonal” we get

$\odot^{2}\mathbb{R}^{3*}\overset{\partial^{0,2}}{\longrightarrow}\mathbb{R}^{3*}\otimes\mathbb{R}^{3*}\overset{\partial^{1,1}}{\longrightarrow}\Lambda^{2}\mathbb{R}^{3*},$

and $\partial^{0,2}$ is just the inclusion, while $\partial^{1,1}$ is the
alternation. So we conclude that $S^{0,2}$ is injective,
$\mathcal{R}(S^{0,2})=\mathcal{N}(S^{1,1})$ and $S^{1,1}$ is surjective. The
next diagonals look uniform for $j=3,\dots,N$. One always gets

$\odot^{j}\mathbb{R}^{3*}\overset{\partial^{0,j}}{\longrightarrow}\mathbb{R}^{3*}\otimes\odot^{j-1}\mathbb{R}^{3*}\overset{\partial^{1,j-1}}{\longrightarrow}\Lambda^{2}\mathbb{R}^{3*}\otimes\odot^{j-2}\mathbb{R}^{3*}\overset{\partial^{2,j-2}}{\longrightarrow}\Lambda^{3}\mathbb{R}^{3*}\otimes\odot^{j-3}\mathbb{R}^{3*}.$

Here $\partial^{0,j}$ is the inclusion while the following maps are induced by
alternations in the first $i+1$ arguments. The first space in this sequence
can be interpreted as homogeneous polyonomials of degree $j$ on
$\mathbb{R}^{n}$. Next, we have one-forms, whose coefficients are homogeneous
polynomials of degree $j-1$ and then two-forms with coefficients that are
homogenous of degree $j-2$. The last space consists of three-forms, whose
coefficients are homogeneous polynomials of degree $j-3$. Therefore, this is
sometimes called a polynomial de-Rham complex at degree $j$. It is well known
(see [8] or Section 6.1 of [60]) that this is exact for $j\geq 3$ with
$\partial^{0,j}$ injective and $\partial^{2,j-2}$ surjective. Hence for
$j=3,\dots,N$, $S^{0,j}$ is injective,
$\mathcal{N}(S^{i,j-i})=\mathcal{R}(S^{i-1,j-i+1})$ for $i=1,2$ and
$S^{2,j-2}$ is surjective. Indeed, this also deals with the remaining
diagonals, namely

$\displaystyle\mathbb{R}^{3*}\otimes\odot^{N}\mathbb{R}^{3*}\overset{\partial^{1,N}}{\longrightarrow}\Lambda^{2}\mathbb{R}^{3*}\otimes\odot^{N-1}\mathbb{R}^{3*}\overset{\partial^{2,N-1}}{\longrightarrow}\Lambda^{3}\mathbb{R}^{3*}\otimes\odot^{N-2}\mathbb{R}^{3*}$
$\displaystyle\Lambda^{2}\mathbb{R}^{3*}\otimes\odot^{N}\mathbb{R}^{3*}\overset{\partial^{2,N}}{\longrightarrow}\Lambda^{3}\mathbb{R}^{3*}\otimes\odot^{N-1}\mathbb{R}^{3*}.$

They both have nontrivial cohomology only in the first position, and this
cohomology is $S^{N+1}\mathbb{R}^{3*}$ for the first sequence. For the second
sequence, the cohomology is just the kernel $W$ of the alternation over the
first three indices in
$\Lambda^{2}\mathbb{R}^{3*}\otimes\odot^{N}\mathbb{R}^{3*}$. Since we
obviously have cohomology $\mathbb{R}$ in degree $(0,0)$ and
$\Lambda^{3}\mathbb{R}^{3*}\otimes\odot^{N}\mathbb{R}^{3*}$ in degree $(3,N)$,
we conclude that the BGG sequence gets the form

(30)
${0}$${H^{q}}$${H^{q-N-1}\otimes(\odot^{N+1}\mathbb{R}^{3*})}$${H^{q-N-2}\otimes
W}$${H^{q-N-3}\otimes\odot^{N}\mathbb{R}^{3*}}$${0.}$$\scriptstyle{D_{0}}$$\scriptstyle{D_{1}}$$\scriptstyle{D_{2}}$

Here $D_{0}$ is just the $(N+1)$st derivative of a function, while the other
two operators are of first order.

##  3\. Algebraic and geometric background

Here we want to briefly sketch the relation of our constructions and of the
examples discussed in Section 2 to the notions of BGG resolutions in geometry
as developed in [21]. The motivation for the construction and the basic input
for the examples comes from representation theory. Representation theory in
particular provides the maps $\partial^{i,j}$ used in Section 2.1 and hence
the operators $S^{i,j}$. It also provides similar maps inducing the operators
${T^{i,j}}$, but these are not needed here. The geometric version works in the
general setting of parabolic subalgebras in semi-simple Lie algebras, but we
focus on two special cases here. In both these cases, one deals with a real
simple Lie algebra $\mathfrak{g}$, which admits a Lie algebra grading of the
form
$\mathfrak{g}=\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$.
This means that $\mathfrak{g}_{-1}$ and $\mathfrak{g}_{1}$ are abelian
subalgebras (so they are just vector spaces) and $\mathfrak{g}_{0}$ is a Lie
subalgebra of $\mathfrak{g}$. Via the bracket, $\mathfrak{g}_{0}$ acts on
$\mathfrak{g}_{-1}$ and $\mathfrak{g}_{1}$, and general results imply that
these two representations are always dual to each other.

The first case originates in projective differential geometry and relates for
example to the elasticity complex, see Section 5 below, and to the higher
order analogs of the Hessian complex discussed in Section 2.4. Here
$\mathfrak{g}=\mathfrak{sl}(n+1,\mathbb{R})$, the Lie algebra of trace-free
$(n+1)\times(n+1)$-matrices, $\mathfrak{g}_{0}=\mathfrak{gl}(n,\mathbb{R})$,
$\mathfrak{g}_{-1}\cong\mathbb{R}^{n}$ and
$\mathfrak{g}_{1}\cong\mathbb{R}^{n*}$. The decomposition of $\mathfrak{g}$
can be simply realized by decomposing (trace-free) matrices of size
$(n+1)\times(n+1)$ into blocks of sizes $1$ and $n$ and viewing elements of
$\mathbb{R}^{n}$ as column vectors and elements of $\mathbb{R}^{n*}$ as row
vectors. Explicitly, denoting by $\mathbb{I}$ the $n\times n$ unit matrix, the
matrix corresponding to $v\in\mathbb{R}^{n}=\mathfrak{g}_{-1}$,
$B\in\mathfrak{gl}(n,\mathbb{R})=\mathfrak{g}_{0}$, and
$\lambda\in\mathbb{R}^{n*}=\mathfrak{g}_{1}$ is given by

(31) $\begin{pmatrix}-\tfrac{1}{n+1}\operatorname{tr}(B)&\lambda\\\
v&B-\tfrac{1}{n+1}\operatorname{tr}(B)\mathbb{I}\end{pmatrix}.$

Observe that this matrix is always trace-free. The non-obvious choice for the
identification of the block diagonal part is motivated by the fact that

$\left[\begin{pmatrix}-\tfrac{1}{n+1}\operatorname{tr}(B)&0\\\
0&B-\tfrac{1}{n+1}\operatorname{tr}(B)\mathbb{I}\end{pmatrix},\begin{pmatrix}0&\lambda\\\
v&0\end{pmatrix}\right]=\begin{pmatrix}0&-\lambda B\\\ Bv&0\end{pmatrix}.$

This shows that the action of $\mathfrak{g}_{0}$ on $\mathfrak{g}_{\pm 1}$ via
the commutator (which is the Lie bracket in $\mathfrak{g}$) coincides with the
standard action on $\mathbb{R}^{n}$ and $\mathbb{R}^{n*}$.

