theories underlying this thesis. On the one hand, this is a version of non-
commutative geometry implemented by Hopf algebras. We reviewed its
mathematical foundations in chapter Quantum Groups and Non-commutative
Geometry and noted that a distinctive feature is the appearance of an observer
independent scale. In this DSR interpretation of the $\kappa$-Poincaré and
other quantum groups the symmetries realized by the algebra of the momenta are
the fundamental objects and the spacetime takes a back seat; its properties as
a smooth manifold are merely emergent. A key function of the coalgebra
structure defined on the momentum space is to induce covariant representations
on the multiparticle states requiring the coassociativity of the coproduct,
which in turn implies the (modified) Yang Baxter equation for the associated
classical r-matrix. A further important constraint arises when we want to
interpret the deformation parameter of the quantum group as an observable
quantity so we have to be able to assign a real value to it. On the other
hand, there is the exciting field of asymptotic symmetries. One of its most
striking features is the existence of distinguishable, degenerate vacua,
intimately related to sub-algebras (embeddings) isomorphic to the Poincaré (or
an analogous bulk symmetry algebra) in the infinite dimensional BMS algebras.
This fact opens a connection to the theory of quantum groups and motivated our
quest to study the deformations on the asymptotic symmetry algebras. In
particular, we were interested in consistently implementing those quantum
groups that are studied and applied in the Poincaré algebra.

After reviewing the relevant asymptotic symmetry algebras in three and four
dimensions, the strategy in chapter Quantum Groups on Asymptotic Symmetry
Algebras to achieve this goal was to start from the Lie bialgebras which, by
the results of Etingof and Kazhdan cited in chapter Quantum Groups and Non-
commutative Geometry, exhausts all possible QUE. Since the cobracket of a LBA
has to satisfy a 1-cocycle condition, a major step in its classification is
the calculation of the corresponding cohomology group. For the asymptotically
(A)dS spacetimes in three dimensions we first proved that
$H^{1}\left(\mathfrak{W}\oplus\mathfrak{W},\bigwedge^{2}\left(\mathfrak{W}\oplus\mathfrak{W}\right)\right)$
vanishes and, together with the simple observation that there are no ad-
invariant elements in (the exterior product of) the
$\mathfrak{W}\oplus\mathfrak{W}$ algebra, we thus reduced the search for LBA
to the classification of triangular r-matrices up to automorphism. Motivated
by the fact that in the interior of the spacetime only the rigid symmetries
(Poincaré or (A)dS) survive, we imposed the constraint that the Hopf algebra
resulting from the deformation should form a sub-Hopf algebra when restricted
to the embedding corresponding to the bulk-symmetry. With such a constraint
the full r-matrix classification up to
$\text{Aut}^{\prime}\left(\mathfrak{W}\oplus\mathfrak{W}\right)$ was achieved
starting from earlier results on the embedded $\mathfrak{o}(4,\mathbb{C})$
algebra.

For the subsequent quantization of the LBA by a Drinfeld twist we again used
results known from the finite dimensional algebras and occasionally
automorphisms from the quotient
$\text{Aut}(\mathfrak{o}(4))/\text{Aut}^{\prime}(\mathfrak{W}\oplus\mathfrak{W})$.
By performing the explicit twisting procedure of the coalgebra sector for
selected LBA, we noticed that except for a specific abelian twist all twists
lead to coproducts containing both infinite power series in the deformation
parameter as well as infinitely many different generators. This peculiarity of
the infinite dimensional algebras makes a specialization of the deformation
parameter to a complex/real value for physical interpretation ill-defined, a
conclusion that could only be averted if we restricted to one-sided Witt
algebras.

As reviewed in section Surface Charges, the charges associated with the
asymptotic symmetries satisfy the Lie algebra of the asymptotic Killing
vectors up to potential central extensions. The inclusion of the central
charges, explicitly calculated in the seminal work by Brown and Henneaux, does
not change the fact that all LBA are triangular and coboundary on
$\mathfrak{W}\oplus\mathfrak{W}$. However, the classification of the
r-matrices is affected in several ways. For once, there are a variety of new
r-matrices containing the central element and we classified which combinations
with the original $\mathfrak{W}\oplus\mathfrak{W}$ r-matrices are allowed. We
furthermore noticed that the Schouten bracket of some r-matrices which are
triangular in the algebra without central extension receive extra terms and
also that the resulting coalgebra structures corresponding to r-matrices
without central elements can nonetheless contain contributions from the
central extension.

While there are some important conceptual differences between AdS and dS in
the study of asymptotic symmetries, in the algebraic description we simply
view them as different real forms of one complex algebra. The corresponding
reality conditions for the r-matrices turned out to be more restrictive for
the asymptotically dS spacetime, excluding r-matrices that have important
contraction limits in the $\mathfrak{B}_{3}$ algebra. But the deformation of
asymptotically (A)dS spacetime symmetries is relevant in its own right for
several reasons. For once, a positive cosmological constant, which after all
seems to be the relevant case in our universe, was also the starting point of
[20], where a connection between a model of quantum gravity (Loop Quantum
Gravity) and the time-like $\kappa$-Poincaré variant was established in a non-
perturbative way. Furthermore, by the AdS/CFT correspondence the asymptotic
symmetries in the bulk become the global symmetries of the boundary theory and
thus a “dual” deformation of the conformal symmetries can be expected. Such
quantum groups have been discussed for example in [137].

In the algebraic description the contraction of the (asymptotically) (A)dS to
the (asymptotically) flat symmetry algebra is quite straightforward. We thus
set out to investigate the properties of the contraction limit on the level of
Lie bialgebras and Hopf algebras in three dimensions. Note that the three
dimensional structures including the Hopf algebras are contained in the four
dimensional case except for the centrally extended algebras. Beyond this
trivial sense we did not explicitly study the deformations of the
asymptotically (A)dS symmetries and its contractions in four dimensions due to
the lack of a notion of quantum groups on Lie algebroids.

For the contraction of the r-matrices of (real forms of)
$\mathfrak{W}\oplus\mathfrak{W}$ we found a remarkably rich behavior. In
particular, depending on the parameter choices and their rescaling, there are
different possibilities of contracting an r-matrix and this process is also
not injective. One could have hoped that if the contraction of triangular
r-matrices is surjective, we can use our knowledge of the twists on
$\mathfrak{W}\oplus\mathfrak{W}$ to be able to construct all Hopf algebras in
$\mathfrak{B}_{3}$. Even though the surjectivity can be established on the
finite dimensional embeddings by including sums of contractions and
contractions along different axis, a result consistent with the findings of
[94], this hope does not quite materialize as the contraction on the level of
twists can not necessarily be performed in the same way as those of the
corresponding r-matrices. The reason behind this is that the triangular
classical r-matrices are in an anti-symmetric form which does not need to hold
for the twists and thus some terms might not cancel during contraction. Note
that the lack of a straightforward contraction limit of the twists is merely a
practical inconvenience; we know that they have to exist even though we did
not explicitly showed how to construct them except for twists related to the
Lie algebra type non-commutativity. Without central charges we showed that
triangular coboundary LBA are all there is on $\mathfrak{B}_{3}$ by
calculating the relevant cohomology group. When taking into account the
central extension this picture is changed considerably; the first cohomology
group does not vanish but consists of a one-parameter family of cocycles which
consequently define a Lie bialgebra that can not be quantized by a twisting
procedure. We nonetheless find the corresponding Hopf algebra by simply
guessing its form. It is interesting that this poses a clear example of how
the asymptotically flat case is more than just the contraction of
asymptotically (A)dS symmetries and their deformations.

A further example of this distinction became apparent when we compared the
process of specialization for Hopf algebras on $\mathfrak{B}_{3}$ with those
in $\mathfrak{W}\oplus\mathfrak{W}$. Even though the contraction limit could
be performed on the level of the LBA, the same limit applied to the twist
($\mathcal{F}_{2^{\prime}}$) was divergent, consistent with the fact that the
twist deformed Hopf algebra in $\mathfrak{B}_{3}$ admits a specialization and
the Hopf algebra deformed with $\mathcal{F}_{2^{\prime}}$ does not. Thus, in
this example the conclusion that the quantum groups on the asymptotically flat
symmetry algebras behave not just as contraction limits can not be seen
already on the infinitesimal Lie bialgebra structure in contrast to the non-
coboundary LBA mentioned above.

We already noted that apart from the central extension the asymptotic symmetry
algebras and their deformations in a three dimensional spacetime are sub
(Hopf) algebras of the algebras in the realistic four dimensional spacetime.
In the asymptotically flat case ($\mathfrak{B}_{4}$) we were able to show that
all LBA are coboundary and triangular just as in $\mathfrak{B}_{3}$. Besides
much more convoluted calculations also a few key differences stood out. First,
it is noteworthy that all the deformations corresponding to a Lie algebra type
non-commutativity single out a Poincaré embedding in $\mathfrak{B}_{4}$
because there is no longer a supertranslation generator that is present in all
of the embeddings. Furthermore, in the four dimensional case it is possible to
find r-matrices and twists containing generators from multiple embeddings and
lead to Hopf algebras with q-analog. In the algebraic sector the possible
central extension appears in a different form but nonetheless calculating the
cohomology group of $\mathfrak{B}_{4c}$ with values in the exterior product of
the adjoint module leads to an almost identical result with just a two
parameter LBA that is not triangular. The different form of the central
extension does, however, have an influence on the classification of triangular
r-matrices which we illustrated on a few examples.

With the classifications of Lie bialgebras and Hopf algebras on the three and
four dimensional BMS at hand in chapter Applications and Phenomenology we
first investigated the fate of one of our initial motivations to study quantum
groups; the non-commutativity of the “dual” spacetime. It turned out that the
requirement that the dual sector of an embedding closes a subalgebra,
interpreted as a deformed Minkwoski spacetime, is a non-trivial constraint. In
particular, the commutation relations dual to the LBA of the extended
Jordanian twist resemble those of the well-known light-cone $\kappa$-Poincaré
but also contain extra terms so that the Minkowski sector is indeed no
subalgebra, posing another technical obstruction for this particular quantum
group. Similarly, other deformations that did not admit a q-analog on the
whole BMS algebra also do not clear this hurdle but the LBA corresponding to
the abelian and Jordanian twist with its dual do and thereby define a non-
commutative Minkowski spacetime of Lie algebra type. This also implies the
introduction of a length scale we again call $1/\kappa$ and in the limit
$\kappa\rightarrow\infty$ the classical commuting spacetime is recovered.

Our hopes of finding phenomenological effects detectable at mesoscopic scales
were mainly resting on the deformed in-vacuo dispersion relations also (but
not only) reported in the context of deformed Hopf algebras, albeit derived in
various ways. In this thesis we employed a construction of wave operators of a
given embedding in terms of a special basis (generating the so-called Quantum
Lie algebra endowed with a star product). As all quantum groups we studied,
with the sole exception of a quantization of the non-coboundary LBA from
sections Deformations of the Surface Charge Algebra and Comparison with 3D
Deformations, are defined by a twisting procedure the quantum R-matrix and
“deformed generators” can be readily derived in order to calculate the
potentially deformed wave operator describing the propagation through the non-
commutative spacetime. By construction it is clear that a deformation
constructed from elements of the same ($n=1$) embedding as the wave operator
does not lead to deformed dispersion relations so in order to obtain an
observable effect we would need to loosen our restriction and allow for
deformations including (soft charge) generators from different embeddings. But
even then the requirement that a q-analog can be defined prohibits any
resulting deformation of the in-vacuo dispersion relation. So far, we thus
return empty-handed from our search for new mesoscopic phenomena implemented
by Hopf algebra structures on the asymptotic symmetry algebras that correspond
to non-commutative geometry of Lie algebra type with a physical scale
$1/\kappa$.

Another phenomenon that was investigated is the deformed Leibniz rule
resulting from the covariant representations of the symmetry algebra on
multiparticle states induced by the coproduct. While this effect occurs in one
form or another in all deformations listed in this thesis, potential
experiments to detect it lack a way to counter-balance the smallness of the
Planck scale. A further obvious possibility that could enable the detection of
signatures of a deformed Leibniz rule would be that $1/\kappa$ is
significantly larger than $l_{\text{Pl}}$ which, however, has the drawback
that it is no longer backed by most of the motivation from quantum gravity
like the thought experiment leading to a maximal resolution due to black hole
creation.

Irrespective of the status of experimental evidence for a $\kappa$-deformed
Leibniz rule in the BMS algebras, a new proposal for its application in the
black hole information loss paradox was brought forward in this thesis. The
main ideas can be summarized as follows. Strominger, Hawking and Perry
suggested that information of the matter falling into the black hole might be
stored in the form of “soft hair”, i.e. supertransformation surface charges
defined on the horizon. Bousso, Poratti and others then critically remarked
that the Poincaré and soft charges decouple and thus concluded that the soft
hair can not retain any “hard” information. If the momentum conservation in
the relevant scattering processes is described using the deformed Leibniz
rule, the soft and hard modes might cease to decouple. In particular, we found
several twist deformations that in this way generate dependencies of the
Poincaré momenta on supertranslation charges but, due to the restriction from
the specialization requirement, not for superrotation charges. Since the
latter act as the soft hair, according to Strominger et.al., we conclude that
on the full $\mathcal{B}_{4}$ the decoupling can not be averted in a way that
rescues the original soft hair proposal.

Finally, we briefly discussed how the exact form of the asymptotic symmetry
algebras might be inferred from experiments and how this affects the rest of
our results. Experimental verification of the existence of ordinary
supertranslations as asymptotic symmetry transformations can be reasonably
expected to be established in the near future which is enough for some of the
most restrictive constraints on consistent quantum groups to persist. I.e. in
the absence of central charges all Hopf algebra structures constructed on said
algebras are quasitriangular and derived from a twist. Whether
supertransformations parametrized by functions of the angular coordinates
without poles have to be included marks an important distinction because if
this is not the case and we can use the one-sided BMS algebras for describing
asymptotic symmetries the specialization of the deformation parameter to real
values is possible for a much wider class of deformations. We reviewed that
this distinction has been linked to the (snapping of) cosmic strings and
experimental evidence consequently constrains the Hopf algebras with
potentially physical interpretation in this sense.

Looking back to what we set out to achieve we made progress in the
classification and analysis of quantum groups (QUE) on a variety of asymptotic
symmetry algebras. Both technical and physical constraints led to the
exclusion of consistent generalizations of quantum groups studied in the
Poincaré algebra. We found interesting behavior in particular by studying the
so-called contraction limit of spacetimes with cosmological constant and the
inclusion of central charges on the level of Lie bialgebras and Hopf algebras.
Somewhat disappointingly, the search for phenomenological signatures of
momentum spaces with the remaining deformed Hopf algebra structures did not
yield promising results at least in a regime accessible by current
experiments.

