# Quantum Groups and Asymptotic Symmetries

Josua Unger

(2021)

Institute for Theoretical Physics November 2021 Thesis Institute for
Theoretical Physics [ ] Jerzy Kowalski-Glikman test Professor of Institute for
Theoretical Physics, University of Wroclaw

Abstract

Acknowledgments

## Introduction

After the dust had settled around the major scientific revolutions of the 20th
century, general relativity and quantum mechanics, these two are standing as
firm as ever. Predictions of general relativity, like the existence of black
holes or gravitational waves, that seemed preposterous when they were first
made, have been spectacularly confirmed. The standard model of particle
physics, an incarnation of a quantum field theory, has passed (almost) all
high-precision tests with flying colors. While attempts to reconcile these
frameworks, a notoriously difficult quest known as quantum gravity, certainly
led to new insights and theoretical progress, especially in String Theory and
Loop Quantum Gravity, distinct predictions that can be tested in current
experiments, remain elusive. A main reason for this is that the scale at which
both general relativity and quantum mechanics are basically guaranteed to be
relevant, the Planck scale (or Planck mass) $m_{\text{Pl}}\sim 10^{19}~{}GeV$,
is orders of magnitude larger than what is currently obtainable in high energy
experiments. This scale has its origin in a famous thought experiment; if we
want to probe ever shorter distances e.g. with photons, their wavelength has
to become shorter and thus the energy increases. When the Planck scale is
reached, with which we could achieve resolutions of about the Planck length
$l_{\text{Pl}}\sim 1/m_{\text{Pl}}$, the energy in the collision region is so
high that a black hole is formed. On the one hand, this means that both
general relativity and quantum mechanics are needed to describe the
experiment, but also that shorter distances than $l_{\text{Pl}}$ are
operationally meaningless.

Some theories and frameworks bundled under the notion of non-commutative
geometry can be seen as attempts to remedy parts of the problems mentioned
above. In particular, the $\kappa$-Poincaré quantum group and related
constructions feature an observer independent scale governing the breakdown of
spacetime as a smooth manifold as heralded by the thought experiment from the
previous paragraph. Its origin lays in the mathematical advances around Hopf
algebras by the mathematical physicists Drinfeld, Fadeev, Jimbo and Woronowicz
([1], [2], [3], [4]) which in turn were used by Lukierski, Ruegg, Majid and
collaborators in [46], [5], [32]. While we leave a more detailed introduction
to chapter Quantum Groups and Non-commutative Geometry, it should be noted
that instead of spacetime itself, its (deformed) symmetries play a fundamental
role in $\kappa$-Poincaré. This feature is also motivated by the fact that an
observer does not have direct access to spacetime coordinates and only
measures the (angular) momenta of incoming particles associated with the
spacetime symmetries via the Noether theorem. On its own, $\kappa$-Poincaré is
not a complete theory of quantum gravity but rather aims to describe effective
phenomena from such a (unspecified) theory. It is expected that in various
ways these phenomena already contribute at some intermediate (“mesoscopic”)
scale and thus provide some empirical clues to the nature of quantum gravity.
An important example, especially in the age of multi-messenger astronomy [6],
is the possibility of deformed in vacuo dispersion relations where effect
sizes can be enhanced by cosmological length scales. As non-commutative
geometry does not explicitly deal with dynamical aspects, a key part of its
study is to consistently formulate general relativity and quantum field theory
in terms of the non-commutative foundation. Considerable progress has been
made in the understanding of QFT on so-called Moyal non-commutative spaces
([7], [8], [9] and references therein) but, despite several valiant efforts
([10],[11], [12], [13], [14], [15], [16], [17], [18]), there was no matching
success for a QFT description in $\kappa$-Poincaré. In [42] a proposal for
expressing the concepts underlying general relativity in a $\kappa$-Poincaré
framework (or at least some variant of it) was brought forward.

Seemingly unrelated, in recent years there has been a renewed interest in
studying the asymptotic symmetries of spacetime (cf. chapter Asymptotic
Symmetries of Spacetime), in part due to new theoretical understanding but
also because of the prospect of testing phenomenological consequences like the
gravitational memory effect. Also in the endeavor to reformulate physics in
the “bulk” of spacetime in terms of a dual theory on the boundary, the so-
called holographic principle with its most prominent example, the AdS/CFT
correspondence [19], the asymptotic symmetries play a vital role. Furthermore,
as reviewed in [61], they are deeply connected to infrared phenomena like the
soft theorems by Weinberg [130].