The second case relates to conformal geometry and is the basis for the
examples discussed in Sections 2.2 and 2.3. Here
$\mathfrak{g}=\mathfrak{so}(n+1,1)$, the Lie algebra of linear maps that are
skew symmetric with respect to a Lorentzian inner product on
$\mathbb{R}^{n+2}$ and $\mathfrak{g}_{0}=\mathfrak{co(n)}$, the Lie algebra of
$n\times n$-matrices spanned by skew symmetric matrices and multiples of the
identity. As above, $\mathfrak{g}_{-1}\cong\mathbb{R}^{n}$ and
$\mathfrak{g}_{1}\cong\mathbb{R}^{n*}$ are just the standard representation of
$\mathfrak{g}_{0}$ and its dual. One can realize $\mathfrak{g}$ as a Lie
algebra of matrices in such a way that the grading corresponds to a block-
decomposition with blocks of sizes $1$, $n$, and $1$, see Section 1.6.3 of
[20].

Now the starting point for the BGG construction is a representation
$\mathbb{V}$ of $\mathfrak{g}$, which can be assumed to be irreducible. By
restriction, $\mathbb{V}$ becomes a representation of $\mathfrak{g}_{0}$, but
this is not irreducible any more. One also obtains actions of
$\mathfrak{g}_{\pm 1}$ on $\mathbb{V}$ and one can decompose $\mathbb{V}$ into
a direct sum $\mathbb{V}=\mathbb{V}_{0}\oplus\dots\oplus\mathbb{V}_{N}$ for
some $N\geq 0$ in such a way that the action of $\mathfrak{g}_{1}$ maps each
$\mathbb{V}_{i}$ to $\mathbb{V}_{i+1}$ while the action of $\mathfrak{g}_{-1}$
maps each $\mathbb{V}_{i}$ to $\mathbb{V}_{i-1}$. (Here we use the convention
that $\mathbb{V}_{i}=\\{0\\}$ if $i<0$ or $i>N$.) The Lie algebra
$\mathfrak{g}$ is semi-simple and the resulting algebra $\mathfrak{g}_{0}$
turns out to be reductive. Hence representations of both algebras can be
understood in terms of so-called highest weights. In this language, there is
also a scheme for describing the possible first operators in a BGG sequence
and the choice of $\mathbb{V}$ that is needed to produce a chosen operator
from that list, see [16]. This is beyond the scope of the current article,
however.

For example, corresponding to the block decomposition of
$\mathfrak{sl}(n+1,\mathbb{R})$ from above, we have to decompose the standard
representation $\mathbb{R}^{n+1}$ of $\mathfrak{g}$ into the top component of
a column vector in $\mathbb{R}^{n+1}$ and the remaining vector in
$\mathbb{R}^{n}$. Restricting to the case of trace-free matrices in
$\mathfrak{g}_{0}$, we see that

$\begin{pmatrix}0&\lambda\\\ v&B\end{pmatrix}\begin{pmatrix}a\\\
w\end{pmatrix}=\begin{pmatrix}\lambda(w)\\\ Bw+av\end{pmatrix}$

This shows that $\mathbb{V}_{0}=\mathbb{R}^{n}$ and
$\mathbb{V}_{1}=\mathbb{R}$ as representations of
$\mathfrak{sl}(n,\mathbb{R})\subset\mathfrak{g}_{0}$, the action
$\mathfrak{g}_{-1}\times\mathbb{V}_{1}\to\mathbb{V}_{0}$ sends $(v,a)$ to $av$
and the action $\mathfrak{g}_{1}\times\mathbb{V}_{0}\to\mathbb{V}_{1}$ is just
the dual pairing $\mathbb{R}^{n*}\times\mathbb{R}^{n}\to\mathbb{R}$.
Similarly, putting $\mathbb{V}=\mathbb{R}^{(n+1)*}$, the dual of the standard
representation, we get $\mathbb{V}_{0}=\mathbb{R}$ and
$\mathbb{V}_{1}=\mathbb{R}^{n*}$ with obvious actions of $\mathfrak{g}_{\pm
1}$. This choice of $\mathbb{V}$ leads to the Hessian complex. For the higher
order analogs discussed in Section 2.4 one has to use
$\mathbb{V}:=\odot^{N}\mathbb{R}^{(n+1)*}$, which accordingly decomposes as
indicated there.

Returning to a general representation $\mathbb{V}$, the action of
$\mathfrak{g}_{1}$ can be interpreted as defining a linear map
$\mathfrak{g}_{1}\otimes\mathbb{V}\to\mathbb{V}$, which maps
$\mathfrak{g}_{1}\otimes\mathbb{V}_{i}$ to $\mathbb{V}_{i+1}$ for each $i$. It
is a purely algebraic fact that this extends to a sequence of linear maps
$\Lambda^{k}\mathfrak{g}_{1}\otimes\mathbb{V}\to\Lambda^{k-1}\mathfrak{g}_{1}\otimes\mathbb{V}$,
which also send $\Lambda^{k}\mathfrak{g}_{1}\otimes\mathbb{V}_{i}$ to
$\Lambda^{k-1}\mathfrak{g}_{1}\otimes\mathbb{V}_{i+1}$. These are Lie algebra
homology differentials, but for historical reasons they are often referred to
as the Kostant codifferential and denoted by $\partial^{*}$. They satisfy
$\partial^{*}\circ\partial^{*}=0$ and hence define a complex, which computes
the Lie algebra homology of $\mathfrak{g}_{1}$ with coefficients in
$\mathbb{V}$.

For the action of $\mathfrak{g}_{-1}$, one uses a different but equivalent
encoding. Rather than as a map
$\mathfrak{g}_{-1}\times\mathbb{V}\to\mathbb{V}$, we view the action as
defining a map $\mathbb{V}\to L(\mathfrak{g}_{-1},\mathbb{V})$. Namely, we
send an element $v\in\mathbb{V}$ to the map $\mathfrak{g}_{-1}\to\mathbb{V}$,
$X\mapsto X\cdot v$. A general algebraic construction extends this to a map
$\partial$ that sends $k$-linear alternating maps
$(\mathfrak{g}_{-1})^{k}\to\mathbb{V}$ to $(k+1)$-linear alternating maps
$(\mathfrak{g}_{-1})^{k+1}\to\mathbb{V}$. This is the Lie algebra cohomology
differential, which in the case of the abelian Lie algebra $\mathfrak{g}_{-1}$
is explicitly given by

$\partial\phi(X_{0},\cdots,X_{k}):=\textstyle\sum_{i=0}^{k}(-1)^{i}X_{i}\cdot\phi(X_{0},\cdots,\widehat{X_{i}},\cdots,X_{k}),$

with the hat denoting omission. The construction via a representation of
$\mathfrak{g}$ ensures that this is equivariant for the action of
$\mathfrak{g}_{0}$ and it is easily seen to satisfy $\partial\circ\partial=0$.
Hence it defines a complex that computes the Lie algebra cohomology of
$\mathfrak{g}_{-1}$ with coefficients in $\mathbb{V}$. Using the duality
between $\mathfrak{g}_{-1}$ and $\mathfrak{g}_{1}$, the space of $k$-linear
maps from above can again be interpreted as
$\Lambda^{k}\mathfrak{g}_{1}\otimes\mathbb{V}$. Thus we can interpret both
$\partial^{*}$ and $\partial$ as acting on
$\Lambda^{\scriptscriptstyle\bullet}\mathfrak{g}_{1}\otimes\mathbb{V}$, one
lowering and one raising the degree in the exterior algebra, and both
preserving the total degree (which is $k+i$ on
$\Lambda^{k}\mathfrak{g}_{1}\otimes\mathbb{V}_{i}$).