Future research may follow up on a variety of points. First, it is conceivable
to relax both some of the mathematical and physical requirements, e.g. one may
consider the $\kappa$-deformed theory to be only an effective theory valid at
the first order on $1/\kappa$ and consequently the specialization can be
applied trivially. Another straightforward extension of our results is to take
into account further newly found variants of BMS-type algebras e.g. from
[138], [139]. To address the problem of identifying phenomenological
signatures on mesoscopic scales despite a deformation parameter of the order
of $l_{\text{Pl}}$, one could turn to deformed discrete symmetries, an idea
recently discussed in the context of $\kappa$-Poincaré [140] (see also [141]).
The mechanisms that could lead to large enough effect sizes range from higher
precision in time measurements to the appearance of the inverse of small mass
differences e.g. in neutral Kaon systems. Last but not least, one could
proceed to explore the application of the Lie bialgebras and Hopf algebras
discussed in this thesis in related branches of pure and applied mathematics.
For example, in the study of integrable systems and in particular in the
quantum inverse scattering method (see [142], [143] and references therein)
quantum R-matrices play a vital role which, incidentally, was one of the first
motivations to introduce quantum groups.

## Proof of the Cohomology Theorems

Proof of Theorem 1
We structure the proof by first noting that the 1-cocycles $\delta$ can be
separated by their degree $d\in\mathbb{Z}$. This degree is derived from the
grading of $\mathfrak{W}$, i.e.

$\displaystyle\delta(L_{m})=L_{i}\wedge L_{j}$ (558)

has degree $d=i+j-m$. The separation by degree follows from the fact that a
cocycle which, applied to elements of $\mathfrak{W}$, results in terms with
different degree can be split into cocycles of homogeneous degree which have
to fulfill the cocycle condition (10) independently. Let us first consider
cocycles of degree $\bm{d\neq 0}$. We will show that all such cocycles
$\delta$ are cohomolog to $0$, i.e. that
$\delta^{\prime}(L_{m})=\delta(L_{m})-(\partial_{0}r)(L_{m})=0$ for all
$m\in\mathbb{Z}$. Let $\delta$ be a cocycle of homogenous degree $d$ such that

$\displaystyle\delta(L_{m})=\sum_{i_{m},j_{m}\in
I_{m}}\alpha^{m}_{i_{m}j_{m}}L_{i_{m}}\wedge L_{j_{m}},$ (559)

where $\alpha^{m}_{i_{m}j_{m}}\in\mathbb{R};i_{m},j_{m}\in\mathbb{Z}$ and
$I_{m}$ are finite subsets of $\mathbb{Z}$. Choose a 0-cochain

$\displaystyle r=-\sum_{i_{0},j_{0}\in
I_{0}}\frac{\alpha^{0}_{i_{0}j_{0}}}{i_{0}+j_{0}}L_{i_{0}}\wedge L_{j_{0}}.$
(560)

Then we have

$\displaystyle\delta^{\prime}(L_{0})$ $\displaystyle=\sum_{i_{0},j_{0}\in
I_{0}}\alpha^{0}_{i_{0},j_{0}}L_{i_{0}}\wedge L_{j_{0}}-[L_{0}\otimes
1+1\otimes L_{0},r]$ $\displaystyle=\sum_{i_{0},j_{0}\in
I_{0}}\alpha^{0}_{i_{0},j_{0}}L_{i_{0}}\wedge L_{j_{0}}-\sum_{i_{0},j_{0}\in
I_{0}}\frac{\alpha^{0}_{i_{0}j_{0}}}{i_{0}+j_{0}}(i_{0}+j_{0})L_{i_{0}}\wedge
L_{j_{0}}=0.$ (561)

From the cocycle condition

$\displaystyle\delta^{\prime}([L_{0},L_{m}])=[L_{0}\otimes 1+1\otimes
L_{0},\delta^{\prime}(L_{m})]-[L_{m}\otimes 1+1\otimes
L_{m},\delta^{\prime}(L_{0})],$ (562)

for $m\neq 0$, we infer

$\displaystyle m\delta(L_{m})$ $\displaystyle=\sum_{i_{m},j_{m}\in
I_{m}}(i_{m}+j_{m})\alpha^{m}_{i_{m}j_{m}}L_{i_{m}}\wedge L_{j_{m}}$
$\displaystyle=(d+m)\delta^{\prime}(L_{m})$ (563)
$\displaystyle\Rightarrow\delta^{\prime}(L_{m})$ $\displaystyle=0,$ (564)

which concludes the proof for cocycles of degree $d\neq 0$.

Next, let us consider cocycles of degree $\bm{d=0}$ which can be written in
the form

$\displaystyle\delta(L_{m})=\sum_{i_{m}\in
I}\gamma^{m}_{i_{m}}L_{m-i_{m}}\wedge L_{i_{m}}.$ (565)

Note that without loss of generality we can restrict the indices $i_{m}$ to be
smaller than $m/2$ due to the following observation. If there were an index
$i_{m}>m/2$, we simply substitute $i^{\prime}_{m}=m-i_{m}$ and
${\gamma^{\prime}}^{m}_{i^{\prime}_{m}}=\gamma^{m}_{i_{m}}-\gamma^{m}_{im-m}$
to describe the same cocycle. We will make repeated use of this restriction in
the rest of the proof.

The conditions

$\displaystyle\delta([L_{0},L_{m}])$ $\displaystyle=[L_{0}\otimes 1+1\otimes
L_{0},\delta(L_{m})]-[L_{m}\otimes 1+1\otimes L_{m},\delta(L_{0})]$
$\displaystyle=(-m)\delta(L_{m})-[L_{m}\otimes 1+1\otimes
L_{m},\delta(L_{0})]$ (566) $\displaystyle\Leftrightarrow 0$
$\displaystyle=[L_{m}\otimes 1+1\otimes L_{m},\delta(L_{0})]$ (567)

implie that all degree $0$ cocycles vanish on $L_{0}$ because it has to hold
for all $m$ and there is no ad-invariant element in
$\bigwedge^{2}\left(\mathfrak{W}\oplus\mathfrak{W}\right)$.

As a next step we show that all cocycles are cohomolog to $\bm{0}$ on
$\bm{L_{\pm 1}}$. Let us assume without loss of generality that the indices of

$\displaystyle\delta(L_{1})=\sum_{i_{1}\in
I_{1}}\gamma^{1}_{i_{1}}L_{1-i_{1}}\wedge L_{i_{1}}$ (568)

are given by $i_{1}\in
I_{1}=\\{-p_{1},-p_{2},...,-p_{n}|p_{1}>p_{2}>...>p_{n}>1,n\in\mathbb{N}\\}$.
From the cocycle condition we get

$\displaystyle\delta([L_{1},L_{-1}])$ $\displaystyle=[L_{1}\otimes 1+1\otimes
L_{1},\delta(L_{-1})]-[L_{-1}\otimes 1+1\otimes L_{-1},\delta(L_{1})]$ (569)
$\displaystyle\Leftrightarrow 0$ $\displaystyle=\sum_{i_{-1}\in
I_{-1}}(\gamma^{-1}_{i_{-1}}(2+i_{-1})\underbrace{L_{-i_{-1}}\wedge
L_{i_{-1}}}_{\text{type}I}+\gamma^{-1}_{i_{-1}}(i_{-1}-1)\underbrace{L_{1+i_{-1}}\wedge
L_{-1-i_{-1}}}_{\text{type}II})$
$\displaystyle\phantom{=}+\sum_{j=1}^{n}(\gamma^{1}_{-p_{j}}(2+p_{j})L_{p_{j}}\wedge
L_{-p_{j}}+\gamma^{1}_{-p_{j}}(p_{j}-1)L_{-1-p_{j}}\wedge L_{1+p_{j}}).$ (570)

Lets focus on the first term in the second line of (570) with $p_{1}$; it can
only be canceled by any of the other $p_{j}$ terms if $p_{2}=p_{1}-1$ which we
discuss below. In the case $\bm{p_{2}\neq p_{1}-1}$ there are two terms that
can contribute, one from the first and the second summand in the first line in
(570) which we will call type $I$ and type $II$ terms respectively 212121Here
and in the following we use the index restriction. Otherwise also e.g. a type
$I$ term with $i_{-1}=p_{j}$ could be used.. The type $II$ term would
correspond to $i_{-1}=-1-p$. If it existed with non-zero $\gamma^{-1}_{-1-p}$
it would imply the existence of a type $I$ term of the form
$\gamma^{-1}_{-1-p}(1-p_{1})L_{1+p_{1}}\wedge L_{-1-p_{1}}$ which in turn can
only be canceled by a type $II$ term with $i_{-1}=-2-p_{1}$. Since also none
of the prefactors $(2+i_{-1})$ and $(i_{-1}-1)$ vanishes if $p_{1}\neq 1$ this
would go on forever so that we need infinitely many terms in $\delta(L_{-1})$
which is not possible. Thus $\gamma^{-1}_{-1-p}=0$ and we need a type $I$ term
with $i_{-1}=-p_{1}$

$\displaystyle\gamma^{-1}_{1-p_{1}}(2+p_{1})L_{p_{1}}\wedge L_{-p_{1}}$ (571)

which implies a type $II$ term with the same $i_{-1}$

$\displaystyle\gamma^{-1}_{-p_{1}}(-1-p_{1})L_{1-p_{1}}\wedge L_{p_{1}-1}.$
(572)

This term can be canceled only by a type $I$ term with $i_{-1}=1-p_{1}$

$\displaystyle\gamma^{-1}_{1-p_{1}}(3-p_{1})L_{1-p_{1}}\wedge L_{p_{1}-1}$
(573)

and the corresponding type $II$ term

$\displaystyle\gamma^{-1}_{1-p_{1}}p_{1}L_{2-p_{1}}\wedge L_{p_{1}}$ (574)

requires again a type $I$ term with $i_{-1}=-p_{1}$

$\displaystyle\gamma^{-1}_{-p_{1}}(2-p_{1})L_{2-p_{1}}\wedge L_{p_{1}}$ (575)

ending the sequence. The cancellation of (574) with (575) implies the
following ratio of the coefficients

$\displaystyle\frac{\gamma^{-1}_{-p_{1}}}{\gamma^{-1}_{1-p_{1}}}=\frac{2-p_{1}}{p_{1}}.$
(576)

When considering the 0-cochain

$\displaystyle r=\gamma_{s}L_{-s}\wedge L_{s}$ (577)

with $s=1-p_{1}$, implying

$\displaystyle(\partial_{0}r)(L_{-1})\equiv\delta_{r}(L_{-1})=\gamma_{1-p_{1}}((-p_{1})L_{-2+p_{1}}\wedge
L_{1-p_{1}}+(-2+p_{1})L_{p_{1}-1}\wedge L_{-p_{1}},$ (578)

we find the same ratio between the two summands. Thus setting
$\gamma_{1-p_{1}}(p_{1})=\gamma^{-1}_{1-p_{1}}$ in the cocycle

$\displaystyle\delta^{\prime}=\delta+\delta_{r}$ (579)

both coefficients
${\gamma^{\prime}}^{-1}_{-p_{1}},{\gamma^{\prime}}^{-1}_{1-p_{1}}$ vanish and
therefore also ${\gamma^{\prime}}^{1}_{p_{1}}$ has to be zero.

Next, we have to consider the case $\bm{p_{2}=p_{1}-1}$. In (570) the term

$\displaystyle\gamma^{1}_{-p_{1}}(p_{1}-1)L_{1+p_{1}}\wedge L_{-1-p_{1}}$
(580)

can be canceled by a type $I$ term with $i_{-1}=1-p_{1}$ or a type $II$ term
with $i_{-1}=-2-p_{1}$. If the second term does not vanish, this implies the
existence of a type $I$ term with the same $i_{-1}$ which can only be
eliminated by a type $II$ term with $i_{-1}=-3-p_{1}$ and so on, so that
infinitely many terms are necessary, ruling out this option. Using the same
cochain as above in (577) with the same choice for $s$ and $\gamma_{s}$ we can
eliminate the coefficient ${\gamma^{\prime}}^{-1}_{1-p_{1}}$ and thus the
possibility to cancel (580) with a type $I$ term is not possible which means
that ${\gamma^{\prime}}^{1}_{-p_{1}}$ has to vanish.

For the rest of the $\bm{p_{j},j>1}$ we can iteratively use the same
argumentation. In particular the arguments with the infinite number of terms
in $\delta(L_{-1})$ can be extended to the higher $j$ as the sequence would
stop at the terms with $i_{-1}=-p_{j-1}$ which already has to vanish.
Furthermore, one has to add coboundaries from the cochains

$\displaystyle r_{j}=\gamma_{j}L_{-(1-p_{j})}\wedge L_{1-p_{j}},\quad j>1$
(581)

with suitable coefficients
$\gamma_{j}(p_{j})={{{\gamma^{\prime}}^{...}}^{\prime}}^{-1}_{1-p_{j}}$ where
we define

$\displaystyle{{\delta^{\prime}}^{\prime}}=\delta^{\prime}+\delta_{r_{2}},...$
(582)

iteratively so that the required terms in
${{\delta^{\prime}}^{...}}^{\prime}(L_{-1})$ are eliminated.

Finally, let us explicitely consider the case $\bm{p_{1}=1}$ that was excluded
in the argumentation above. In that case

$\displaystyle\delta(L_{1})=\gamma^{1}_{-1}L_{0}\wedge L_{1}$ (583)

and from the cocycle condition we infer that

$\displaystyle\delta(L_{-1})=\gamma^{1}_{-1}L_{-2}\wedge L_{1}.$ (584)

On $L_{\pm 1}$ $\delta$ then coincides with $\delta_{r}$, where
$r=\gamma^{1}_{-1}/2L_{1}\wedge L_{-1}$ and thus
$\delta^{\prime}=\delta-\delta_{r}$ is zero on these elements. This concludes
the proof that $\delta$ is cohomolog to $0$ on $L_{1}$.

In the next step it will be shown that $\bm{\delta(L_{1})=0}$ implies that
$\bm{\delta(L_{m})=0}$ for $\bm{m>1}$. Starting from

$\displaystyle\delta(L_{2})=\sum_{i_{2}\in
I_{2}}\gamma^{2}_{i_{2}}L_{2-i_{2}}\wedge L_{i_{2}},$ (585)

one explicitly obtains by using (10) with $m=1,n=2$; $m=1,n=3$ and $m=1,n=4$

$\displaystyle\delta(L_{3})=$ $\displaystyle-\sum_{i_{2}\in
I_{2}}\gamma^{2}_{i_{2}}((i_{2}-1)L_{3-i_{2}}\wedge
L_{i_{2}}-(1-i_{2})L_{2-i_{2}}\wedge L_{i_{2}+1})$ (586)
$\displaystyle\delta(L_{4})=$ $\displaystyle\sum_{i_{2}\in
I_{2}}\frac{\gamma^{2}_{i_{2}}}{2}(i_{2}-1)((i_{2}-2)L_{4-i_{2}}\wedge
L_{i_{2}}+2(1-i_{2})L_{3-i_{2}}\wedge L_{i_{2}+1}-i_{2}L_{2-i_{2}}\wedge
L_{i_{2}+2})$ (587) $\displaystyle\delta(L_{5})=$
$\displaystyle-\sum_{i_{2}\in
I_{2}}\frac{\gamma^{2}_{i_{2}}}{6}(i_{2}-1)\bigg{(}(i_{2}-2)(i_{2}-3)L_{5-i_{2}}\wedge
L_{i_{2}}+3(1-i_{2})(i_{2}-2)L_{4-i_{2}}\wedge L_{i_{2}+1}$
$\displaystyle+3(1-i_{2})i_{2}L_{3-i_{2}}\wedge
L_{i_{2}+2}+i_{2}(1+i_{2})L_{2-i_{2}}\wedge L_{i_{2}+3}\bigg{)}.$ (588)

Using the same argumentation as above, we can restrict $i_{2}$ to be bigger
than $1$ and we consider the largest index $i^{\prime}_{2}$. Then, (10) with
$m=2,n=3$ yields

$\displaystyle 0=$ $\displaystyle\sum_{i_{2}\in
I_{2}}\bigg{(}L_{5-i_{2}}\wedge
L_{i_{2}}\gamma^{2}_{i_{2}}\left(\frac{1}{6}(i_{2}-1)(i_{2}-2)(i_{2}-3)-(i_{2}-1)^{2}-(i^{\prime}_{2}+1)\right)$
$\displaystyle+L_{4-i_{2}}\wedge
L_{i_{2}+1}\gamma^{2}_{i_{2}}\left(\frac{1}{2}(i_{2}-1)^{2}(2-i_{2})+(i_{2}-1)i_{2}\right)$
$\displaystyle+L_{3-i_{2}}\wedge
L_{i_{2}+2}\gamma^{2}_{i_{2}}\left(\frac{1}{2}(i_{2}-1)^{2}i_{2}+(i_{2}-1)(i_{2}-2)\right)$
$\displaystyle+L_{2-i_{2}}\wedge
L_{i_{2}+3}\left((i_{2}-1)^{2}-(3-i_{2})-\frac{(i_{2}-1)(i_{2}+1)i_{2}}{6}\right)\bigg{)}$
(589) $\displaystyle\Rightarrow\quad 0$
$\displaystyle=L_{2-i^{\prime}_{2}}\wedge
L_{i^{\prime}_{2}+3}\left((i^{\prime}_{2}-1)^{2}-(3-i^{\prime}_{2})-\frac{(i^{\prime}_{2}-1)(i^{\prime}_{2}+1)i^{\prime}_{2}}{6}\right)$
(590)

and (590) implies for $i^{\prime}_{2}>1,\gamma^{2}_{i^{\prime}_{2}}\neq 0$ the
solutions $i^{\prime}_{2}=3,4$. $i_{2}$ can therefore only take the values
$i_{2}=2,3,4$ and one can calculate explicitly that e.g. the term proportional
to $L_{1}\wedge L_{5}$ in (589) does not vanish so
$\gamma^{1}_{i_{2}^{\prime}}=0$. Thus $\delta(L_{2})=0$ and iteratively one
shows that (10) with $m=1$ implies $\delta(L_{n})=0$ for $n>2$. For arbitrary
positive $m$ one finds

$\displaystyle\delta([L_{-1},L_{m}])$ $\displaystyle=-[L_{m}\otimes 1+1\otimes
L_{m},\delta(L_{-1})]$ (591) $\displaystyle\Rightarrow 0$
$\displaystyle=-\sum_{i_{-1}\in
I_{-1}}\gamma^{-1}_{i_{-1}}((m+1+i_{-1})L_{m-1-i_{-1}}\wedge
L_{i_{-1}}+(m-i_{-1})L_{-1-i_{-1}}\wedge L_{i_{-1}+m})$ (592)

which yields $\gamma^{-1}_{i^{\prime}_{-1}}=0$ for the largest index
$i^{\prime}_{-1}$ and thus $\delta(L_{-1})=0$.

Finally, one shows explicitly that (10) with $m=1,n=-2$ results in
$\delta(L_{-2})=0$ and, similarly to the case of positive $m$ that can be used
to show that $\bm{\delta(L_{m})=0}$ for all $\bm{m<-2}$, completing the proof
of the first theorem.

Proof of Theorem 2
Note that a 1-cocycle $\delta$ applied to an element of $\mathfrak{W}$ can be
split into three parts $\delta^{I},\delta^{II},\delta^{III}$, mapping to
$\mathfrak{W}\wedge\mathfrak{W}$,
$\overline{\mathfrak{W}}\wedge\overline{\mathfrak{W}}$ or
$\mathfrak{W}\wedge\overline{\mathfrak{W}}$ respectively, which have to
satisfy the cocycle condition separately222222In other words the cocycles
decompose into
$Z(\mathfrak{W}\oplus\overline{\mathfrak{W}},\bigwedge^{2}\mathfrak{W})\oplus
Z(\mathfrak{W}\oplus\overline{\mathfrak{W}},\bigwedge^{2}\overline{\mathfrak{W}})\oplus
Z(\mathfrak{W}\oplus\overline{\mathfrak{W}},\mathfrak{W}\wedge\overline{\mathfrak{W}})$..
From the previous theorem it follows that $\delta^{I}$ is cohomolog to zero
and from (10) one can easily see that $\delta^{II}$ has to vanish. Thus we
only need to consider the part $\bm{\delta^{III}}$ which again can be
separated by the degree $d$, which we define such that

$\displaystyle\delta(L_{m})=L_{i}\wedge\bar{L}_{j}$ (593)

has $d=i-m$. A general cocycle of homogenous degree $\bm{d\neq 0}$ is given by

$\displaystyle\delta(L_{0})=\sum_{j_{0}\in
I_{0}}\alpha^{0}_{j_{0}}L_{d}\wedge\bar{L}_{j_{0}}$ (594)

on $L_{0}$. Setting

$\displaystyle r=\sum_{j_{0}\in
I_{0}}\frac{\alpha^{0}_{j_{0}}}{d}L_{d}\wedge\bar{L}_{j_{0}}$ (595)

we then have

$\displaystyle\delta^{\prime}(L_{0})=\delta(L_{0})-\delta_{r}(L_{0})=0.$ (596)

Using this in

$\displaystyle\delta^{\prime}([L_{0},L_{m}])=[L_{0}\otimes 1+1\otimes
L_{0},\delta^{\prime}(L_{m})]-[L_{m}\otimes 1+1\otimes
L_{m},\delta^{\prime}(L_{0})]$ (597)

it follows that

$\displaystyle-m\delta^{\prime}(L_{m})=-(d+m)\delta^{\prime}(L_{m})\Rightarrow\delta^{\prime}(L_{m})=0$
(598)

concluding the proof for $d\neq 0$.

A general degree $\bm{0}$ cocycle has the form

$\displaystyle\delta(L_{m})=\sum_{i_{m}\in
I_{m}}\gamma^{m}_{i_{m}}L_{m}\wedge\bar{L}_{i_{m}}$ (599)

and by choosing

$\displaystyle r=\sum_{i_{1}\in
I_{1}}\gamma_{i_{1}}L_{0}\wedge\bar{L}_{i_{1}}$ (600)

it follows that

$\displaystyle\delta^{\prime}(L_{1})=\delta(L_{1})-\delta_{r}(L_{1})=0.$ (601)

Then for $m\neq 1$

$\displaystyle\delta^{\prime}([L_{m},L_{1}])$ $\displaystyle=-[L_{1}\otimes
1+1\otimes L_{1},\delta^{\prime}(L_{m})]$ (602)
$\displaystyle\Rightarrow\sum_{i_{m+1}\in
I_{m+1}}(m-1)\gamma^{m+1}_{i_{m+1}}L_{m+1}\wedge\bar{L}_{i_{m+1}}$
$\displaystyle=\sum_{i_{m}\in
I_{m}}(m-1)\gamma^{m}_{i_{m}}L_{m+1}\wedge\bar{L}_{i_{m}}$ (603)

and it follows that

$\displaystyle\gamma^{m+1}_{i_{m+1}}=\gamma^{m}_{i_{m}}.$ (604)

If $m=0$ in (602) we conclude

$\displaystyle 0=-\sum_{i_{0}\in
I_{0}}\gamma^{0}_{i_{0}}L_{1}\wedge\bar{L}_{i_{0}}$ (605)

and thus $\gamma^{0}_{i_{0}}=0$. Because of (604)
$\gamma^{m}_{i_{m}}=\gamma^{0}_{i_{0}}$ for $m<0$ and for $m>0$ all
coefficients are given by $\gamma^{m}_{i_{m}}=\gamma^{2}_{i_{2}}$. However,
from (10) with $m=2,n=3$ we find

$\displaystyle-\sum_{i_{2}\in
I_{2}}\gamma^{2}_{i_{2}}L_{5}\wedge\bar{L}_{i_{2}}=\sum_{i_{2}\in
I_{2}}\gamma^{2}_{i_{2}}(-2)L_{5}\wedge\bar{L}_{i_{2}}$ (606)

and thus $\gamma^{2}_{i_{2}}=0$, concluding the proof.

Proof of Theorem 3

As a first step we will now show that
$\bm{H^{1}\left(\mathfrak{W}_{+},\bigwedge^{2}\mathfrak{W}_{+}\right)=\\{0\\}}$.
Before starting the actual proof of the theorem let us note that we can not
simply reuse the proof of theorem 1 because there we made use of the index
restriction which is not possible in the one-sided Witt algebra. However, for
cocoycles of degree $d\neq 0$ this does not apply and thus only the $\bm{d=0}$
part is left. Vanishing degree of a cocycle $\delta$ on $\mathfrak{W}_{+}$
implies that

$\displaystyle\delta(L_{0})$ $\displaystyle=\gamma^{0}_{1}L_{-}1\wedge
L_{1},\quad\delta(L_{-1})=\gamma^{-1}_{1}L_{0}\wedge L_{-1},$ (607)
$\displaystyle\delta(L_{1})$ $\displaystyle=\gamma^{1}_{2}L_{-1}\wedge
L_{2}+\gamma^{1}_{1}L_{0}\wedge
L_{1},\quad\delta(L_{2})=\gamma^{2}_{3}L_{-1}\wedge
L_{3}+\gamma^{2}_{2}L_{0}\wedge L_{2},$ (608) $\displaystyle\delta(L_{3})$
$\displaystyle=\gamma^{3}_{4}L_{-1}\wedge L_{4}+\gamma^{3}_{3}L_{0}\wedge
L_{3}+\gamma^{3}_{2}L_{1}\wedge L_{2},$ (609)

for the first few generators. The only possible 0-cochain that induces a
coboundary of degree $0$ is of the form $r=\alpha L_{1}\wedge L_{-1}$. Setting
$\alpha=-\gamma^{1}_{1}/2$ the new cocycle $\delta-\delta_{r}$ (which is
cohomolog to $\delta$) has $\gamma^{1}_{1}=0$. By imposing the cocycle
condition

$\displaystyle\delta([L_{1},L_{0}])$ $\displaystyle=[L_{1}\otimes 1+1\otimes
L_{1},\delta(L_{0})]-[L_{0}\otimes 1+1\otimes L_{0},\delta(L_{1})],$ (610)
$\displaystyle\Rightarrow\delta(L_{1})$
$\displaystyle=2\gamma^{0}_{1}L_{0}\wedge L_{1}+\delta(L_{1}),$ (611)

we infer $\delta(L_{0})=0$. Furthermore, the cocycle conditions with
$\delta([L_{1},L_{-1}]),\delta([L_{2},L_{-1}]),$ $\delta([L_{1},L_{2}])$ yield
the equations

$\displaystyle\gamma^{-1}_{1}+3\gamma^{1}_{2}=0,$ (612) $\displaystyle
3\gamma^{1}_{2}=-4\gamma^{2}_{3}-\gamma^{2}_{2}+2\gamma^{-1}_{1},\quad-3\gamma^{2}_{2}-3\gamma^{-1}_{1}=0,$
(613)
$\displaystyle-\gamma^{3}_{4}=-2\gamma^{2}_{3},\quad-\gamma^{3}_{3}=2\gamma^{2}_{3}-\gamma^{2}_{2},\quad-\gamma^{3}_{2}=\gamma^{2}_{2}-3\gamma^{1}_{2},$
(614)

which do not have a non-trivial solution and thus $\delta(L_{-1,1,2,3})=0$.
From cocycle conditions with $\delta([L_{1},L_{m}])$ it is then easy to
iteratively show that $\delta(L_{m})=0,m\geq-1$. This concludes the first part
of the proof.

Showing that
$H^{1}\left(\mathfrak{W}_{+}\oplus\mathfrak{W}_{+},\bigwedge^{2}(\mathfrak{W}_{+}\oplus\mathfrak{W}_{+})\right)=\\{0\\}$
with the help of the first part is then completely analogous to the proof of
theorem 2 since there is no use of the index restriction.

Proof of Theorem 4
We first prove that
$\bm{H^{1}\left(\mathfrak{Vir},\bigwedge^{2}\mathfrak{Vir}\right)=\\{0\\}}$.
Let $\delta$ be a 1-cocycle on $\mathfrak{Vir}$ with values in
$\bigwedge^{2}\mathfrak{Vir}$. It can be decomposed into
$\delta=\delta_{\mathfrak{W}}+\delta_{c}$, where
$\delta_{\mathfrak{W}}:\mathfrak{Vir}\rightarrow\bigwedge^{2}\mathfrak{W}$ and
$\delta_{c}:\mathfrak{Vir}\rightarrow c_{L}\bigwedge\mathfrak{W}$. Let
$X_{c}=X+\alpha c_{L}$ and $Y_{c}=Y+\alpha^{\prime}c_{L}$ be generic elements
in $\mathfrak{Vir}$ with
$X,Y\in\mathfrak{W},\alpha,\alpha^{\prime}\in\mathbb{C}$. The cocycle
condition then reads

$\displaystyle\delta([X_{c},Y_{c}])=[X_{c}\otimes 1+1\otimes
X_{c},\delta(Y_{c})]-[Y_{c}\otimes 1+1\otimes Y_{c},\delta(X_{c})],$ (615)

resulting in the separate equations

$\displaystyle\delta_{\mathfrak{W}}([X,Y])$
$\displaystyle=[\delta_{\mathfrak{W}}(X_{c}),Y\otimes 1+1\otimes
Y]_{\mathfrak{W}}-[\delta_{\mathfrak{W}}(Y_{c}),X\otimes 1+1\otimes
X]_{\mathfrak{W}},$ (616) $\displaystyle\delta_{c}([X,Y])$
$\displaystyle=[\delta_{\mathfrak{W}}(X_{c}),Y\otimes 1+1\otimes
Y]_{c}-(X\leftrightarrow Y)+[\delta_{c}(X_{c}),Y\otimes+1\otimes
Y]_{\mathfrak{W}}-(X\leftrightarrow Y),$ (617)

where $[,]_{\mathfrak{W}}$ denotes the part of the induced bracket on
$\otimes^{2}\left(\mathfrak{Vir}\right)$ that conserves the number of central
elements in the input (e.g. mapping from
$\otimes^{4}\left(\mathfrak{W}\right)$ to
$\otimes^{2}\left(\mathfrak{W}\right)$) and $[,]_{c}$ the rest (e.g. mapping
$\otimes^{4}\left(\mathfrak{W}\right)$ to
$\mathbb{C}c_{L}\otimes\mathfrak{W}$).

From (615) with the commutator $[c_{L},Y]=0$ on the lhs it is easy to see that
$\bm{\delta(c_{L})=0}$ as there are no ad-invariant elements (cf. section Lie
Bialgebra Structures on $\mathfrak{W}\oplus\mathfrak{W}$). Thus (616) is
independent from the central charge and one can apply theorem 1 to show that
$\delta_{\mathfrak{W}}$ is a coboundary from a cochain
$r\in\bigwedge^{2}\left(\mathfrak{W}\right)$.