The whole setup was introduced (in the setting of general parabolic
subalgebras) in B. Kostant’s work [51], where it was next shown that
$\partial$ and $\partial^{*}$ are adjoint with respect to an appropriate inner
product on $\Lambda^{*}\mathfrak{g}_{1}\otimes\mathbb{V}$. Defining the
Kostant Laplacian
$\square:=\partial^{*}\circ\partial+\partial\circ\partial^{*}$ one obtains a
map that sends each $\Lambda^{k}\mathfrak{g}_{1}\otimes\mathbb{V}$ to itself.
The constructions easily imply that both $\partial$ and $\partial^{*}$ are
$\mathfrak{g}_{0}$-equivariant, and one obtains an algebraic Hodge
decomposition

(32)
$\Lambda^{k}\mathfrak{g}_{1}\otimes\mathbb{V}=\mathcal{R}(\partial)\oplus\mathcal{N}(\square)\oplus\mathcal{R}(\partial^{*})$

into $\mathfrak{g}_{0}$-invariant subspaces for each $k$. This has the
property that the first two summands add up to $\mathcal{N}(\partial)$ while
the last two summands add up to $\mathcal{N}(\partial^{*})$. This easily
implies that $\partial$ restricts to an isomorphism between the
$\mathcal{R}(\partial^{*})$-component in
$\Lambda^{k}\mathfrak{g}_{1}\otimes\mathbb{V}$ and the
$\mathcal{R}(\partial)$-component in
$\Lambda^{k+1}\mathfrak{g}_{1}\otimes\mathbb{V}$, and $\partial^{*}$ restricts
to an isomorphism in the opposite direction. The next step in [51] is showing
that the map $\square$ can be nicely interpreted in terms of representation
theory. This leads to an explicit description of $\mathcal{N}(\square)$ and
hence of the Lie algebra (co)homology spaces described above in representation
theory terms, which is known as Kostant’s version of the Bott-Borel-Weyl
theorem.

This setup can be translated to geometry, more precisely to projective
differential geometry in dimension $n\geq 2$ in the case that
$\mathfrak{g}=\mathfrak{sl}(n+1,\mathbb{R})$ and to conformal differential
geometry of dimension $n\geq 3$ for $\mathfrak{g}=\mathfrak{so}(n+1,1)$. The
key issue here is that representations of $\mathfrak{gl}(n,\mathbb{R})$
correspond to natural vector bundles on smooth manifolds of dimension $n$.
Similarly, representations of $\mathfrak{co}(n)$ give rise to natural vector
bundles on $n$-dimensional manfiolds endowed with a Riemannian metric.
(Indeed, conformal equivalence class of such metrics suffices.) In both cases,
the standard representation $\mathbb{R}^{n}\cong\mathfrak{g}_{-1}$ corresponds
to the tangent bundle $TM$ and its dual $\mathbb{R}^{n*}\cong\mathfrak{g}_{1}$
corresponds to the cotangent bundle $T^{*}M$. General representations of
$\mathfrak{gl}(n,\mathbb{R})$ respectively vector representations of
$\mathfrak{co}(n)$ can then be obtained from these two representations via
constructions that can also be applied to vector bundles. Moreover,
$\mathfrak{g}_{0}$-equivariant linear maps between representations induce
natural bundle maps between the corresponding natural vector bundles.

We only describe the next steps in the simpler setting of a manifold $M$
endowed with a linear connection $\nabla$ on $TM$ respectively with a
Riemannian metric for which $\nabla$ is the Levi-Civita connection. Then there
is an induced connection on any of the natural vector bundles described above
and we will denote all these connections by $\nabla$. Given a representation
$\mathbb{V}$ of $\mathfrak{g}$ as above, we can simply view it as a
representation of $\mathfrak{g}_{0}$ and then consider the corresponding
natural bundle on some manifold $M$, which we denote by $\mathcal{V}M\to M$.
(To deal with the setting of projective or conformal differential geometry, a
different interpretation is needed, but we don’t go into these aspects here.)
The representations $\Lambda^{k}\mathfrak{g}_{1}\otimes\mathbb{V}$ then
correspond to the bundles $\Lambda^{k}T^{*}M\otimes\mathcal{V}M$ whose
sections are $\mathcal{V}M$-valued $k$-forms. Hence the maps $\partial$ and
$\partial^{*}$ from above give rise to natural bundle maps on these bundles of
differential forms. On the other hand, the connection $\nabla$ on
$\mathcal{V}M$ can be coupled to the exterior derivative on differential forms
to obtain the covariant exterior derivative on $\mathcal{V}M$-valued
differential forms.

Motivated by ideas from projective and conformal differential geometry (which
generalize to so-called parabolic geometries) one can use the connection
$\nabla$, the bundle map $\partial$ and some components of the curvature of
$\nabla$ to defined a new connection $\nabla^{\mathcal{V}}$ on the bundle
$\mathcal{V}M$. This can be arranged in such a way that $\nabla^{\mathcal{V}}$
is flat, even if the given connection $\nabla$ on $TM$ is only projectively
flat respectively the given Riemannian metric is only conformally flat. The
connection $\nabla^{\mathcal{V}}$ again can be coupled to the exterior
derivative to obtain an operation on $\mathcal{V}M$-valued forms that defines
a complex generalizing the de-Rham complex. Together with the bundle maps
induced by $\partial^{*}$, this operation can be used to obtain a BGG complex
of higher order natural differential operator from this twisted de-Rham
complex.

Things simplify considerably in the setting we use in this article. We
restrict to the case of the flat connection, respectively the flat metric, on
$\mathbb{R}^{n}$ or an open subset $U\subset\mathbb{R}^{n}$ and do not require
projective or conformal invariance. This allows us to ignore the maps
$\partial^{*}$ and focus on the maps $\partial$ (which can be modifed as long
as the kernel and the image remain unchanged). The map $\partial^{i,j}$ used
in Section 2.1 is then the restriciton of $\partial$ to
$\Lambda^{i}\mathbb{R}^{n*}\otimes\mathbb{V}_{j}$, which by construction has
values in $\Lambda^{i+1}\mathbb{R}^{n*}\otimes\mathbb{V}_{j-1}$. The algebraic
Hodge decomposition (32) gets replaced by (11) and we can define the operators
$T$ as partial inverses of $S$. Since the tangent bundle of $\mathbb{R}^{n}$
is trivialized by sections that are parallel for $\nabla$, the same holds for
any of the natural bundles described above. Hence $\mathcal{V}U$ can be
identified with $U\times\mathbb{V}$ and hence the space of
$\mathcal{V}U$-valued forms can be identified with
$\Omega^{k}(U)\otimes\mathbb{V}$ (and further with functions
$U\to\Lambda^{k}\mathbb{R}^{n*}\otimes\mathbb{V}$). Under this identification,
the covariant exterior derivative associated to $\nabla$ simply becomes
$d\otimes\operatorname{id}_{\mathbb{V}}$. Further, as a representation of
$\mathfrak{g}_{0}$, we have
$\mathbb{V}=\mathbb{V}_{0}\oplus\dots\oplus\mathbb{V}_{N}$, which gives the
splitting into rows that we use. Since $\nabla$ has trivial curvatures, we
only need the operators $S$ to construct $\nabla^{\mathcal{V}}$ and the
corresponding covariant exterior derivative $d_{V}$.