Redefining $\delta\rightarrow\delta-\delta_{r}$ we find that
$\bm{\delta_{\mathfrak{W}}(L_{m})=0}$ without affecting $\delta_{c}(L_{0,\pm
1})$. Without loss of generality the remaining part of the cocycle is of the
form

$\displaystyle\delta_{c}(L_{m})=\sum_{i_{m}\in
I_{m}}\gamma^{m}_{i_{m}}c_{L}\wedge L_{i_{m}},$ (618)

which again can be separated by the degree $d=m-i_{m}$. We still have the
freedom to choose a cochain $r\in c_{L}\bigwedge\mathfrak{W}$ and for
$\bm{d\neq 0}$ we redefine the cocycle as
$\delta\rightarrow\delta-\delta_{r_{c}}$. Setting $r_{c}=\sum_{i_{0}\in
I_{0}}\frac{-\gamma^{0}_{i_{0}}}{i_{0}}c_{L}\wedge L_{i_{0}}$ results in
$\delta_{c}(L_{0})=0$ except if there is one index $i_{0}=0$. In that case,
consider the cocycle condition (617) with $X=L_{1},Y=L_{0}$, yielding

$\displaystyle\sum_{i_{1}\in I_{1}}\gamma^{1}_{i_{1}}c_{L}\wedge
L_{i_{1}}=\sum_{i_{1}\in
I_{1}}\gamma^{1}_{i_{1}}c_{L}\wedge(i_{1})L_{i_{1}}-\gamma^{0}_{0}c_{L}\wedge(-1)L_{1}.$
(619)

The last term can only cancel if there is a $i_{1}=1$ but then
$\gamma^{0}_{0}=0$ follows nonetheless. Thus we can set $\delta_{c}(L_{0})=0$.

Next, we set $X=L_{m},Y=L_{0}$ in (617) and because $d=m-i_{m}\neq 0$ we find
$\delta_{c}(L_{m})=0$.

For $\bm{d=0}$ we instead use the freedom of choosing a cochain to set
$\delta_{c}(L_{1})=0$ which can be achieved with
$r_{c}=\gamma^{1}_{1}c_{L}\wedge L_{0}$. Then (617) can be used with
$X=L_{1},Y=L_{m}$ and one finds that the resulting equations are equivalent to
those in the last part of the proof of theorem 2. Thus we find
$\delta_{c}(L_{m})=0$ in the $d=0$ case as well. Note that this means that
after we set $\delta_{\mathfrak{W}}=0$ the remaining part of the proof is
basically equivalent to $H^{1}\left(\mathfrak{W},\mathfrak{W}\right)=\\{0\\}$.

Now we are ready to prove the actual theorem and set
$c_{L}=c_{\bar{L}}$232323This is physically motivated. If $c_{L}\neq
c_{\bar{L}}$ a bit more work is needed because also terms of the form
$c_{L}\wedge c_{\bar{L}}$ can appear.. Let $\delta$ be a 1-cocycle on
$\mathfrak{Vir}\oplus\mathfrak{Vir}$ with values in
$\bigwedge^{2}\left(\mathfrak{Vir}\oplus\mathfrak{Vir}\right)$ and it again
splits according to
$\delta=\delta_{\mathfrak{W}\oplus\mathfrak{W}}+\delta_{c}+\delta_{\bar{c}}$
where $\delta_{c}$ maps to $c_{L}\wedge\mathfrak{W}$ and $\delta_{\bar{c}}$ to
$c_{L}\wedge\overline{\mathfrak{W}}$. The cocycle condition results in
separate equations and the analogue of (616) implies that
$\delta_{\mathfrak{W}\oplus\mathfrak{W}}$ is a coboundary due to theorem 2.
Redefining the cocycle accordingly causes
$\delta_{\mathfrak{W}\oplus\mathfrak{W}}$ to vanish and the analogue of (617)
used with $X=L_{m},Y=L_{n}$ implies an equation with terms in
$c_{L}\wedge\mathfrak{W}$ that can be used to show that $\delta_{c}$ is
cohomolog to zero as in the first part of the proof. Similarly with
$X=\bar{L}_{m},Y=\bar{L}_{n}$ we infer $\delta_{\bar{c}}$ is derived from a
cochain concluding the proof.

Proof of Theorem 5
Following [98] we start with some general statements on semi-direct products
of the form $\mathfrak{g}=\mathfrak{n}\rtimes\mathfrak{h}$, where
$\mathfrak{h}$ is semi-simple and $\mathfrak{n}$ is an ideal. If
$\mathfrak{h}$ admits only coboundary cocycles then for each cocycle
$\delta\in Z(\mathfrak{g},\bigwedge^{2}\mathfrak{g})$ there exists an
$r\in\bigwedge^{2}\mathfrak{h}$ such that $\delta_{0}=\delta-\partial_{0}r$
satisfies ${\delta_{0}}_{|_{\mathfrak{h}}}=0$ and it is a cocycle if and only
if ${\delta_{0}}_{|_{\mathfrak{n}}}\in
Z(\mathfrak{n},\bigwedge^{2}\mathfrak{g})\cap\text{Mor}_{\mathfrak{h}}(\mathfrak{n},\bigwedge^{2}\mathfrak{g})$,
where $\text{Mor}_{\mathfrak{h}}(\mathfrak{n},\bigwedge^{2}\mathfrak{g})$
denotes the morphisms between the $\mathfrak{h}$ modules
$\mathfrak{n},\bigwedge^{2}\mathfrak{g}$, i.e.

$\displaystyle\delta(x\triangleright n)=x\triangleright\delta(n),\quad
x\in\mathfrak{h},n\in\mathfrak{n}.$ (620)

The latter follows from the fact that the cocycle condition involving
$[\mathfrak{h},\mathfrak{n}]$ reduces to (620) if
${\delta_{0}}_{|_{\mathfrak{h}}}=0$.

Here we have
$\mathfrak{g}\equiv\mathfrak{B}_{3},\mathfrak{h}\equiv\mathfrak{W},\mathfrak{n}\equiv
V$ and the plan for the proof is as follows. First, we show that on
$\mathfrak{W}$ all cocycles are indeed coboundary, i.e.
$\bm{Z(\mathfrak{W},\bigwedge^{2}\mathfrak{B}_{3})=B(\mathfrak{W},\bigwedge^{2}\mathfrak{B}_{3})}$.
In the second step it is shown that
$\bm{Z(V,\bigwedge^{2}\mathfrak{B}_{3})\cap\text{{Mor}}_{\mathfrak{W}}(V,\bigwedge^{2}\mathfrak{B}_{3})}$
vanishes.

The cocycles (and similarly the coboundaries) can be decomposed according to

$Z(\mathfrak{W},\bigwedge^{2}\mathfrak{B}_{3})=Z(\mathfrak{W},\bigwedge^{2}\mathfrak{W})\oplus
Z(\mathfrak{W},\bigwedge^{2}V)\oplus Z(\mathfrak{W},V\wedge\mathfrak{W}).$

From theorem 1 it is known that
$Z(\mathfrak{W},\bigwedge^{2}\mathfrak{W})=B(\mathfrak{W},\bigwedge^{2}\mathfrak{W})$
and we can use the same arguments to show
$Z(\mathfrak{W},\bigwedge^{2}V)=B(\mathfrak{W},\bigwedge^{2}V)$. Namely, all
the formula in the proof of theorem 1 with $L_{m}\rightarrow T_{m}$ in all
terms with a wedge product, including the r-matrices, and $L_{m}\equiv l_{m}$
in the rest can be used verbatim to establish the latter statement.

Defining the degree of a cocycle $\delta\in
Z(\mathfrak{W},V\wedge\mathfrak{W})$ to be $i_{m}+j_{m}-m$ for

$\displaystyle\delta(l_{m})=l_{i_{m}}\wedge T_{j_{m}}$ (621)

the statement
$Z(\mathfrak{W},V\wedge\mathfrak{W})=B(\mathfrak{W},V\wedge\mathfrak{W})$
follows for cocycles of degree $\bm{d\neq 0}$ similarly as in the proof of
theorem 1. For degree zero cocycles we again have $\delta(l_{0})=0$ from the
cocycle condition with $l_{0},l_{m}$. Considering the cocycle condition

$\displaystyle\delta([l_{1},l_{-1}])=$ $\displaystyle\sum_{i_{-1}\in
I_{-1}}\gamma^{-1}_{i_{-1}}\bigg{(}(2+i_{-1})l_{-i_{-1}}\wedge
T_{i_{-1}}+(1-i_{-1})l_{-1-i_{-1}}\wedge T_{1+i_{-1}}\bigg{)}$
$\displaystyle-\sum_{i_{1}\in
I_{1}}\gamma^{1}_{i_{1}}\bigg{(}(-2+i_{1})l_{-i_{1}}\wedge
T_{i_{1}}+(-1-i_{1})l_{1-i_{1}}\wedge T_{-1+i_{1}}\bigg{)},$ (622)
$\displaystyle=$ $\displaystyle 0,$ (623)

where we used the ansatz

$\displaystyle\delta(l_{1})=\sum_{i_{1}\in
I_{1}}\gamma^{1}_{i_{1}}l_{1-i_{1}}\wedge
T_{i_{1}},\quad\delta(l_{-1})=\sum_{i_{-1}\in
I_{-1}}\gamma^{-1}_{i_{-1}}l_{-1-i_{-1}}\wedge T_{i_{-1}},$ (624)

the situation is a bit different than in the proof of theorem 1 as we can not
make use of the index restriction. Instead, we have to distinguish two
different cases for the ordered set of non-zero indices
$i_{1}=-p_{j},j=1,...,n$. If the largest index is bigger than zero we call it
$p_{1}\geq p_{j}$. Then the corresponding fourth term (type $IV$ term) of
(622) can not be canceled by a type $III$ term. Thus it can be only canceled
by a type $I$ or $II$ term and we use a coboundary to eliminate the potential
type $I$ term analogously to the proof of theorem 1. The type $II$ term would
require $i_{-1}=-2-p_{1}$ to cancel the first type $IV$ term which implies the
existence of a type $I$ term of the form $l_{2+p_{1}}\wedge T_{-2-p_{1}}$
which necessitates again a type $II$ term with $i_{-1}=-3-p_{1}$. This would
lead to an infinite tower of terms which is not allowed as $I_{-1}$ has to be
a finite set. The only caveat arises if $p_{1}=1$ because a factor in the
first line of (622) vanishes in that case, which we discuss below. If the
largest index is smaller or equal to zero we call the smallest index $p_{1}$
and use the coboundary to eliminate the type $II$ term. Then a type $I$ term
would be necessary to cancel the type $IV$ term with $p_{1}$. Again this would
lead to an infinite tower except for the case $p_{1}=0$. Using the same
iterative arguments as in the proof of theorem 1 yields that
$\gamma^{1}_{-p_{j}}=0$ except for the two special cases $p_{1}=1,0$, i.e.

$\displaystyle\delta(l_{1})=l_{0}\wedge T_{1},\quad\delta(l_{1})=l_{1}\wedge
T_{0}.$ (625)

One can use the r-matrices

$\displaystyle r=l_{-1}\wedge T_{1},\quad r=l_{1}\wedge T_{-1}$ (626)

with appropriate scaling to explicitly show that also in these cases $\delta$
is cohomolog to zero on $l_{1}$.

When proving that $\delta(l_{2})$ is cohomolog to zero we can reuse the
equations (585)-(590) where every right tensor leg is replaced by $T$ and the
left with $l$. The only difference is that without the index restriction there
is also the solution $i^{\prime}_{2}=-1$ to (590). Plugging it into (589) one
finds that e.g. the term proportional to $l_{4}\wedge T_{1}$ does not vanish
and thus $\delta(l_{2})=0$. The rest of the proof of
$Z(\mathfrak{W},V\wedge\mathfrak{W})=B(\mathfrak{W},V\wedge\mathfrak{W})$ is
completely analogous to that of theorem 1.

In the second step we start by noting that

$Z(V,\bigwedge^{2}\mathfrak{B}_{3})=Z(V,\bigwedge^{2}\mathfrak{W})\oplus
Z(V,\bigwedge^{2}V)\oplus Z(V,V\wedge\mathfrak{W}),$

and similar for the morphisms. For a $\bm{\delta\in
Z(V,\bigwedge^{2}\mathfrak{W})}$ we make the ansatz

$\displaystyle\delta(T_{m})=\sum_{i_{m},j_{m}}\alpha^{m}_{i_{m},j_{m}}l_{i_{m}}\wedge
l_{j_{m}}$ (627)

and if the degree of $\delta$, i.e. $d=i_{m}+j_{m}-m$,does not vanish then the
condition (from (620)) yields

$\displaystyle l_{0}\triangleright\delta(T_{m})$
$\displaystyle=\delta(l_{0}\triangleright T_{m}),$ (628)
$\displaystyle\sum_{i_{m},j_{m}}(-i_{m}-j_{m})\alpha^{m}_{i_{m},j_{m}}l_{i_{m}}\wedge
l_{j_{m}}$
$\displaystyle=\sum_{i_{m},j_{m}}(-m)\alpha^{m}_{i_{m},j_{m}}l_{i_{m}}\wedge
l_{j_{m}}$ (629)

and $\delta$ has to be zero. For a degree zero cocycle we set

$\displaystyle\delta(T_{m})=\sum_{i_{m}}\alpha^{m}_{i_{m}}l_{m-i_{m}}\wedge
l_{i_{m}},$ (630)

with the index restriction $i_{m}>m/2$. From the cocycle condition

$\displaystyle 0=\delta([T_{0},T_{1}])=[T_{0}\otimes 1+1\otimes
T_{0},\delta(T_{1})]-[T_{1}\otimes 1+1\otimes T_{1},\delta(T_{0})]$ (631)

we get

$\displaystyle 0=$
$\displaystyle\sum_{i_{0}}\alpha^{0}_{i_{0}}((1+i_{0})T_{1-i_{0}}\wedge
l_{i_{0}}+(1-i_{0})l_{-i_{0}}\wedge T_{1+i_{0}})$
$\displaystyle-\sum_{i_{1}}\alpha^{1}_{i_{1}}((i_{1}-1)T_{1-i_{1}}\wedge
l_{i_{1}}-i_{1}l_{1-i_{1}}\wedge T_{i_{1}}.$ (632)

Canceling the term proportional to $T_{1-i_{0}}\wedge l_{i_{0}}$ with the
largest index $i^{\prime}_{0}\geq i_{0}$ we find that it can be done with the
first term in the first line of (632) with $i_{1}=i^{\prime}_{0}$. A
cancellation with the second term and $i_{1}=1-i^{\prime}_{0}$ is not possible
because of the index restriction. Thus we have

$\displaystyle\alpha^{0}_{i^{\prime}_{0}}(1+i^{\prime}_{0})=\alpha^{1}_{i^{\prime}_{0}}(1-i^{\prime}_{0}).$
(633)

From (628) with $m=1$ and $m$ and $0$ exchanged on the other hand one has

$\displaystyle\sum_{i_{0}}\alpha^{0}_{i_{0}}((1+i_{0})l_{1-i_{0}}\wedge
l_{i_{0}}+(1-i_{0})l_{-i_{0}}\wedge
l_{i_{0}+1})=\sum_{i_{1}}\alpha^{1}_{i_{1}}l_{1-i_{1}}\wedge l_{i_{1}}.$ (634)

Cancellation of the first term on the lhs with the largest index can only be
done by the rhs and we find

$\displaystyle(1+i^{\prime}_{0})\alpha^{0}_{i^{\prime}_{0}}=\alpha^{1}_{i^{\prime}_{0}},$
(635)

which would imply $i^{\prime}_{0}=0$ together with (633) violating the index
restriction. Thus, we conclude that for $\delta\in
Z(V,\bigwedge^{2}\mathfrak{W})\cap\text{Mor}_{\mathfrak{W}}(V,\bigwedge^{2}\mathfrak{W})$
$\delta(T_{0})=0$ and by (628) also $\delta(T_{m})=0$ for all $m$.

Next, consider a cocycle $\bm{\delta\in
Z(V,V\wedge\mathfrak{W})\cap\text{{Mor}}_{\mathfrak{W}}(V,V\wedge\mathfrak{W})}$.
If its degree is not zero then the first line in (628) results in
$\delta(T_{m})=0$ similar to the case before. A degree zero cocycle has the
form

$\displaystyle\delta(T_{m})=\sum_{i_{m}}\alpha^{m}_{i_{m}}l_{m-i_{m}}\wedge
T_{i_{m}}$ (636)

and from the cocycle condition (631) we get

$\displaystyle 0=\sum_{i_{0}}\alpha^{0}_{i_{0}}(m+i_{0})T_{m-i_{0}}\wedge
T_{i_{0}}-\sum_{i_{m}}\alpha^{m}_{i_{m}}(i_{m}-m)T_{m-i_{m}}\wedge T_{i_{m}}.$
(637)

If we set $m$ sufficiently large, the term with the smallest index
$i^{\prime}_{0}$ on the lhs has to be canceled by $i_{m}=i^{\prime}_{0}$
leading to

$\displaystyle(m+i^{\prime}_{0})\alpha^{0}_{i^{\prime}_{0}}=(i^{\prime}_{0}-m)\alpha^{m}_{i^{\prime}_{0}}.$
(638)

From the condition that $\delta$ is a morphism of modules ((628) with $m$ and
$0$ exchanged) we have furthermore

$\displaystyle\sum_{i_{m}}\alpha^{m}_{i_{1}}ml_{m-i_{m}}\wedge
T_{i_{m}}=\sum_{i_{0}}\alpha^{0}_{i_{0}}((m+i_{0})l_{m-i_{0}}\wedge
T_{i_{0}}+(m-i_{0})l_{-i_{0}}\wedge T_{i_{0}+m})$ (639)

leading to

$\displaystyle\alpha^{m}_{i^{\prime}_{0}}m=\alpha^{0}_{i_{0}}(m+i^{\prime}_{0}),$
(640)

which can not be satisfied for all (large enough) $m$ together with (638).
Thus, $\delta(T_{0})=0$ and by (628) the cocycle vanishes also on all other
generators of $V$.

Finally, a cocycle $\bm{\delta\in
Z(V,\bigwedge^{2}V)\cap\text{{Mor}}_{\mathfrak{W}}(V,\bigwedge^{2}V)}$ of
degree $d\neq 0$ vanishes also by virtue of (628). If the degree is zero it
can be written as

$\displaystyle\delta(T_{m})=\sum_{i_{m}}\alpha^{m}_{i_{m}}T_{m-i_{m}}\wedge
T_{i_{m}}$ (641)

and from the morphism condition one extracts

$\displaystyle\sum_{i_{m}}\alpha^{m}_{i_{m}}mT_{m-i_{m}}\wedge
T_{i_{m}}=\sum_{i_{0}}\alpha^{0}_{i_{0}}((m+i_{0})T_{m-i_{0}}\wedge
T_{i_{0}}+(m-i_{0})T_{-i_{0}}\wedge T_{i_{0}+m}).$ (642)

With the index restriction $i_{m}<m/2$ and $m<0$ the second term on the rhs
with a fixed index $i^{\prime}_{0}$ can be canceled by the first with some
$i_{0}=i^{\prime}_{0}+m$ or by the lhs with $i_{m}=i^{\prime}_{0}$, requiring
that $i^{\prime}_{0}<m/2$. Since the set of indices $i_{0}$ is finite these
two conditions can not be met for sufficiently large $|m|$ and thus all
$\delta\in Z(V,\bigwedge^{2}V)\cap\text{Mor}_{\mathfrak{W}}(V,\bigwedge^{2}V)$
have to vanish concluding the proof.

Proof of Theorem 6

We will structure the proof in the same way as the proof of theorem 5 and as
already noted in the proof of theorem 3 all results that do not use the index
restriction carry over directly. For the first part we are then left to show
that
$Z\left(\mathfrak{W}_{+},\bigwedge^{2}V_{+}\right)=B\left(\mathfrak{W}_{+},\bigwedge^{2}V_{+}\right)$
which follows from the proof of
$H^{1}\left(\mathfrak{W}_{+},\bigwedge^{2}\mathfrak{W}_{+}\right)=\\{0\\}$ if
in all wedge products we replace $L_{m}\rightarrow T_{m}$ and
$L_{m}\rightarrow l_{m}$ in the single generators.

In the second part of the proof we need to show that degree $0$ cocycles in
$Z\left(V_{+},\bigwedge^{2}\mathfrak{W}_{+}\right)\cap\text{Mor}_{\mathfrak{W}_{+}}\left(V_{+},\bigwedge^{2}\mathfrak{W}_{+}\right)$
and
$Z\left(V_{+},\bigwedge^{2}V_{+}\right)\cap\text{Mor}_{\mathfrak{W}_{+}}\left(V_{+},\bigwedge^{2}V_{+}\right)$
vanish. The former cocycle has the form

$\displaystyle\delta(T_{0})=\gamma^{0}_{1}l_{0}\wedge
l_{1},\quad\delta(T_{1})=\gamma^{1}_{2}l_{-1}\wedge
l_{2}+\gamma^{1}_{1}l_{0}\wedge l_{1}$ (643)

and imposing the cocycle condition yields

$\displaystyle\delta([T_{0},T_{1}])$ $\displaystyle=[T_{0}\otimes 1+1\otimes
T_{0},\delta(T_{1})]-[T_{1}\otimes 1+1\otimes T_{1},\delta(T_{0})],$ (644)
$\displaystyle\Rightarrow\gamma^{1}_{2}$
$\displaystyle=0,\quad\gamma^{1}_{1}=-2\gamma^{0}_{1}.$ (645)

From the morphism condition on the other hand we have

$\displaystyle l_{1}\triangleright\delta(T_{0})=\delta(l_{1}\triangleright
T_{0}),\quad\Rightarrow 2\gamma^{0}_{1}=\gamma^{1}_{1},$ (646)

which is only possible if $\delta(T_{0})=0$. Then we can use cocycle
conditions

$\displaystyle 0=\delta([T_{0},T_{m}])=-m\delta(T_{m})$ (647)

for $m\geq-1$ to show that $\delta$ has to vanish on $V_{+}$.

Finally, for a degree $0$ cocycle $\delta\in
Z\left(V_{+},\bigwedge^{2}V_{+}\right)$ we use the same argumentation as above
with $l_{m}\rightarrow T_{m}$ in all wedge products to conclude the proof.

Proof of Theorem 7

Similarly to the proof of theorem 4 we start by splitting a cocycle $\delta\in
Z\left(\mathfrak{B}_{3c},\bigwedge^{2}\mathfrak{B}_{3c}\right)$ into
$\delta=\delta_{\mathfrak{B}}+\delta_{c}$ so that $\delta_{\mathfrak{B}}$ maps
to $\bigwedge^{2}\mathfrak{B}_{3}$ and $\delta_{c}$ to
$c_{L}\bigwedge\mathfrak{B}_{3}$. Then the cocycle condition (615) reads

$\displaystyle\delta_{\mathfrak{B}}([X,Y])$
$\displaystyle=[\delta_{\mathfrak{B}}(X),Y\otimes 1+1\otimes
Y]_{\mathfrak{B}}-[\delta_{\mathfrak{B}}(Y),X\otimes 1+1\otimes
X]_{\mathfrak{B}},$ (648) $\displaystyle\delta_{c}([X,Y])$
$\displaystyle=[\delta_{\mathfrak{B}}(X),Y\otimes 1+1\otimes
Y]_{c}-(X\leftrightarrow Y)+[\delta_{c}(X),Y\otimes+1\otimes
Y]_{\mathfrak{B}}-(X\leftrightarrow Y),$ (649)

where the brackets are defined analogously to the proof of theorem 4 and we
already omitted any parts of $X,Y\in\mathfrak{B}_{3}$ containing the central
element. Equation (648) then implies with theorem 5 that
$\bm{\delta_{\mathfrak{B}}}$ is a coboundary derived from a cochain
$r\in\bigwedge^{2}\mathfrak{B}_{3}$ and we redefine
$\delta\rightarrow\delta-\delta_{r}$. Thus the first terms on the rhs of (649)
vanish and for $\bm{\delta_{c}}$ we proceed similarly as in the proof of
theorem 5. This means we first consider cocycles in
$Z\left(\mathfrak{W},c_{L}\bigwedge\mathfrak{B}_{3}\right)=Z\left(\mathfrak{W},c_{L}\bigwedge\mathfrak{W}\right)\oplus
Z\left(\mathfrak{W},c_{L}\bigwedge V_{3}\right)$. For the first summand the
proposition
$Z\left(\mathfrak{W},c_{L}\bigwedge\mathfrak{W}\right)=B\left(\mathfrak{W},c_{L}\bigwedge\mathfrak{W}\right)$
follows as in the last part of the proof of theorem 2 by replacing
$L_{m}\rightarrow l_{m}$ and $\bar{L}_{m}\rightarrow c_{L}$ in all the
formulas. The same formulas can also be used with the replacement
$L_{m}\rightarrow T_{m},\,\bar{L}_{m}\rightarrow c_{L}$ in all wedge products
including the r-matrices and $L_{m}\rightarrow l_{m},\,\bar{L}_{m}\rightarrow
c_{L}$ in the single tensors to show that $Z\left(\mathfrak{W},c_{L}\bigwedge
V_{3}\right)=B\left(\mathfrak{W},c_{L}\bigwedge V_{3}\right)$.

After redefining the cocycle so that $\delta_{c}|_{\mathfrak{W}}=0$ the
cocycle condition involving $l_{n},T_{m}$ reduces to the morphism condition.
In particular, for $\bm{\delta_{c}\in
Z\left(V_{3},c_{L}\bigwedge\mathfrak{W}\right)}$ we additionally have to
demand

$\displaystyle\delta_{c}(l_{n},T_{m}])=-[l_{n}\otimes 1+1\otimes
l_{n},\delta_{c}(T_{m})].$ (650)

This equation and the cocycle condition

$\displaystyle\delta_{c}([T_{m},T_{n}])=[T_{m}\otimes 1+1\otimes
T_{m},\delta_{c}(T_{n})]-[T_{n}\otimes 1+1\otimes T_{n},\delta_{c}(T_{m})]$
(651)

have to be satisfied separately for cocycles with different degree which we
define to be $d=i_{m}-m$ for

$\displaystyle\delta_{c}(T_{m})=\alpha^{m}_{i_{m}}c_{L}\wedge l_{i_{m}}.$
(652)

For $d\neq 0$ the morphism condition (650) with $n=0$ implies
$-m\delta_{c}(T_{m})=-i_{m}\delta_{c}(T_{m})\rightarrow\delta_{c}(T_{m})=0$.
If $d=0$ the morphism condition with $l_{m},T_{0}$ yields

$\displaystyle m\alpha^{m}_{m}c_{L}\wedge T_{m}=m\alpha^{0}_{0}c_{L}\wedge
T_{m}$ (653)

which contradicts the cocycle condition with $T_{0},T_{m}$ leading to
$\alpha^{0}_{0}=\alpha^{m}_{m}$ unless $\delta_{c}$ vanishes.

Finally, we have to deal with cocycles in $\bm{Z\left(V_{3},c_{L}\bigwedge
V_{3}\right)}$. The degree is defined analogously and for $d\neq 0$ the
morphism condition (650) with $n=0$ and the ansatz
$\delta_{c}(T_{m})=\alpha^{m}_{i_{m}}c_{L}\wedge T_{i_{m}}$ yields
$-m\delta_{c}(T_{m})=-i_{m}\delta_{c}(T_{m})\rightarrow\delta_{c}(T_{m})=0$.
For $d=0$ on the other hand the morphism condition with $l_{m},T_{0}$ implies
$\alpha^{m}_{m}=\alpha^{0}_{0}$ for all $m\in\mathbb{Z}$. As the other cocycle
conditions are trivially satisfied we showed that

$\displaystyle\delta_{\alpha}:l_{m}\mapsto 0,\;T_{m}\mapsto\alpha c_{L}\wedge
T_{m}$ (654)

is indeed a cocycle in
$Z\left(\mathfrak{B}_{3},\bigwedge^{2}\mathfrak{B}_{3}\right)$. Is it easy to
see that it can not be derived from a cochain since the only possibility that
has the correct grading ($r\sim c_{L}\wedge l_{0}$) produces the wrong
coefficients. Thus we showed that
$H^{1}\left(\mathfrak{B}_{3},\bigwedge^{2}\mathfrak{B}_{3}\right)=\delta_{\alpha}$.

Proof of Theorem 8

By the arguments from the proof of theorem 5 for the four dimensional BMS
algebra $\mathfrak{B}_{4}=(\mathfrak{W}\oplus\overline{\mathfrak{W}})\rtimes
V_{4}$ it suffices to show that

$\displaystyle
Z((\mathfrak{W}\oplus\overline{\mathfrak{W}}),\bigwedge^{2}\mathfrak{B}_{4})=B((\mathfrak{W}\oplus\overline{\mathfrak{W}}),\bigwedge^{2}\mathfrak{B}_{4})$
(655)

and

$\displaystyle
Z(V_{4},\bigwedge^{2}\mathfrak{B}_{4})\cap\text{Mor}_{\mathfrak{W}\oplus\overline{\mathfrak{W}}}(V_{4},\bigwedge^{2}\mathfrak{B}_{4})=0.$
(656)

In order to show (655) one can decompose

$\displaystyle
Z((\mathfrak{W}\oplus\overline{\mathfrak{W}}),\bigwedge^{2}\mathfrak{B}_{4})=$
$\displaystyle
Z((\mathfrak{W}\oplus\overline{\mathfrak{W}}),\bigwedge^{2}\mathfrak{W})\oplus
Z((\mathfrak{W}\oplus\overline{\mathfrak{W}}),\bigwedge^{2}\overline{\mathfrak{W}})\oplus
Z((\mathfrak{W}\oplus\overline{\mathfrak{W}}),\bigwedge^{2}V_{4})$
$\displaystyle\oplus
Z((\mathfrak{W}\oplus\overline{\mathfrak{W}}),\mathfrak{W}\wedge\overline{\mathfrak{W}})\oplus
Z((\mathfrak{W}\oplus\overline{\mathfrak{W}}),\mathfrak{W}\wedge V_{4})\oplus
Z((\mathfrak{W}\oplus\overline{\mathfrak{W}}),\overline{\mathfrak{W}}\wedge
V_{4})$ (657)

and from Theorem 1,2 and 5 we know that

$\displaystyle
Z(\mathfrak{W},\bigwedge^{2}\mathfrak{W})=B(\mathfrak{W},\bigwedge^{2}\mathfrak{W}),\quad
Z(\mathfrak{W},\mathfrak{W}\wedge\overline{\mathfrak{W}})=B(\mathfrak{W},\mathfrak{W}\wedge\overline{\mathfrak{W}}),$
(658) $\displaystyle Z(\mathfrak{W},\bigwedge^{2}\overline{\mathfrak{W}})=0$
(659)

and similar for $\mathfrak{W}\leftrightarrow\overline{\mathfrak{W}}$.
Considering a cocycle $\bm{\delta\in Z(\mathfrak{W},\mathfrak{W}\wedge
V_{4}))}$ we make the ansatz

$\displaystyle\delta(l_{m})=\sum_{i_{m},j_{m},k_{m}}\alpha^{m}_{i_{m},j_{m},k_{m}}l_{i_{m}}\wedge
T_{j_{m},k_{m}}.$ (660)

Choosing the cochain

$\displaystyle
r=\sum_{i_{0},j_{0},k_{0}}\alpha^{0}_{i_{0},j_{0},k_{0}}\frac{-1}{i_{0}+j_{0}-1/2}l_{i_{0}}\wedge
T_{j_{0},k_{0}}$ (661)

the cocycle $\delta^{\prime}=\delta-\partial_{0}r$242424We redefine the ansatz
so that (660) holds for $\delta^{\prime}$ except that $\alpha^{0}$ vanishes.
vanishes on $l_{0}$ and thus the cocycle condition

$\displaystyle\delta^{\prime}([l_{0},l_{m}])$ $\displaystyle=[l_{0}\otimes
1+1\otimes l_{0},\delta^{\prime}(l_{m})]-[l_{m}\otimes 1+1\otimes
l_{m},\delta^{\prime}(l_{0})],$ (662)
$\displaystyle\rightarrow(-i_{m}+1/2-j_{m})\delta^{\prime}(l_{m})$
$\displaystyle=-m\delta^{\prime}(l_{m})$ (663)

implies that it is indeed zero on all generators irrespective of the degree as
the indices take only values in $\mathbb{Z}$.