We use the convention that $\nabla^{\mathcal{V}}=\nabla-S$ and then
$d_{V}=d-S$. The fact that $\nabla^{\mathcal{V}}$ is flat is then equivalent
to $d_{V}\circ d_{V}=0$ and hence to (5) and (6). Flatness of the connection
$\nabla^{\mathcal{V}}$ implies that the bundle $\mathcal{V}\mathbb{R}^{n}$ can
be globally trivialized by sections that are parallel for
$\nabla^{\mathcal{V}}$. This trivialization is the basis for the isomorphism
in Theorem 1. In the special case of $\mathbb{R}^{n}$, there is a neat
description of this isomorphism using the operators we call $K$ in (3). This
description has no analog in the general theory and is not projectively or
conformally invariant.

##  4\. Conformal Korn inequalities in 2D

In this section, we use $\|u\|:=(\int_{U}u^{2}\,dx)^{\frac{1}{2}}$ and
$\|u\|_{1}:=(\|u\|^{2}+\|\nabla u\|^{2})^{\frac{1}{2}}$ to denote the $L^{2}$
and $H^{1}$ norms, respectively, and similarly for higher Sobolev norms. The
(second) Korn inequality refers to

$\|u\|_{1}\leq C\|\operatorname{def}u\|,\quad\forall u\in
H^{1}(\Omega)\otimes\mathbb{R}^{n},\int_{\Omega}u\cdot q\,dx=0,~{}\forall
q\in\mathrm{RM},$

where $C$ is a constant not depending on $u$, and
$\mathrm{RM}:=\mathcal{N}(\operatorname{def})$ is the finite dimensional space
of Killing vector fields, i.e. infinitesimal rigid body motions (see, e.g.,
[32]). The orthogonality condition against $\mathrm{RM}$ is to exclude the
kernel of $\operatorname{def}$. There are other ways to fix the kernel. For
example, one has the first Korn inequality with boundary conditions.

The following generalization of the Korn inequality, sometimes referred to as
the conformal (trace-free) Korn inequality, holds

(33) $\|u\|_{1}\leq C\|\mathrm{dev}\operatorname{def}u\|,\quad\forall u\in
H^{1}(\Omega)\otimes\mathbb{R}^{n},~{}n\geq 3,~{}\int_{\Omega}u\cdot
q=0,~{}\forall q\in\mathcal{N}(\mathrm{dev}\operatorname{def}),$

where $\mathcal{N}(\mathrm{dev}\operatorname{def})$ is the space of conformal
Killing vectors, i.e. infinitesimal conformal motions. The trace-free Korn
inequality has various applications in, e.g., general relativity [37] and
continuum theory with microstructures [49]. The trace-free Korn inequality
only holds in $n$D for $n\geq 3$. The failure of this inequality in 2D is
discussed in [37].

The framework in [11] provides a systematic cohomological approach for
establishing Poincaré type inequalities. The key is that the differential
operators have closed range, which is further because the cohomology has
finite dimension. In this framework, the trace-free Korn inequality (33) holds
in 3D because the conformal deformation complex has finite dimensional
cohomology on Lipschitz domains, see [11]. Of course, this folows from the
results in 2.3. Different from the three-row diagram (28) with de-Rham
complexes there, [11] used either the first and the second rows or the second
and the third rows of the following diagram

(34)
${0}$${H^{q}\otimes\mathbb{V}}$${H^{q-1}\otimes\mathbb{T}}$${H^{q-2}\otimes\mathbb{S}}$${H^{q-4}}$${0}$${0}$${H^{q-1}\otimes\mathbb{V}}$${H^{q-2}\otimes\mathbb{S}}$${H^{q-4}\otimes\mathbb{S}}$${H^{q-5}\otimes\mathbb{V}}$${0}$${0}$${H^{q-2}}$${H^{q-4}\otimes\mathbb{S}}$${H^{q-5}\otimes\mathbb{T}}$${H^{q-6}\otimes\mathbb{V}}$${0.}$$\scriptstyle{\mathrm{dev}\operatorname{grad}}$$\scriptstyle{\operatorname{sym}\operatorname{curl}}$$\scriptstyle{\operatorname{div}\operatorname{div}}$$\scriptstyle{\operatorname{def}}$$\scriptstyle{-\operatorname{mskw}}$$\scriptstyle{\operatorname{inc}}$$\scriptstyle{\mathcal{S}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{\operatorname{tr}}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\iota}$$\scriptstyle{\operatorname{curl}}$$\scriptstyle{\mathcal{S}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{2\operatorname{vskw}}$

It is also observed in [11] that the two dimensional version of (34), i.e.,

(35)
${0}$${H^{q}\otimes\mathbb{V}}$${H^{q-1}\otimes\mathbb{S}}$${H^{q-3}}$${0}$${0}$${H^{q-1}}$${H^{q-3}\otimes\mathbb{S}}$${H^{q-4}}$${0,}$$\scriptstyle{{\operatorname{def}}}$$\scriptstyle{\operatorname{rot}\operatorname{rot}}$$\scriptstyle{{\operatorname{hess}}}$$\scriptstyle{\iota}$$\scriptstyle{\operatorname{rot}}$$\scriptstyle{\operatorname{tr}}$

does not satisfy the injectivity/surjectivity conditions of [11]. This is
consistent with the fact that the trace-free Korn inequality does not hold in
two space dimensions.

Based on the construction in this paper, we now fix the trace-free Korn
inequality in two space dimensions by adding terms involving third order
operators. In the language of Section 3, the conformal deformation complex can
be derived in any dimension $n\geq 3$ starting from
$\mathfrak{g}=\mathfrak{so}(n+1,1)$ and the adjoint representation
$\mathbb{V}=\mathfrak{g}$. We use the straightforward analog of this
construction obtained from $\mathfrak{g}=\mathfrak{so}(3,1)$ and
$\mathbb{V}=\mathfrak{g}$. We know from Section 3 that
$\mathfrak{g}_{0}=\mathfrak{co}(n)$, so for $n=2$, this just gives
$\mathbb{C}$ viewed as a subspace of $M_{2}(\mathbb{R})$. Defining the
operators $K^{i,j}$ and $S^{i,j}$ as in higher dimensions, one immediately
concludes that $S^{1,0}$and $S^{2,0}$ are injective while $S^{1,1}$ and
$S^{1,2}$ are surjective. Consequently, in degree one the ”cohomology”
$\Upsilon^{1}=\mathcal{R}(S^{2})^{\perp}\cap\mathcal{N}(S^{1})$ has two
components which both are functions with values in a two-dimensional space.
One component sits in the top row and hence gives rise to a first order
operator, which can be most easily interpreted as the Cauchy-Riemann operator.
The other component sits in the bottom row and thus determines a third order
operator. In degree two, the cohomology sits in the bottom row, so one obtains
operators of order 3 and 1. Passing to vector proxies the resulting diagram
looks as follows.

(36)
${0}$${H^{q}\otimes\mathbb{V}}$${H^{q-1}\otimes\mathbb{M}}$${H^{q-2}\otimes\mathbb{V}}$${0}$${0}$${H^{q-1}\otimes(\mathbb{R}\times\mathbb{R})}$${H^{q-2}\otimes(\mathbb{V}\times\mathbb{V})}$${H^{q-3}\otimes(\mathbb{R}\times\mathbb{R})}$${0}$${0}$${H^{q-2}\otimes\mathbb{V}}$${H^{q-3}\otimes\mathbb{M}}$${H^{q-4}\otimes\mathbb{V}}$${0.}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{\operatorname{rot}}$$\scriptstyle{T^{1,0}}$$\scriptstyle{T^{2,0}}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{S^{0,1}}$$\scriptstyle{\operatorname{rot}}$$\scriptstyle{T^{1,1}}$$\scriptstyle{S^{1,1}}$$\scriptstyle{T^{2,1}}$$\scriptstyle{\operatorname{grad}}$$\scriptstyle{S^{0,2}}$$\scriptstyle{\operatorname{rot}}$$\scriptstyle{S^{1,2}}$

Here $\operatorname{rot}(u_{1},u_{2})=-\partial_{2}u_{1}+\partial_{1}u_{2}$,
$S^{0,1}=(\iota,\operatorname{mskw})^{T}$, $S^{1,1}=(-\perp,I)^{T}$,
$S^{0,2}=(I,\perp)$, and $S^{1,2}=(2\operatorname{sskw},-\operatorname{tr})$,
with $\perp u{=\perp(u_{1},u_{2})}:=(-u_{2},u_{1})$,
$\operatorname{sskw}\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right):=\frac{1}{2}(b-c)$,
$\operatorname{mskw}u:=\left(\begin{array}[]{cc}0&-u\\\
u&0\end{array}\right).$ These operators can be obtained by (4) with
$K^{i,1}(u,v):=x\otimes u+x^{\perp}\otimes v$, and $K^{i,2}(u):=(x\cdot
u,-x^{\perp}\cdot u)$. For (36),
$T^{1,0}=(\frac{1}{2}\operatorname{tr},-\operatorname{sskw})$,
$T^{2,0}={\frac{1}{2}}(\perp,I)$, $T^{1,1}={\frac{1}{2}}(I,-\perp)^{T}$, and
$T^{2,1}=(-\operatorname{mskw},-\frac{1}{2}\iota)$.

From the matrix form of the operators (48), the BGG operators have the form

$\displaystyle D^{0}\left(\begin{array}[]{c}u\\\ 0\\\
0\end{array}\right)=\left(\begin{array}[]{c}P_{\mathcal{R}^{\perp}}du\\\ 0\\\
(dT)^{2}du\end{array}\right)$
$\displaystyle=\left(\begin{array}[]{c}\mathrm{dev}\operatorname{def}\operatorname{grad}u\\\
0\\\
\operatorname{grad}T^{1,1}\operatorname{grad}T^{1,0}\operatorname{grad}u\end{array}\right),$

and

$\displaystyle D^{1}\left(\begin{array}[]{c}u\\\ 0\\\
v\end{array}\right)=\left(\begin{array}[]{c}0\\\ 0\\\
(dT)^{2}du+d{v}\end{array}\right),$

where

$\displaystyle(dT)^{2}$ $\displaystyle
du+du=\operatorname{rot}T^{2,1}\operatorname{rot}T^{2,0}\operatorname{rot}u+\operatorname{rot}v.$

The output complex is

(37)
${0}$${H^{q}\otimes\mathbb{V}}$${\left(\begin{array}[]{c}H^{q-1}\otimes(\mathbb{S}\cap\mathbb{T})\\\
H^{q-3}\otimes(\mathbb{S}\cap\mathbb{T})\end{array}\right)}$${H^{q-4}}$${0,}$$\scriptstyle{{{D}^{0}}}$$\scriptstyle{{D}^{1}}$

where

${D}^{0}:=\left(\begin{array}[]{c}\mathrm{dev}\operatorname{def}\operatorname{grad}\\\
\operatorname{grad}T^{1,1}\operatorname{grad}T^{1,0}\operatorname{grad}\end{array}\right),\quad{D}^{1}=\left(\begin{array}[]{c}\operatorname{rot}T^{2,1}\operatorname{rot}T^{2,0}\operatorname{rot}\\\
\operatorname{rot}\end{array}\right)^{T}.$

To summarize, we get the following.

######  Theorem 3 (Conformal Korn inequality in two space dimensions).

Let $\Omega\subset\mathbb{R}^{2}$ be a bounded Lipschitz domain. There exists
a positive constant $C$, such that

$\|u\|_{3}\leq
C(\|\mathrm{dev}\operatorname{def}\operatorname{grad}u\|_{2}+\|\operatorname{grad}T^{1,1}\operatorname{grad}T^{1,0}\operatorname{grad}u\|).$

######  Remark 2.

In the classical approaches for the Korn inequalities, the proof for results
in $L^{p}$ with $p\neq 2$ are more difficult than the case of $p=2$. With
small modifications, the complex-based argument in [11] and in this paper
leads to a unified proof for inequalities in a broad range of function spaces,
i.e., those spaces in [36] for the de-Rham complexes. It is known that the
Korn and conformal Korn inequalities do not hold for $W^{1,p}$ with $p=1$ or
$p=\infty$ [38, Chapter 7], [17, 18]. This is consistent with the fact that
the results in [36] exclude these two cases.

As indicated above, the main difference between the 2D complex and conformal
deformation complexes in higher dimensions lies in the location of (Lie
algebraic) cohomology, which affects the order of the operators in the
resulting BGG sequences. This already occurs in dimensions $n\geq 3$, where it
reflects the fact that the fundamental conformal invariants change. While in
dimension $3$, this fundamantal invariant is the Cotton-York tensor that
depends on third derivatives of a metric in the conformal class, the
fundamental invariant for $n\geq 4$ is the Weyl curvature, which only depends
on second derivatives of such a metric. The special form of the complex in
dimension $2$ also has a geometric background. This is not related to
conformal geometry but to a refinement. Indeed, the isomorphism
$\mathfrak{co}(2)=\mathbb{C}$ observed above implies that in dimension $2$,
conformal geometry reduces to complex analysis, which also explains the
occurrence of the Cauchy-Riemann operator in our complex. To obtain a two
dimensional analog of higher-dimensional conformal geometry, one has to add to
a conformal structure an additional ingredient called a Möbius-structure. This
is given by a symmetric tensor of rank two, which plays the role of an
abstract version of the Schouten tensor (a trace-adjustment of the Ricci
curvature) in higher dimensions. The third order part in $D^{0}$ describes the
infinitesimal change of that tensor under a diffeomorphism, while the Cauchy-
Riemann operator describes the infinitesimal change of the conformal
structure. In analogy to higher dimensional conformal geometry, a Möbius-
structure always has a finite-dimensional automorphism group with dimension
bounded above by $\dim(\mathfrak{so}(3,1))=6$.

##  5\. Continuum with microstructures

In this section, we establish a cohomological approach for the linear Cosserat
models and generalize these models from this new perspective.

###  5.1. Linear Cosserat model as a Hodge-Laplacian

We will show that the linear Cosserat model corresponds to a Hodge-Laplacian
boundary value problem of the twisted de-Rham complex that corresponds to the
elasticity complex. To make this statement precise, we will follow the
framework of Hilbert complexes [4] and investigate the $L^{2}$ twisted de-Rham
complex with unbounded operators and its domain complex. The Sobolev
regularity required for this purpose is slightly different from the setting in
[11] for deriving the elasticity complex. The difference is similar to the
situation discussed in Section 2.1.

In the setting of Sections 2.1 and 3, we take $n=3$,
$\mathfrak{g}=\mathfrak{sl}(n+1,\mathbb{R})$ and
$\mathbb{V}=\Lambda^{2}\mathbb{R}^{(n+1)*}$. This leads to $N=2$,
$\mathbb{V}_{0}=\mathbb{R}^{3*}$ and
$\mathbb{V}_{1}=\Lambda^{2}\mathbb{R}^{3*}$. We denote elements of $Z^{i,0}$
as triples $(\phi_{1},\phi_{2},\phi_{3})$ of $i$-forms and elements of
$Z^{i,1}$ as families $\psi_{jk}$ of $i$-forms with two-skew symmetric
indices, i.e. $\psi_{jj}=0$ and $\psi_{kj}=-\psi_{jk}$. Since there are just
two rows, we just have one K-operator, which is given by
$K(\psi)_{j}=\sum_{\ell}x^{\ell}\psi_{\ell j}$, which leads to
$S(\psi)_{j}=\sum_{\ell}dx^{\ell}\wedge\psi_{\ell j}$. In this case, no
additional identities need to be verified, so our machintery applies. The form
of [11] is then obtained by passing to vector proxies. This looks as follows.