Similar, for $\bm{\delta\in Z(\mathfrak{W},\overline{\mathfrak{W}}\wedge
V_{4}))}$ with the ansatz

$\displaystyle\delta(l_{m})=\sum_{i_{m},j_{m},k_{m}}\alpha^{m}_{i_{m},j_{m},k_{m}}\bar{l}_{i_{m}}\wedge
T_{j_{m},k_{m}}$ (664)

and

$\displaystyle
r=\sum_{i_{0},j_{0},k_{0}}\alpha^{0}_{i_{0},j_{0},k_{0}}\frac{-1}{j_{0}-1/2}\bar{l}_{i_{0}}\wedge
T_{j_{0},k_{0}}$ (665)

the cocycle $\delta^{\prime}=\delta-\partial_{0}r$ vanishes on $l_{0}$. The
cocycle condition (662) again yields

$\displaystyle(1/2-j_{m})\delta^{\prime}(l_{m})=-m\delta^{\prime}(l_{m})$
(666)

proving that $Z(\mathfrak{W},\overline{\mathfrak{W}}\wedge
V_{4}))=B(\mathfrak{W},\overline{\mathfrak{W}}\wedge V_{4}))$.

This leaves only $\bm{Z(\mathfrak{W},\bigwedge^{2}V_{4}))}$ in the first step
of the proof. For a cocycle $\delta$ of the form

$\displaystyle\delta(l_{m})=\sum_{i_{m},j_{m},k_{m},s_{m}}\alpha^{m}_{i_{m},j_{m},k_{m},s_{m}}T_{i_{m},j_{m}}\wedge
T_{k_{m},s_{m}}$ (667)

we define the (left) degree $d_{L}$ to be $d_{L}=m-i_{m}-k_{m}$ and
analogously the right degree $d_{R}$. Then for a cocycle of degree $d_{L}\neq
1$ (or similarly for $d_{R}\neq 1$) the cochain

$\displaystyle
r=\sum_{i_{0},j_{0},k_{0},s_{0}}\alpha^{0}_{i_{0},j_{0},k_{0},s_{0}}\frac{-1}{1-i_{0}-k_{0}}T_{i_{0},j_{0}}\wedge
T_{k_{0},s_{0}}$ (668)

can be used to show that $\delta^{\prime}=\delta-\partial_{0}r$ vanishes on
$l_{0}$. In that case the cocycle condition (662) yields

$\displaystyle-m\delta^{\prime}(l_{m})=(1-i_{m}-k_{m})\delta^{\prime}(l_{m}),$
(669)

which has to vanish.

A cocycle of (left) degree one can be written as

$\displaystyle\delta(l_{m})=\sum_{i_{m},j_{m},s_{m}}\alpha^{m}_{i_{m},j_{m},s_{m}}T_{m-i_{m},j_{m}}\wedge
T_{i_{m}+1,s_{m}},$ (670)

with $i_{m}<m/2$. From (662) one obtains

$\displaystyle-m\delta(l_{m})=-m\delta(l_{m})-[l_{m}\otimes 1+1\otimes
l_{m},\delta(l_{0})],$ (671)

which has to hold for all $m$ and thus $\delta(l_{0})=0$. Then the cocycle
condition

$\displaystyle\delta([l_{1},l_{-1}])=[l_{1}\otimes 1+1\otimes
l_{1},\delta(l_{-1})]-[l_{-1}\otimes 1+1\otimes l_{-1},\delta(l_{1})]$ (672)

yields the equation

$\displaystyle 0=$
$\displaystyle\sum_{i_{1},j_{1},s_{1}}\alpha^{1}_{i_{1},j_{1},s_{1}}\left((i_{1}-1)T_{-i_{1},j_{1}}\wedge
T_{i_{1}+1s_{1}}-(i_{1}+1)T_{1-i_{1},j_{1}}\wedge T_{i_{1},s_{1}}\right)$
$\displaystyle-\sum_{i_{-1},j_{-1},s_{-1}}\alpha^{-1}_{i_{-1},j_{-1},s_{-1}}\left((2-i_{-1})T_{-i_{-1},j_{1}}\wedge
T_{i_{-1}+1,s_{-1}}-i_{-1}T_{-1-i_{-1},j_{-1}}\wedge
T_{i_{-1}+2,s_{-1}}\right).$ (673)

Comparing with (570) shows that the index structure (neglecting the second
index of the $T$) is almost the same, except that the index of the $T$ on the
rhs of the wedge product and the prefactors on the second terms in the first
and second line of (673) are always higher by one. Because of the index
restriction, the ordering of the wedge product is fixed (so that a left and a
right term never have to coincide) and thus the same argumentation can be
used. This is also consistent with the prefactors as any commutators with the
right wedge product legs are higher by one compared to the situation in the
proof of theorem 1. Therefore $\delta(l_{1})=0$ and we use the same strategy
to show that $\delta(l_{2})=0$, i.e. we iteratively use the cocycle condition
to compute $\delta(l_{5})$ and then compare with the cocycle condition with
$l_{2}$ and $l_{3}$. As a result, we obtain in analogy with (590)

$\displaystyle 0=T_{2-i^{\prime}_{2},j_{2}}\wedge
T_{i^{\prime}_{2}+4,s_{2}}\left(-\left(\frac{1}{2}+i^{\prime}_{2}\right)i^{\prime}_{2}+(i^{\prime}_{2}-1)-\frac{1}{6}i^{\prime}_{2}(1+i^{\prime}_{2})(2+i^{\prime}_{2})\right)$
(674)

for the largest index $i^{\prime}_{2}$ and since this equation has no solution
in $\mathbb{Z}$, $\delta(l_{2})=0$. Then $\delta(l_{m})=0,m>0$ follows easily
and

$\displaystyle 0=\delta([l_{-1},l_{m}])=-[l_{m}\otimes 1+1\otimes
l_{m},\delta(l_{-1})]$ (675)

for all positive $m$ yields $\delta(l_{-1})=0$. Showing that
$\delta(l_{-2})=0$ and subsequently for all negative $m$ is again obtained in
an analogous way and thus
$Z(\mathfrak{W},\bigwedge^{2}V_{4}))=B(\mathfrak{W},\bigwedge^{2}V_{4}))$.

In the second step it will be shown that
$\bm{Z(V_{4},\bigwedge^{2}\mathfrak{B}_{4})\cap\text{{Mor}}_{\mathfrak{W}\oplus\overline{\mathfrak{W}}}(V_{4},\bigwedge^{2}\mathfrak{B}_{4})=\\{0\\}}$.
Using a decomposition as in (657) we start with
$\bm{\text{{Mor}}_{\mathfrak{W}\oplus\overline{\mathfrak{W}}}(V_{4},\bigwedge^{2}\mathfrak{W})}$.
The general ansatz

$\displaystyle\delta(T_{mk})=\sum_{i,j}l_{i}\wedge l_{j}$ (676)

and the morphism condition

$\displaystyle l_{0}\triangleright\delta(T_{mk})=\delta(l_{0}\triangleright
T_{mk})$ (677)

yield

$\displaystyle\sum_{i,j}\left(\frac{1}{2}-m\right)l_{i}\wedge
l_{j}=\sum_{i,j}(-i-j)l_{i}\wedge l_{j}$ (678)

and thus
$\text{Mor}_{\mathfrak{W}\oplus\overline{\mathfrak{W}}}(V_{4},\bigwedge^{2}\mathfrak{W})=\\{0\\}$.
Similarly, one has
$\bm{\text{{Mor}}_{\mathfrak{W}\oplus\overline{\mathfrak{W}}}(V_{4},\bigwedge^{2}\overline{\mathfrak{W}})=\\{0\\}}$
and
$\bm{\text{{Mor}}_{\mathfrak{W}\oplus\overline{\mathfrak{W}}}(V_{4},\mathfrak{W}\wedge\overline{\mathfrak{W}})=\\{0\\}}$.

For a cocycle
$\bm{\delta\in\text{{Mor}}_{\mathfrak{W}\oplus\overline{\mathfrak{W}}}(V_{4},\bigwedge^{2}V_{4})}$
(677) results in

$\displaystyle\sum_{i_{mk},j_{mk},k_{mk},s_{mk}}\alpha^{mk}_{i_{mk},j_{mk},k_{mk},s_{mk}}\left(\frac{1}{2}-m\right)T_{i_{mk},j_{mk}}\wedge
T_{k_{mk},s_{mk}}$
$\displaystyle=\sum_{i_{mk},j_{mk},k_{mk},s_{mk}}\alpha^{mk}_{i_{mk},j_{mk},k_{mk},s_{mk}}(1-i_{mk}-k_{mk})T_{i_{mk},j_{mk}}\wedge
T_{k_{mk},s_{mk}},$ (679)

with the ansatz

$\displaystyle\delta(T_{mk})=\sum_{i_{mk},j_{mk},k_{mk},s_{mk}}\alpha^{mk}_{i_{mk},j_{mk},k_{mk},s_{mk}}T_{i_{mk},j_{mk}}\wedge
T_{k_{mk},s_{mk}}$ (680)

and therefore
$\text{Mor}_{\mathfrak{W}\oplus\overline{\mathfrak{W}}}(V_{4},\bigwedge^{2}V_{4})=\\{0\\}$.

Finally, for a cocycle
$\bm{\delta\in\text{{Mor}}_{\mathfrak{W}\wedge\overline{\mathfrak{W}}}(V_{4},V_{4}\oplus\mathfrak{W})}$
we define the degree such that

$\displaystyle\delta(T_{mk})=\sum_{i_{mk},j_{mk},k_{mk}}\alpha^{mk}_{i_{mk},j_{mk},k_{mk}}l_{i_{mk}}\wedge
T_{j_{mk},k_{mk}}$ (681)

has degree $d=i_{mk}+j_{mk}-m$. If $d\neq 0$ the equation

$\displaystyle\left(\frac{1}{2}-m\right)\sum_{i_{mk},j_{mk},k_{mk}}\alpha^{mk}_{i_{mk},j_{mk},k_{mk}}l_{i_{mk}}\wedge
T_{j_{mk},k_{mk}}$
$\displaystyle=\sum_{i_{mk},j_{mk},k_{mk}}\alpha^{mk}_{i_{mk},j_{mk},k_{mk}}\left(\frac{1}{2}-i_{mk}-j_{mk}\right)l_{i_{mk}}\wedge
T_{j_{mk},k_{mk}}$ (682)

from (677) implies $\delta=0$. Otherwise, i.e. for a degree zero cocycle, we
set

$\displaystyle\delta(T_{mk})=\sum_{i_{mk},s_{mk}}\alpha^{mk}_{i_{mk},s_{mk}}l_{m-i_{mk}}\wedge
T_{i_{mk},s_{mk}}$ (683)

and the cocycle condition

$\displaystyle 0=\delta([T_{0k},T_{mk^{\prime}}])=[T_{0k}\otimes 1+1\otimes
T_{0k},\delta(T_{mk^{\prime}})][T_{mk^{\prime}}\otimes 1+1\otimes
T_{mk^{\prime}},\delta(T_{0k})]$ (684)

results in

$\displaystyle 0=$
$\displaystyle\sum_{i_{mk^{\prime}},s_{mk^{\prime}}}\alpha^{mk^{\prime}}_{i_{mk^{\prime}},s_{mk^{\prime}}}\left(-\frac{m-i_{mk}+1}{2}T_{m-i_{mk^{\prime}}}\wedge
T_{i_{mk^{\prime}},s_{mk^{\prime}}}\right)$
$\displaystyle-\sum_{i_{0k},s_{0k}}\alpha^{0k}_{i_{0k},s_{0k}}\left(\left(-\frac{1-i_{0k}}{2}-m\right)T_{m-i_{0k},k}\wedge
T_{i_{0k},s_{ok}}\right),$ (685)
$\displaystyle\rightarrow\alpha^{mk}_{i^{\prime}_{0k},s_{0k}}\frac{m-i^{\prime}_{0k}+1}{2}=$
$\displaystyle\alpha^{0k}_{i^{\prime}_{0k},s_{0k}}\left(\frac{1-i^{\prime}_{0k}}{2}-m\right)$
(686)

for the smallest index $i^{\prime}_{0k}$ and sufficiently large $m$.
Additionally, the morphism condition yields

$\displaystyle l_{m}\triangleright\delta(T_{0k})=$
$\displaystyle\delta(l_{m}\triangleright T_{0k}),$ (687)
$\displaystyle\sum_{i_{0k},s_{0k}}\alpha^{0k}_{i_{0k},s_{0k}}\left((m+i_{0k})l_{m-i_{0k}}\wedge
T_{i_{0k},s_{0k}}+\left(\frac{m+1}{2}-i_{0k}\right)l_{-i_{0k}}\wedge
T_{i_{0k}+m,s_{0k}}\right)$
$\displaystyle=\sum_{i_{mk},s_{mk}}\alpha^{mk}_{i_{mk},s_{mk}}\left(\frac{m+1}{2}l_{m-i_{mk}}\wedge
T_{i_{mk},s_{mk}}\right),$ (688)
$\displaystyle\rightarrow\alpha^{0k}_{i^{\prime}_{0k},s_{0k}}(m+i^{\prime}_{0k})=\alpha^{mk}_{i^{\prime}_{0k},s_{0k}}\frac{m+1}{2}$
(689)

under the same conditions. Since the equations (686) and (689) can not be
satisfied simultaneously for all $m$ we have
$\text{Mor}_{\mathfrak{W}\wedge\overline{\mathfrak{W}}}(V_{4},V_{4}\wedge\mathfrak{W})=\\{0\\}$
and analogously
$\text{Mor}_{\mathfrak{W}\wedge\overline{\mathfrak{W}}}(V_{4},V_{4}\wedge\overline{\mathfrak{W}})=\\{0\\}$.
This completes the proof.