Let

$d_{V}^{i}:=\left(\begin{array}[]{cc}d^{i}&-S^{i}\\\
0&d^{i}\end{array}\right),$

where $d^{i}=\operatorname{grad},\operatorname{curl},\operatorname{div}$ for
$i=0,1,2$, $S^{0}=-\operatorname{mskw}$, $S^{1}u:=u^{T}-\operatorname{tr}(u)I$
and $S^{2}=2\operatorname{vskw}$. We have $d^{i+1}S^{i}=-S^{i+1}d^{i}$, for
$i=0,1$ and this implies $d_{V}^{i+1}\circ d_{V}^{i}=0$, $i=0,1$.

For any linear operator $D$, define $H(D):=\\{u\in L^{2}:Du\in L^{2}\\}$. By
defintion, we have $H(d_{V}^{i})=H(d^{i})\oplus H(d^{i})$. Below we focus on
the following $L^{2}$ twisted de-Rham complex with unbounded linear operators

(38) ${0}$${\left(\begin{array}[]{c}L^{2}\otimes\mathbb{V}\\\
L^{2}\otimes\mathbb{V}\end{array}\right)}$${\left(\begin{array}[]{c}L^{2}\otimes\mathbb{M}\\\
L^{2}\otimes\mathbb{M}\end{array}\right)}$${\left(\begin{array}[]{c}L^{2}\otimes\mathbb{M}\\\
L^{2}\otimes\mathbb{M}\end{array}\right)}$${\left(\begin{array}[]{c}L^{2}\otimes\mathbb{V}\\\
L^{2}\otimes\mathbb{V}\end{array}\right)}$${0,}$$\scriptstyle{d_{V}^{0}}$$\scriptstyle{{d_{V}^{1}}}$$\scriptstyle{d_{V}^{2}}$

and its domain complex

(39)
${0}$${H(d_{V}^{0})}$${H(d_{V}^{1})}$${H(d_{V}^{2})}$${H(d_{V}^{3})}$${0.}$$\scriptstyle{d_{V}^{0}}$$\scriptstyle{{d_{V}^{1}}}$$\scriptstyle{d_{V}^{2}}$

Using a similar argument as [11, Theorem 1], we conclude that the cohomology
of (39) is isomorphic to the cohomology of the Sobolev complex

(40) ${0}$${\left(\begin{array}[]{c}H^{q}\otimes\mathbb{V}\\\
H^{q}\otimes\mathbb{V}\end{array}\right)}$${\left(\begin{array}[]{c}H^{q-1}\otimes\mathbb{M}\\\
H^{q-1}\otimes\mathbb{M}\end{array}\right)}$${\left(\begin{array}[]{c}H^{q-2}\otimes\mathbb{M}\\\
H^{q-2}\otimes\mathbb{M}\end{array}\right)}$${\left(\begin{array}[]{c}H^{q-3}\otimes\mathbb{V}\\\
H^{q-3}\otimes\mathbb{V}\end{array}\right)}$${0,}$$\scriptstyle{d_{V}^{0}}$$\scriptstyle{{d_{V}^{1}}}$$\scriptstyle{d_{V}^{2}}$

which further has isomorphic cohomology as the sum of the de-Rham complexes
for any real number $q$ by applying the argument in Section 1.2.

We briefly recall the abstract setting of the Hodge-Laplacian boundary value
problems [4, 8]. Let
$(W^{{\scriptscriptstyle\bullet}},\mathscr{D}^{{\scriptscriptstyle\bullet}})$
be a Hilbert complex with unbounded linear operators
$\mathscr{D}^{{\scriptscriptstyle\bullet}}$, and
$(V^{{\scriptscriptstyle\bullet}},\mathscr{D}^{{\scriptscriptstyle\bullet}})$
be its domain complex where $\mathscr{D}^{{\scriptscriptstyle\bullet}}$
becomes bounded. Let $V^{\ast}_{k}$ be the domain of $\mathscr{D}^{\ast}_{k}$,
the adjoint of ${\mathscr{D}^{k-1}}$. The Hodge-Laplacian operator is
$L^{k}:=\mathscr{D}^{\ast}_{k+1}\mathscr{D}_{k}+\mathscr{D}_{k-1}\mathscr{D}^{\ast}_{k}$,
with the domain $D(L^{k})=\\{u\in V^{k}\cap V_{k}^{\ast}:\mathscr{D}u\in
V_{k+1}^{\ast},\mathscr{D}^{\ast}u\in V^{k-1}\\}$.

From this perspective, once we specify a closed Hilbert complex, the Hodge-
Laplacian problems and their well-posedness follow from general arguments.

In our case, we choose (38) as
$(W^{{\scriptscriptstyle\bullet}},\mathscr{D}^{{\scriptscriptstyle\bullet}})$
and choose (39) as the domain complex. Our main claim is that with suitable
inner products, the Hodge-Laplacian problem at index 0 is the linear Cosserat
elasticity model. Next, we verify this claim by comparing to the formulation
in [53]. In fact, with certain metric $C$ for the space at index 1, we have

$\displaystyle(d_{V}^{0}(u,\omega),d_{V}^{0}(u,\omega))_{C}:$
$\displaystyle=\|\operatorname{grad}u+\operatorname{mskw}\omega\|_{C_{1}}^{2}+\|\operatorname{grad}\omega\|_{C_{2}}^{2}$
$\displaystyle:=\mu\|\operatorname{sym}\operatorname{grad}u\|^{2}+\frac{\mu_{c}}{2}\|2\operatorname{vskw}(\operatorname{grad}u+\operatorname{mskw}\omega)\|^{2}+\frac{\lambda}{2}\|\operatorname{div}u\|^{2}$
(41)
$\displaystyle\quad+\frac{\gamma+\beta}{2}\|\operatorname{sym}\operatorname{grad}\omega\|^{2}+\frac{\gamma-\beta}{4}\|\operatorname{curl}\omega\|^{2}+\frac{\alpha}{2}\|\operatorname{div}\omega\|^{2},$

where we have used the identities
$2\operatorname{vskw}\operatorname{grad}u=\operatorname{curl}u$ and
$\operatorname{vskw}\circ\operatorname{mskw}=\operatorname{mskw}^{-1}\circ\operatorname{skw}\circ\operatorname{mskw}=I$.
Here the metric $C$ is defined by the physical parameters. We refer to [4,
Chapter 8] for the use of weighted inner products for the Maxwell equations
and linear elasticity.

Now we see that (5.1) exactly corresponds to the linear Cosserat model in
[53]. In particular,
$\mu\|\operatorname{sym}\operatorname{grad}u\|^{2}+\frac{\mu_{c}}{2}\|2\operatorname{vskw}(\operatorname{grad}u+\operatorname{mskw}\omega)\|^{2}+\frac{\lambda}{2}\|\operatorname{div}u\|^{2}$
is the strain energy and
$\frac{\gamma+\beta}{2}\|\operatorname{sym}\operatorname{grad}\omega\|^{2}+\frac{\gamma-\beta}{4}\|\operatorname{curl}\omega\|^{2}+\frac{\alpha}{2}\|\operatorname{div}\omega\|^{2}$
is the curvature energy. Also note that $\operatorname{vskw}$ restricted on
skew-symmetric matrices corresponds to $\mathrm{axl}$ in [53], and $\omega$
corresponds to $-\phi$ in [53]. Moreover,
$\operatorname{grad}u+\operatorname{mskw}\omega$ is the infinitesimal first
Cosserat stretch tensor, which corresponds to the $d_{V}^{0}$ operator in the
twisted complex; $-\operatorname{grad}\omega$ is the micropolor curvature
tensor; $\mu,\lambda$ are the classical Lamé moduli; $\alpha,\beta,\gamma$ are
additional micropolar moduli with dimension of a force; $\mu_{c}$ is the
Cosserat couple modulus.