Proof of Theorem 9

The proof is structured similarly as for theorem 7. A generic cocycle in
$Z\left(\mathfrak{B}_{4c},\bigwedge^{2}\mathfrak{B}_{4c}\right)$ can be split
into the parts
$\delta=\delta_{\mathfrak{B}}+\delta_{c}+\delta_{\bar{c}}+\delta_{c,\bar{c}}$
mapping to $\bigwedge^{2}\mathfrak{B}_{4}$, $c_{l}\bigwedge\mathfrak{B}_{4}$,
$c_{\bar{l}}\bigwedge\mathfrak{B}_{4}$ and $c_{l}\bigwedge
c_{\bar{l}}\mathbb{C}$ respectively. The cocycle condition with
$X,Y\in\mathfrak{B}_{4}$

$\displaystyle\delta([X,Y])=[\delta(X),Y\otimes 1+1\otimes
Y]-(X\leftrightarrow Y)$ (690)

splits into

$\displaystyle\delta_{\mathfrak{B}}([X,Y]_{\mathfrak{B}})$
$\displaystyle=[\delta_{\mathfrak{B}}(X),Y\otimes 1+1\otimes
Y]_{\mathfrak{B}}-(X\leftrightarrow Y),$ (691)
$\displaystyle\delta_{c}([X,Y]_{\mathfrak{B}})$
$\displaystyle=[\delta_{\mathfrak{B}}(X),Y\otimes 1+1\otimes
Y]_{c}+[\delta_{c}(X),Y\otimes 1+1\otimes Y]_{\mathfrak{B}}-(X\leftrightarrow
Y),$ (692) $\displaystyle\delta_{\bar{c}}([X,Y]_{\mathfrak{B}})$
$\displaystyle=[\delta_{\mathfrak{B}}(X),Y\otimes 1+1\otimes
Y]_{\bar{c}}+[\delta_{\bar{c}}(X),Y\otimes 1+1\otimes
Y]_{\mathfrak{B}}-(X\leftrightarrow Y),$ (693)
$\displaystyle\delta_{c,\bar{c}}([X,Y]_{\mathfrak{B}})$
$\displaystyle=[\delta_{c}(X),Y\otimes 1+1\otimes
Y]_{\bar{c}}+[\delta_{\bar{c}}(X),Y\otimes 1+1\otimes Y]_{c}-(X\leftrightarrow
Y),$ (694)

where $[,]_{\mathfrak{B}}$, $[,]_{c}$ and $[,]_{\bar{c}}$ conserves or
respectively increases the number of central charges in the input. Equation
(691) implies with theorem 8 that $\bm{\delta_{\mathfrak{B}}}$ is a coboundary
with a cochain $r\in\bigwedge^{2}\mathfrak{B}_{4}$. After redefining
$\delta\rightarrow\delta-\delta_{r}$ we next turn to the semi-simple “factor”
$\mathfrak{W}\oplus\mathfrak{W}$ of the semi-direct product $\mathfrak{B}$
using equations (692)-(693) where the first term on the rhs vanishes. We
further decompose

$\displaystyle
Z\left(\mathfrak{W}\oplus\mathfrak{W},c_{l}\bigwedge\left(\mathfrak{W}\oplus\mathfrak{W}\right)\right)=$
$\displaystyle Z\left(\mathfrak{W},c_{l}\bigwedge\mathfrak{W}\right)\oplus
Z\left(\mathfrak{W},c_{l}\bigwedge\overline{\mathfrak{W}}\right)$
$\displaystyle\oplus
Z\left(\overline{\mathfrak{W}},c_{l}\bigwedge\overline{\mathfrak{W}}\right)\oplus
Z\left(\overline{\mathfrak{W}},c_{l}\bigwedge\mathfrak{W}\right)$ (695)

and (692) for $\bm{\delta_{c}\in
Z\left(\mathfrak{W},c_{l}\bigwedge\mathfrak{W}\right)}$ can be used as in the
last part of the proof of theorem 2 by replacing $L_{m}\rightarrow
l_{m},\bar{L}_{m}\rightarrow c_{l}$ to show that $\delta_{c}$ has to be a
coboundary from a cochain $r\in c_{l}\bigwedge\mathfrak{W}$. Similarly
$\bm{Z\left(\overline{\mathfrak{W}},c_{l}\bigwedge\overline{\mathfrak{W}}\right)=B\left(\overline{\mathfrak{W}},c_{l}\bigwedge\overline{\mathfrak{W}}\right)}$
and the remaining parts have to vanish because e.g. the rhs of (692) is alwas
zero for $\delta_{c}\in
Z\left(\mathfrak{W},c_{l}\bigwedge\overline{\mathfrak{W}}\right)$. Equation
(693) completely analogously asserts the same statements for $c_{l}\rightarrow
c_{\bar{l}}$.

Next we consider cocycles $\bm{\delta_{c}\in
Z\left(\mathfrak{W}\oplus\mathfrak{W},c_{l}\bigwedge V_{4}\right)}$. With the
general ansatz

$\displaystyle\delta_{c}(l_{m})=\sum_{p_{m},q_{m}}\alpha^{m}_{p_{m},q_{m}}c_{l}\wedge
T_{p_{m},q_{m}}$ (696)

the cochain
$r=\sum_{p_{0},q_{0}}\alpha^{0}_{p_{0},q_{0}}\frac{1}{1/2-p_{0}}c_{l}\wedge
T_{p_{0},q_{0}}$ can be used so that
$\delta_{c}\rightarrow\delta_{c}-\delta_{r}$ vanishes on $l_{0}$. Then the
cocycle condition (692) with $X=l_{m},Y=l_{0}$ gives

$\displaystyle m\delta_{c}(l_{m})=\left(p-\frac{1}{2}\right)\delta_{c}(l_{m})$
(697)

and thus $\delta_{c}(l_{m})=0$ and similarly also
$\delta_{c}(\bar{l}_{m})=0=\delta_{\bar{c}}(l_{m})=\delta_{\bar{c}}(\bar{l}_{m})$.

Equation (692) for $\bm{\delta_{c}\in
Z\left(V_{4},c_{l}\bigwedge\mathfrak{W}\right)}$ (and similarly for
$\mathfrak{W}\rightarrow\overline{\mathfrak{W}}$) with $X=l_{0},Y=T_{pq}$ is
just a morphism condition and with the ansatz

$\displaystyle\delta_{c}(T_{pq})=\sum_{i_{pq}}\alpha^{pq}_{i_{pq}}c_{l}\wedge
l_{i_{pq}}$ (698)

it yields

$\displaystyle\left(\frac{1}{2}-p\right)\delta_{c}(T_{pq})=\delta_{c}(T_{pq})i_{pq}$
(699)

which thus has to vanish.

This leaves us with $\bm{\delta_{c}\in Z\left(V_{4},c_{l}\bigwedge
V_{4}\right)}$ where the morphism condition with $X=l_{0},Y=T_{pq}$ and the
ansatz

$\displaystyle\delta_{c}(T_{pq})=\sum_{i_{pq},j_{pq}}\alpha^{pq}_{i_{pq},j_{pq}}c_{l}\wedge
T_{i_{pq},j_{pq}}$ (700)

results in

$\displaystyle\left(p-\frac{1}{2}\right)\delta_{c}(T_{pq})=\delta_{c}(T_{pq})\left(i_{pq}-\frac{1}{q}\right),$
(701)

which implies $\delta_{c}(T_{pq})=0$ unless the left degree $d_{L}\equiv
i_{pq}-m$ is zero. Similarly one finds that any cocycle with right degree
$d_{R}\equiv j_{pq}-m\neq 0$ has to vanish, leaving only the possibility

$\displaystyle\delta_{\alpha,\bar{\alpha}}:l_{m}\mapsto
0,\quad\bar{l}_{m}\mapsto 0,\quad T_{pq}\mapsto(\alpha
c_{l}+\bar{\alpha}c_{\bar{l}})\wedge T_{pq},$ (702)

which satisfies all the cocyle conditions and can not be written as a
coboundary because the only r-matrices with the correct grading ($r\sim
c_{l/\bar{l}}\wedge l_{0},\;r\sim c_{l/\bar{l}}\wedge\bar{l}_{0}$) do not
reproduce the coefficients.

Finally, we have to consider cocycles mapping to $\bm{c_{l}\bigwedge
c_{\bar{l}}\mathbb{C}}$. To this end we simply use the cocycle condition (694)
with $X=l_{m},Y=l_{n}$ (respectively $X=\bar{l}_{m},Y=\bar{l}_{n}$) and that
$\delta_{c}$ ($\delta_{\bar{c}}$) is cohomolog to zero to show that
$\delta_{c,\bar{c}}(l_{m})$ ($\delta_{c,\bar{c}}(\bar{l}_{m})$) has to vanish.
The same equations with $X=l_{m},Y=T_{pq}$ imply
$\delta_{c,\bar{c}}(T_{pq})=0$ if we take into account that on the rhs
$\delta_{c}(T_{pq})$ (or $\delta_{\bar{c}}(T_{pq})$) is cohomolog to at most
$\delta_{\alpha,\bar{\alpha}}$ and thus the brackets $[,]_{c},[,]_{\bar{c}}$
vanish.

From cocycle conditions with $X=c_{l}$ and $Y\in\mathfrak{B}_{4}$ we again
conclude that $\delta(c_{l})$ (or $\delta(c_{\bar{l}})$) has to be ad-
invariant. Contrary to the proof of theorems 4 and 7 this at first sight
leaves open the possibility that $\delta(c_{l})\sim c_{l}\wedge c_{\bar{l}}$.
However, from

$\displaystyle\delta([l_{n},l_{-n}]_{c})=[\delta(l_{n}),l_{-n}\otimes
1+1\otimes l_{-n}]_{c}-(n\leftrightarrow-n)$ (703)

and the previous results that $\delta$ is cohomolog to zero on $\mathfrak{W}$
we see that $\delta(c_{l})=\delta(c_{\bar{l}})=0$ concluding the proof.

## Classification of $r$-matrices

### Classification of Triangular r-matrices in $\mathfrak{o}(4)$

First, note that since
$\mathfrak{o}(4,\mathbb{C})=\mathfrak{sl}(2)\oplus\bar{\mathfrak{sl}}(2)$ and
$\bigwedge^{2}\mathfrak{o}(4,\mathbb{C})=\mathfrak{sl}(2)\wedge{\mathfrak{sl}}(2)\oplus\mathfrak{sl}(2)\wedge\bar{\mathfrak{sl}}(2)\oplus\bar{\mathfrak{sl}}(2)\wedge\bar{\mathfrak{sl}}(2)$
each r-matrix can be split according to

$\displaystyle r=a+\bar{a}+b,\quad
a\in\mathfrak{sl}(2)\wedge{\mathfrak{sl}}(2),\,\bar{a}\in\bar{\mathfrak{sl}}(2)\wedge\bar{\mathfrak{sl}}(2),\,b\in\mathfrak{sl}(2)\wedge\bar{\mathfrak{sl}}(2).$
(704)

Starting with a generic

$\displaystyle a=\alpha_{+}L_{1}\wedge L_{0}+\alpha_{0}L_{1}\wedge
L_{-1}+\alpha_{-}L_{-1}\wedge L_{0},$ (705)

triangularity $[[a,a]]=0$ enforces

$\displaystyle\alpha_{0}^{2}=-\alpha_{+}\alpha_{-}.$ (706)

Using the automorphism (233) with
$\gamma=\sqrt{-\frac{\alpha_{-}}{\alpha_{+}}},\epsilon=1$ in the case
$\alpha_{0}\neq 0$ and with $\epsilon=-1(+1)$ if $\alpha_{-}=0(\alpha_{+}=0)$
we find that there are two one-parameter r-matrices in
$\mathfrak{sl}(2)\wedge{\mathfrak{sl}}(2)$

$\displaystyle a_{1}=\alpha(L_{1}\wedge L_{0}+L_{1}\wedge L_{-1}-L_{-1}\wedge
L_{0}),$ (707) $\displaystyle a_{2}=\alpha L_{1}\wedge L_{0}$ (708)

and similar for $\bar{\mathfrak{sl}}(2)\wedge\bar{\mathfrak{sl}}(2)$

$\displaystyle\bar{a}_{1}=\bar{\alpha}(\bar{L}_{1}\wedge\bar{L}_{0}+\bar{L}_{1}\wedge\bar{L}_{-1}-\bar{L}_{-1}\wedge\bar{L}_{0}),$
(709) $\displaystyle\bar{a}_{2}=\bar{\alpha}\bar{L}_{1}\wedge\bar{L}_{0}.$
(710)

For r-matrices that only contain terms of type $b$ one has to demand
$[[b,b]]=0$ and the general result (before applying any automorphisms) as
obtained in [95] reads

$\displaystyle(\beta_{+}L_{1}+\beta_{0}L_{0}+\beta_{-}L_{-1})\wedge(\bar{\beta}_{+}\bar{L}_{1}+\bar{\beta}_{0}\bar{L}_{0}+\bar{\beta}_{-}\bar{L}_{-1}).$
(711)

Taking into account the automorphisms (233), (234) one can represent this as
eleven r-matrices with up to four parameters

$\displaystyle b_{1}=$ $\displaystyle(\beta L_{1}+\beta_{0}L_{0}+\beta
L_{-1})\wedge(\bar{\beta}\bar{L}_{1}+\bar{\beta}_{0}\bar{L}_{0}+\bar{\beta}\bar{L}_{-1}),$
(712) $\displaystyle b_{2}=$
$\displaystyle(L_{1}+L_{0})\wedge(\bar{\beta}\bar{L}_{1}+\bar{\beta}_{0}\bar{L}_{0}+\bar{\beta}\bar{L}_{-1}),$
(713) $\displaystyle b_{3}=$
$\displaystyle(L_{1}+L_{-1})\wedge(\bar{\beta}\bar{L}_{1}+\bar{\beta}_{0}\bar{L}_{0}+\bar{\beta}\bar{L}_{-1}),$
(714) $\displaystyle b_{4}=$
$\displaystyle\beta(L_{1}+L_{0})\wedge(\bar{L}_{1}+\bar{L}_{0}),\quad
b_{5}=\beta(L_{1}+L_{-1})\wedge(\bar{L}_{1}+\bar{L}_{-1}),$ (715)
$\displaystyle b_{6}=$
$\displaystyle\beta(L_{1}+L_{-1})\wedge(\bar{L}_{1}+\bar{L}_{0}),\quad
b_{7}=L_{1}\wedge(\bar{\beta}\bar{L}_{1}+\bar{\beta}_{0}\bar{L}_{0}+\bar{\beta}\bar{L}_{-1}),$
(716) $\displaystyle b_{8}=$ $\displaystyle
L_{1}\wedge\bar{\beta}(\bar{L}_{1}+\bar{L}_{0}),\quad
b_{9}=L_{1}\wedge\bar{\beta}(\bar{L}_{1}+\bar{L}_{-1}),$ (717) $\displaystyle
b_{10}=$ $\displaystyle L_{1}\wedge\bar{L}_{1},\quad
b_{11}=L_{1}\wedge\bar{L}_{0}.$ (718)