We also observe that the above discussions for the Hessian complex and its
twisted de-Rham version corresponds to the Kirchhoff and modified Reissner-
Mindlin plate models, respectively.

The twisted de-Rham complex corresponding to the Hessian complex in 2D [11]
is:

(42) ${0}$${\left(\begin{array}[]{c}H^{1}\\\
H^{1}\otimes\mathbb{V}\end{array}\right)}$${\left(\begin{array}[]{c}H(\operatorname{rot})\\\
H(\operatorname{rot})\otimes\mathbb{V}\end{array}\right)}$${\left(\begin{array}[]{c}L^{2}\\\
L^{2}\otimes\mathbb{V}\end{array}\right)}$${0,}$$\scriptstyle{d_{V}^{0}}$$\scriptstyle{{d_{V}^{1}}}$

where

$d_{V}^{0}=\left(\begin{array}[]{cc}\operatorname{grad}&-I\\\
0&\operatorname{grad}\end{array}\right),\quad
d_{V}^{1}=\left(\begin{array}[]{cc}\operatorname{rot}&-\mathrm{sskw}\\\
0&\operatorname{rot}\end{array}\right).$

The Hodge-Laplacian problem at index zero has the energy functional

(43)
$\|\operatorname{grad}u-\phi\|_{A}^{2}+\|\operatorname{grad}\phi\|_{B}^{2}.$

With proper inner products, this corresponds to a modified Reissner-Mindlin
plate model [6, p.1279]. The only difference between (43) and the original
Reissner-Mindlin model [42, (10)] is that $\|\operatorname{def}\phi\|_{B}^{2}$
is replaced by $\|\operatorname{grad}\phi\|_{B}^{2}$, which is equivalent due
to the Korn inequality.

The Hessian complex in 2D is

(44)
${0}$${H^{2}}$${H(\operatorname{rot};\mathbb{S})}$${L^{2}\otimes\mathbb{V}}$${0.}$$\scriptstyle{\operatorname{hess}}$$\scriptstyle{\operatorname{rot}}$

The first nontrivial Hodge-Laplacian problem of (44) describes the Kirchhoff
plate, which is the $\Gamma$-limit of the Reissner-Mindlin plate as the
thickness tends to zero. This connection, although not formulated in terms of
the twisted complex, was used to derive a discretization for the biharmonic
equation [5], [57, Chapter 5].

Our results on finite dimensional uniform representation of cohomology imply
the Poincaré inequalities for the twisted de-Rham complexes (although these
inequalities readily follow from a straightforward argument in this particular
case), and hence the well-posedness of the variational problem. More
importantly, fitting the linear Cosserat model in the cohomological framework
makes it possible to bring in tools and general results from the abstract
theory. For example, the Cosserat version of the “rigid body motion” consists
of functions in the kernel of $d_{V}^{0}$, which has the form $(a,b\wedge x)$,
where $a,b\in\mathbb{R}^{3}$ are constant vectors. This can be obtained by
transforming the kernel of $\operatorname{grad}$ in the sum de-Rham complex by
(9). Once we fix the domain complex, the general theory also determines the
boundary conditions and the formulations are automatically well-posed.
Moreover, we may obtain different formulations, e.g., Cosserat versions of the
primal formulation, the Hellinger-Reissner principle, the intrinsic
formulation [34], and the Hu-Washizu principle [61]. More importantly, the
idea of modelling defects of linear elasticity by the violation of the Saint-
Venant condition (vanishing of the linearized curvature encoded in the
elasticity complex) [1] provides a plausible way to investigate defects of
Cosserat models by generalizing the “Cosserat strain” $(e,p):=d_{V}^{0}(u,w)$
which satisfies the compatibility $d_{V}^{1}(e,p)=0$ to more general functions
that violate this condition. In fact, discussions in this direction can be
found in, e.g., [46, (2.14a, 2.15a)], where $d_{V}^{1}$ in the twisted de-Rham
complex is involved. However, a detailed discussion of the above issues is
beyond the scope of this article.

###  5.2. Potential generalizations

For potential generalizations, we first consider diagrams that lead to the
conformal deformation complex. In [11], the following diagram is used to
derive the conformal Hessian complex:

${0}$${H^{q}\otimes\mathbb{V}}$${H^{q-1}\otimes\mathbb{T}}$${H^{q-2}\otimes\mathbb{S}^{1}}$${H^{q-4}\otimes\mathbb{R}}$${0}$${0}$${H^{q-1}\otimes\mathbb{V}}$${H^{q-2}\otimes\mathbb{S}}$${H^{q-4}\otimes\mathbb{S}}$${H^{q-5}\otimes\mathbb{V}}$${0,}$$\scriptstyle{\mathrm{dev}\operatorname{grad}}$$\scriptstyle{\operatorname{sym}\operatorname{curl}}$$\scriptstyle{\operatorname{div}\operatorname{div}}$$\scriptstyle{\operatorname{def}}$$\scriptstyle{-\operatorname{mskw}}$$\scriptstyle{\operatorname{inc}}$$\scriptstyle{\mathcal{S}}$$\scriptstyle{\operatorname{div}}$$\scriptstyle{\operatorname{tr}}$

where we recall that $S^{1}:=u^{T}-\operatorname{tr}(u)I$. The motivation was
to eliminate the skew-symmetric part from $\mathrm{dev}\operatorname{grad}u$
to get the operator in the conformal Korn inequality.

From this diagram, we propose a generalized elasticity model with the energy

(45)
$\alpha\|\mathrm{dev}\operatorname{grad}\phi+\operatorname{mskw}u\|_{C}^{2}+\|\operatorname{def}u\|_{A}^{2},$

where $\|\cdot\|_{A}$ and $\|\cdot\|_{C}$ are weighted $L^{2}$ norms. In
particular, we are interested in the case where $\|\cdot\|_{A}$ is the
standard inner product in elasticity, i.e.,

$\|\operatorname{def}u\|_{A}^{2}:=\mu\|\operatorname{sym}\operatorname{grad}u\|^{2}+\frac{\lambda}{2}\|\operatorname{div}u\|^{2}.$

Then the formal limit $\alpha\rightarrow 0$ in (45) corresponds to the
standard elasticity.

Another possible generalization of the Cosserat model is from the three-row
diagram (28). In particular, the energy defined by the first operator
$d_{V}^{0}$ in the twisted complex is

(46)
$\|\operatorname{grad}u-\iota\sigma+\operatorname{mskw}\omega\|_{C_{1}}^{2}+\|{(\operatorname{grad}\sigma-\phi)+(\operatorname{grad}\omega+\operatorname{mskw}\phi)}\|_{C_{2}}^{2}+\|\operatorname{grad}\phi\|_{C_{3}}^{2}.$

Now the first term
$\|\operatorname{grad}u-\iota\sigma+\operatorname{mskw}\omega\|_{C_{1}}$
contains an additional $\iota\sigma$ term compared to the Cosserat model. This
term can be interpreted as a pointwise dilation, in addition to the rotational
degrees of freedom.