When combining $a$, $\bar{a}$ and $b$ terms there are two different cases,
$[[b,b]]=0$ and $[[b,b]]=-2[[b,a]]-2[[b,\bar{a}]]\neq 0$, that will be
analyzed separately. In the first case (for the moment considering only $a$
terms) one infers $[[b,a]]=0$ and $[[a,a]]=0$. With the general ansatz (705)
for $a$ and

$\displaystyle b=$
$\displaystyle\beta_{1}L_{1}\wedge\bar{L}_{1}+\beta_{2}L_{1}\wedge\bar{L}_{0}+\beta_{3}L_{1}\wedge\bar{L}_{-1}$
$\displaystyle+\beta_{4}L_{0}\wedge\bar{L}_{1}+\beta_{5}L_{0}\wedge\bar{L}_{0}+\beta_{6}L_{0}\wedge\bar{L}_{-1}$
$\displaystyle+\beta_{7}L_{-1}\wedge\bar{L}_{1}+\beta_{8}L_{-1}\wedge\bar{L}_{0}\beta_{9}L_{-1}\wedge\bar{L}_{-1}$
(719)

we extract the equations

$\displaystyle-2\beta_{1}\alpha_{0}+\beta_{4}\alpha_{1}$
$\displaystyle=0,\quad-\beta_{4}\alpha_{-1}-2\beta_{7}\alpha_{0}=0,$ (720)
$\displaystyle\beta_{1}\alpha_{-1}+\beta_{7}\alpha_{1}$
$\displaystyle=0,\quad-2\beta_{2}\alpha_{0}+\beta_{5}\alpha_{1}=0,$ (721)
$\displaystyle-\beta_{5}\alpha_{-1}-2\beta_{8}\alpha_{0}$
$\displaystyle=0,\quad\beta_{2}\alpha_{-1}+\beta_{8}\alpha_{1}=0,$ (722)
$\displaystyle-2\beta_{3}\alpha_{0}+\beta_{6}\alpha_{1}$
$\displaystyle=0,\quad-\beta_{6}\alpha_{-1}-2\beta_{9}\alpha_{0}=0,$ (723)
$\displaystyle\beta_{3}\alpha_{-1}+\beta_{9}\alpha_{1}$ $\displaystyle=0$
(724)

from $[[b,a]]=0$. For the coefficients of $a$ triangularity entails (706) and
for $b$ we additionally use the automorphisms to bring them in the form
(712)-(718). For $b=b_{1}$, implying

$\beta_{1}=\beta_{3}=\beta_{7}=\beta_{9},\quad\beta_{2}=\beta_{8},\quad\beta_{4}=\beta_{6},$

the equations (720)-(724) yield

$\alpha_{-1}=-\alpha_{1},\quad\beta_{4}=2\beta_{1},\quad\beta_{5}=2\beta_{2},$

resulting in

$\displaystyle
r=(L_{1}+L_{-1}+2L_{0})\wedge(\beta_{1}(\bar{L}_{1}+\bar{L}_{-1})+\beta_{2}\bar{L}_{0})+a_{1}.$
(725)

Similarly, for the other r-matrix components of type $b$ one has

$\displaystyle r\equiv b_{2}+a=$ $\displaystyle
L_{1}\wedge(\beta_{1}(\bar{L}_{1}+\bar{L}_{-1})+\beta_{2}\bar{L}_{0})+a_{2},$
(726) $\displaystyle r\equiv b_{3}+a=$ $\displaystyle
b_{3}+\alpha(L_{1}-L_{-1})\wedge L_{0},$ (727) $\displaystyle r\equiv
b_{4}+a=$ $\displaystyle\beta L_{1}\wedge(\bar{L}_{1}+\bar{L}_{0})+a_{2},$
(728) $\displaystyle r\equiv b_{4}+a=$
$\displaystyle\beta(L_{1}+L_{0})\wedge\bar{L}_{1}+a_{2},$ (729) $\displaystyle
r\equiv b_{5}+a=$ $\displaystyle b_{5}+\alpha(L_{1}-L_{-1})\wedge L_{0},$
(730) $\displaystyle r\equiv b_{6}+a=$ $\displaystyle b_{6}+a_{1},$ (731)
$\displaystyle r\equiv b_{7}+a=$ $\displaystyle L_{1}\wedge\bar{L}_{0}+a_{2},$
(732) $\displaystyle r\equiv b_{8}+a=$ $\displaystyle
L_{1}\wedge(\bar{L}_{1}+\bar{L}_{0})+a_{2},$ (733) $\displaystyle r\equiv
b_{9}+a=$ $\displaystyle L_{1}\wedge(\bar{L}_{1}+\bar{L}_{-1})+a_{2},$ (734)
$\displaystyle r\equiv b_{10}+a=$ $\displaystyle
L_{1}\wedge\bar{L}_{1}+a_{2}.$ (735)

To classify r-matrices of the form $b+\bar{a}$ one can use (234) and that the
coefficients of $\bar{b}$ with (719) are just the transposed coefficients (if
they are represented by a $3\times 3$ matrix) of $b$ and a global minus sign.
In the symmetric cases $b_{1},b_{4},b_{5},b_{10}$ the results are automorphic
to (726)-(731) with $\varphi^{\prime}$ and for the rest one has

$\displaystyle r=-\bar{b}_{2}+a=$
$\displaystyle\beta(L_{1}+L_{-1}+2L_{0})\wedge(\bar{L}_{1}+\bar{L}_{0})+a_{1},$
(736) $\displaystyle r=-\bar{b}_{4}+a=$
$\displaystyle\beta(L_{1}+L_{-1}+2L_{0})\wedge(\bar{L}_{1}+\bar{L}_{-1})+a_{1},$
(737) $\displaystyle r=-\bar{b}_{6}+a=$ $\displaystyle b_{6}+\alpha
L_{1}\wedge(L_{0}+2L_{-1}),$ (738) $\displaystyle r=-\bar{b}_{7}+a=$
$\displaystyle
L_{1}\wedge(\bar{L}_{1}+\bar{L}_{-1}+2\bar{L}_{0})+\bar{a}_{1}.$ (739)

While the $r$ in (727) and (738) are solutions of $[[b,a]]=0$, the $a$ part is
not triangular so they have to be discarded. Combining the previous results
(and explicitly calculating some ”overlaps” of the form $[[b,\bar{a}]]$), we
find for $b+a+\bar{a}$ the following possibilities

$\displaystyle r\equiv$ $\displaystyle
b_{1}+a+\bar{a}=(L_{1}+L_{-1}+2L_{0})\wedge(\bar{L}_{1}+\bar{L}_{1}+2\bar{L}_{0})+a_{1}+\bar{a}_{1},$
(740) $\displaystyle r\equiv$ $\displaystyle b_{2}+a+\bar{a}=\beta
L_{1}\wedge(\bar{L}_{1}+\bar{L}_{-1}+2\bar{L}_{0})+a_{2}+\bar{a}_{1},$ (741)
$\displaystyle r\equiv$ $\displaystyle
b_{4}+a+\bar{a}=L_{1}\wedge(\bar{L}_{1}+\bar{L}_{0})+a_{2}+\bar{a}_{2},$ (742)
$\displaystyle r\equiv$ $\displaystyle
b_{10}+a+\bar{a}=L_{1}\wedge\bar{L}_{1}+a_{2}+\bar{a}_{2}.$ (743)

In the case $[[b,b]]\neq 0$ we again make use of the results found in [95]. In
particular, the general solution for the equation

$\displaystyle 0\neq[[b,b]]=-2[[b,a]]-2[[b,\bar{a}]]$ (744)

up to $\text{Aut}(\mathfrak{o}(4,\mathbb{C}))$ has the form

$\displaystyle\alpha L_{1}\wedge
L_{-1}-\alpha\bar{L}_{1}\wedge\bar{L}_{-1}+b,\quad\alpha L_{1}\wedge
L_{0}+\alpha\bar{L}_{1}\wedge\bar{L}_{0}+b^{\prime},$ (745)

with specific $b,b^{\prime}$ that are not of interest for now. The first
r-matrix in (745) is quasitriangular with ad-invariant (in
$\mathfrak{o}(4,\mathbb{C})$) element containing
$\Omega=4\alpha^{2}L_{1}\wedge L_{0}\wedge L_{-1}+...$ and since the solutions
of (744) up to $\text{Aut}(\mathfrak{W}\oplus\mathfrak{W})$ are in the orbits
of $\mathfrak{o}(4,\mathbb{C})$ automorphisms $\varphi$ containing (745) we
would need

$\displaystyle\varphi(\Omega)=4\alpha^{2}\varphi(L_{1}\wedge L_{0}\wedge
L_{-1})+...=0$ (746)

to obtain a triangular solution. This, however, would entail that the matrix
of the coefficients of $\varphi$ has determinant zero but then it would not be
invertible and thus $\varphi$ no automorphism. Furthermore, there can be no
$\mathfrak{o}(4,\mathbb{C})$ automorphism that maps the $a$ terms of the
second solution of (745) to $a_{1}$ because $a_{1}$ can not be written in the
form
$(\alpha_{1}L_{1}+\alpha_{2}L_{0}+\alpha_{3}L_{-1})\wedge(\alpha^{\prime}_{1}L_{1}+\alpha^{\prime}_{2}L_{0}+\alpha^{\prime}_{3}L_{-1})$.
We conclude that only r-matrices of the form $b+a_{2}+\bar{a}_{2}$ have to be
considered. To this end we extract the equations

$\displaystyle\beta_{1}\beta_{5}-\beta_{2}\beta_{4}+\beta_{4}2\alpha$
$\displaystyle=0,$ $\displaystyle-\beta_{4}\beta_{8}+\beta_{7}\beta_{5}$
$\displaystyle=0,$ (747)
$\displaystyle\beta_{1}\beta_{8}-\beta_{2}\beta_{7}+\beta_{7}2\alpha$
$\displaystyle=0,$ $\displaystyle
2\beta_{1}\beta_{6}-2\beta_{3}\beta_{4}+\beta_{5}2\alpha$ $\displaystyle=0,$
(748) $\displaystyle-2\beta_{4}\beta_{9}+2\beta_{6}\beta_{7}$
$\displaystyle=0,$ $\displaystyle
2\beta_{1}\beta_{9}-2\beta_{3}\beta_{7}+\beta_{8}2\alpha$ $\displaystyle=0,$
(749) $\displaystyle-\beta_{3}\beta_{5}+\beta_{2}\beta_{6}+\beta_{6}2\alpha$
$\displaystyle=0,$ $\displaystyle-\beta_{5}\beta_{9}+\beta_{6}\beta_{8}$
$\displaystyle=0,$ (750)
$\displaystyle-\beta_{3}\beta_{8}+\beta_{2}\beta_{9}+\beta_{9}2\alpha$
$\displaystyle=0$ (751)

from (744). Additionally, we also get the same equations with
$\alpha\rightarrow-\bar{\alpha}$ and in the terms proportional to
$\bar{\alpha}$ the coefficients of $b$ are transposed. Solving these equations
yields only the solution

$\displaystyle\beta_{1}L_{1}\wedge\bar{L}_{1}+\beta_{2}(L_{1}\wedge\bar{L}_{0}+\bar{L}_{1}\wedge
L_{0})+\beta_{2}L_{1}\wedge L_{0}+\beta_{2}\bar{L}_{1}\wedge\bar{L}_{0},$
(752)

i.e. the same as in [95]. After removing duplicates, all the r-matrices we
found can be casted into the classes (235)-(242).

### Triangular r-matrices in $\mathfrak{P}_{4}$

In table 1 in [98] the classification of r-matrices in the four dimensional
Poincaré algebra is given. It is not quite complete, however, only certain
quasitriangular r-matrices are missing. After calculating the Schouten
brackets we obtain the following triangular part of this classification.

$\displaystyle r_{1}$ $\displaystyle=\bar{k}_{0}\wedge k_{0}+\alpha
S_{11}\wedge S_{00},\qquad r_{2}=\bar{k}_{1}\wedge k_{1}+\beta
S_{11}\wedge\bar{k}_{0},$ (753) $\displaystyle r_{3}$
$\displaystyle=\bar{k}_{1}\wedge k_{1}+\alpha S_{11}\wedge S_{01},\qquad
r_{4}=\alpha_{1}S_{11}\wedge S_{01}+\alpha_{2}S_{11}\wedge A_{01},$ (754)
$\displaystyle r_{5}$ $\displaystyle=k_{0}\wedge
k_{1}-\bar{k}_{0}\wedge\bar{k}_{1}+\gamma\bar{k}_{1}\wedge k_{1},\qquad
r_{6}=k_{0}\wedge k_{1},$ (755) $\displaystyle r_{7}$
$\displaystyle=S_{11}\wedge k_{0}+S_{01}\wedge
k_{1}+A_{01}\wedge\bar{k}_{1}+\beta S_{11}\wedge k_{1},$ (756) $\displaystyle
r_{8}$ $\displaystyle=S_{11}\wedge(k_{0}+k_{1})+\alpha S_{11}\wedge A_{01},$
(757) $\displaystyle r_{9}$
$\displaystyle=S_{01}\wedge\bar{k}_{1}+S_{11}\wedge
k_{1}+\alpha_{1}S_{00}\wedge S_{01}+\alpha_{2}S_{11}\wedge A_{01},$ (758)
$\displaystyle r_{10}$ $\displaystyle=A_{01}\wedge
k_{1}+\alpha_{1}S_{11}\wedge S_{01}+\alpha_{2}S_{00}\wedge A_{01},$ (759)
$\displaystyle r_{11}$ $\displaystyle=S_{11}\wedge k_{1}+S_{00}\wedge(\alpha
S_{11}+\alpha_{1}S_{01}+\alpha_{2}A_{01})+\tilde{\alpha}S_{11}\wedge A_{01},$
(760) $\displaystyle r_{12}$
$\displaystyle=(S_{11}+S_{00})\wedge\bar{k}_{0}+\alpha_{1}S_{11}\wedge
S_{00}+\alpha_{2}S_{01}\wedge A_{01},$ (761) $\displaystyle r_{13}$
$\displaystyle=(S_{11}-S_{00})\wedge\bar{k}_{0}+\alpha_{1}S_{11}\wedge
S_{00}+\alpha_{2}S_{01}\wedge A_{01},$ (762) $\displaystyle r_{14}$
$\displaystyle=S_{11}\wedge\bar{k}_{0}+\alpha_{1}S_{11}\wedge
S_{00}+\alpha_{2}S_{01}\wedge A_{01},$ (763) $\displaystyle r_{15}$
$\displaystyle=S_{01}\wedge k_{0}+\alpha_{1}S_{11}\wedge
S_{00}+\alpha_{2}S_{01}\wedge A_{01},$ (764) $\displaystyle r_{16}$
$\displaystyle=S_{11}\wedge k_{0}+\alpha_{1}S_{11}\wedge
S_{00}+\alpha_{2}S_{01}\wedge A_{01},$ (765) $\displaystyle r_{17}$
$\displaystyle=S_{11}\wedge(k_{0}+\beta\bar{k}_{0})+\alpha S_{01}\wedge
A_{01}.$ (766)

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