From the 2D diagram (36), we also obtain a generalization of the plate models,
with the energy

(47)
$\|\operatorname{grad}u-\iota\sigma-\operatorname{mskw}\omega\|_{C_{1}}^{2}+\|{(\operatorname{grad}\sigma-\phi)+(\operatorname{grad}\omega-\perp\phi)}\|_{C_{2}}^{2}+\|\operatorname{grad}\phi\|_{C_{3}}^{2}.$

For each of the models proposed above, analytic results, e.g. Poincaré
inequalities and well-posedness follow from standard argument based on
complexes [11, Chapter 2].

Other generalizations of the linear Cosserat model exist, e.g., [54] with a
conformally invariant curvature energy.

##  6\. Conclusion

In this paper, we established some BGG complexes with weak regularity and
computed their cohomology. The generality and the geometric and algebraic
structures allow us to fix the 2D conformal Korn inequality by adding a third
order term in addition to the Cauchy-Riemann operator and achieve
generalizations of the linear Cosserat elasticity model from a cohomological
perspective. These applications show a deep connection between analysis (e.g.
inequalities), geometry, homological algebra and continuum mechanics.

This relaxation of the injectivity/surjectivity conditions in [11] renders a
feasible approach to fit finite element spaces in the BGG diagrams. Therefore
numerical schemes with weakly imposed constraints (symmetry, trace-freeness
etc.) with standard de-Rham finite elements should be promising (c.f., [9]).
We proposed several model problems as generalizations of the Cosserat model
from a cohomological perspective. With the idea of the Erlangen program in
mind, we leave it as future work to investigate the mechanical significance,
numerics and nonlinear versions of the generalized models in Section 5.
Applications of these complexes in (numerical) general relativity, e.g.,
potential-based formulations in the direction of [15], can be also further
directions.

The complexes we have constructed have a close relation to general relativity
(in fact, the Cosserat model and general relativity were Cartan’s motivation
for his development of the concept of torsion [59]). For example, the
conformal deformation complex encodes the York split and the transverse-trace-
free (TT) gauge in the description of gravitational waves. Beig and Chrusciel
[15] discussed the applications of complexes in the linearized Einstein
constraint equations with the conformal deformation complex and the conformal
Hessian complexes playing a key role. Our results describe the cohomology (and
thus analytic properties) of these complexes. In [15] the authors proceed to
define the so-called momentum complex, by a construction that looks similar to
repeated application of the “two-step construction” in [11]. While the
momentum complex is not an example of a BGG sequence, we expect that our
methods can be applied to the study of this complex. As other applications to
general relativity, this will be taken up elsewhere.

## Acknowledgement

The first author is supported by the Austrian Science Fund (FWF): P33559-N.
The second author is supported by a Hooke Research Fellowship.

The authors would like to thank Douglas N. Arnold, Andrea Dziubek and Endre
Suli for helpful discussions.

## Appendix: Matrix form of operators

The operators involved in the theorems and proofs in Section 1 can be given in
an explicit component-wise form, which are listed below:

${G}=\sum_{k=0}^{\infty}(Td)^{k}T=\left(\begin{array}[]{cccccc}0&0&0&0&\cdots&0\\\
T&0&0&0&\cdots&0\\\ TdT&T&0&0&\cdots&0\\\ (Td)^{2}T&TdT&T&0&\cdots&0\\\
&\cdots&\cdots&\cdots&&\end{array}\right),$
$A=I-Hd_{V}=\left(\begin{array}[]{cccccc}I&0&0&0&\cdots&0\\\
Td&P_{\mathcal{N}}&0&0&\cdots&0\\\
(Td)^{2}&TdP_{\mathcal{N}}&P_{\mathcal{N}}&0&\cdots&0\\\
(Td)^{3}&(Td)^{2}P_{\mathcal{N}}&TdP_{\mathcal{N}}&P_{\mathcal{N}}&\cdots&0\\\
&\cdots&\cdots&\cdots&&\end{array}\right),$

and

$d_{V}A=\left(\begin{array}[]{cccccc}P_{\mathcal{R}^{\perp}}d&0&0&0&\cdots&0\\\
P_{\mathcal{R}^{\perp}}dTd&P_{\mathcal{R}^{\perp}}dP_{\mathcal{N}}&0&0&\cdots&0\\\
P_{\mathcal{R}^{\perp}}(dT)^{2}d&P_{\mathcal{R}^{\perp}}dTdP_{\mathcal{N}}&P_{\mathcal{R}^{\perp}}dP_{\mathcal{N}}&0&\cdots&0\\\
P_{\mathcal{R}^{\perp}}(dT)^{3}d&P_{\mathcal{R}^{\perp}}(dT)^{2}dP_{\mathcal{N}}&P_{\mathcal{R}^{\perp}}dTdP_{\mathcal{N}}&P_{\mathcal{R}^{\perp}}dP_{\mathcal{N}}&\cdots&0\\\
&\cdots&\cdots&\cdots&&\end{array}\right).$

Restricted to $\Upsilon$, $D={P_{\mathcal{N}}}d_{V}A$ becomes

(48)
$D=P_{\Upsilon}d_{V}A=\left(\begin{array}[]{cccccc}P_{\Upsilon}d&0&0&0&\cdots&0\\\
P_{\Upsilon}dTd&P_{\Upsilon}d&0&0&\cdots&0\\\
P_{\Upsilon}(dT)^{2}d&P_{\Upsilon}dTd&P_{\Upsilon}d&0&\cdots&0\\\
P_{\Upsilon}(dT)^{3}d&P_{\Upsilon}(dT)^{2}d&P_{\Upsilon}dTd&P_{\Upsilon}d&\cdots&0\\\
&\cdots&\cdots&\cdots&&\end{array}\right).$

Furthermore, we have

$B=P_{\Upsilon}(I-d_{V}H)=\left(\begin{array}[]{cccccc}P_{\Upsilon}&0&0&0&\cdots&0\\\
P_{\Upsilon}dT&P_{\Upsilon}&0&0&\cdots&0\\\
P_{\Upsilon}(dT)^{2}&P_{\Upsilon}dT&P_{\Upsilon}&0&\cdots&0\\\
P_{\Upsilon}(dT)^{3}&P_{\Upsilon}(dT)^{2}&P_{\Upsilon}dT&P_{\Upsilon}&\cdots&0\\\
&\cdots&\cdots&\cdots&&\end{array}\right).$

From Theorem 1 and Theorem 2, the composition $B\circ F$ gives a cohomology-
preserving map from the input complexes to the BGG complex. The component-wise
form of $B\circ F$ is the following:

$B\circ
F=\left(\begin{array}[]{ccccc}P_{\Upsilon}&P_{\Upsilon}K&\frac{1}{2}P_{\Upsilon}K^{2}&\frac{1}{6}P_{\Upsilon}K^{3}&\cdots\\\
P_{\Upsilon}dT&P_{\Upsilon}dTK+P_{\Upsilon}&\frac{1}{2}P_{\Upsilon}dTK^{2}+P_{\Upsilon}K&\frac{1}{6}P_{\Upsilon}dTK^{3}+\frac{1}{2}P_{\Upsilon}K^{2}&\cdots\\\
P_{\Upsilon}(dT)^{2}&P_{\Upsilon}(dT)^{2}K+P_{\Upsilon}dT&\frac{1}{2}P_{\Upsilon}(dT)^{2}K^{2}+P_{\Upsilon}dTK+P_{\Upsilon}&\frac{1}{6}P_{\Upsilon}(dT)^{2}K^{3}+\frac{1}{2}P_{\Upsilon}dTK^{2}+P_{\Upsilon}K&\cdots\\\
&\cdots&\cdots&\cdots&\end{array}\right).$

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