A hallmark of the corresponding symmetry algebras is that they are infinite
dimensional but always contain the Poincaré algebra (or its analogon in
spacetimes with constant curvature). Thus a natural question appears which
will be at the heart of this thesis: How can we consistently formulate
deformed quantum groups like the $\kappa$-Poincaré if they are to be embedded
in the asymptotic symmetry algebras? And more generally, what kind of quantum
groups are possible on the infinite dimensional algebras? Armed with the
technical constraints coming from this analysis we can then investigate
potential phenomenological consequences, some of which are known from the
study of quantum groups, while others arise only in the setting of asymptotic
symmetries. So far, the particular combination of non-commutative geometry and
asymptotic symmetry algebras has not been studied from a physical point of
view, however, several contributions to the relevant mathematical
underpinnings have been made, e.g. by [92].

Even though our universe exhibits a four dimensional spacetime, it is
interesting to study the three dimensional case as well beyond the obvious
reason that many things are technically easier. On the gravity side this has
been particularly fruitful as the three dimensional general relativity has no
dynamical degrees of freedom and allows for quantization [84]. Similarly, the
study of three dimensional spacetime revealed connections between quantum
gravity and $\kappa$-Poincaré [20].

This thesis is structured as follows. In the second chapter, we start by
motivating some of the mathematical ideas behind quantum groups (Hopf
algebras) and develop its framework mainly following [21], [24]. We put
special emphasis on deformations that can be obtained from a so-called twist.
Then the mathematical structures are endowed with a physical interpretation
focusing on the $\kappa$-Poincaré. In chapter Asymptotic Symmetries of
Spacetime, the derivation of the symmetry algebra of asymptotically flat four
dimensional spacetime is reviewed and related to the algebra of the conserved
soft charges. In a similar way the more recent constructions of symmetries of
asymptotically (anti) de Sitter spacetimes are introduced in three and four
dimensions and in the former case we highlight the possibility of viewing the
asymptotically flat case as a contraction limit.

After these introductory parts, the content of which is mostly well-known,
chapter Quantum Groups on Asymptotic Symmetry Algebras features mainly
original research by the author based on the publications [62], [63] and [64].
In particular, we classify Lie bialgebras on symmetry algebras of
asymptotically flat and (A)dS spacetimes in three and four dimensions building
on recent work by [87], [95], [94] in the finite dimensional cases. This
rather technical part is supplemented by theorems about certain cohomology
groups for which we give elementary proofs in Appendix Proof of the Cohomology
Theorems. Some of the theorems are already known in the mathematical
literature [92], [89]. We then continue by explicitly constructing several
Hopf algebras in all orders of the deformation parameter and analyze them.
Thereby a special focus lies on the possibility of the so-called
specialization, a requirement that turns out to be rather restricting in the
infinite dimensional setting. In chapter Applications and Phenomenology, we
apply the previously obtained structures to non-commutative geometry and study
potential phenomenological consequences. While the latter draws from results
known in the $\kappa$-Poincaré literature, its application to asymptotic
symmetries constitutes novel research. Furthermore, we review a potential
solution to the black hole information loss paradox brought forward by
Strominger, Hawking and Perry in [115] and some critical responses to which we
in turn reply with a new proposal that could remedy the original argument.

Finally, in chapter Conclusion the results are summarized and put in the
context of the main goals of the thesis. Some open questions and further
research directions are highlighted.

## Quantum Groups and Non-commutative Geometry

In this chapter, the underlying ideas of quantum groups are introduced,
following [21], and the mathematical framework is developed [21],[24]. We
present the connection to non-commutative geometry, focussing on the so-called
$\kappa$-Poincaré formulation and highlight conceptual differences in various
approaches to non-commutativity.

### Quantum Groups

In classical mechanics the state of a system at time $t$ is described as a
point $m(t)$ on a Poisson manifold $M$. The evolution of functions $f$ of this
state, representing an observable, is governed by the equation

$\displaystyle\frac{df}{dt}(m(t))=\\{\mathcal{H},f\\}(m(t))$ (1)

where $\mathcal{H}$ is the classical Hamiltonian and $\\{,\\}$ denotes the
Poisson bracket defined on smooth functions on $M$ (cf. next section for more
details). For the coordinate functions, e.g. the canonical momentum $p$ and
position $q$ of a particle in one dimension it is given by

$\displaystyle\\{p,q\\}=1$

and (1) become Hamilton’s equation. In contrast, the description of a state in
quantum mechanics is given by a (complex) ray in a Hilbert space $V$ and
observables of the system are represented by linear operators $A$ on $V$.
Their time evolution is governed by the commutator

$\displaystyle\frac{dA}{dt}=[\mathcal{H},A]$

with the quantum Hamiltonian $\mathcal{H}$. Passing from the classical to the
quantum mechanical description is known as the problem of quantization. Of
course, there is no one to one correspondence between these descriptions but
one possibility, due to Moyal [22], is to reformulate the problem in terms of
finding a “deformed” product $\star_{h}$ on the algebra of functions of a
manifold so that the commutative product is obtained in the limit of vanishing
deformation parameter $h$ and the first order deformation satisfies

$\displaystyle\underset{h\rightarrow
0}{\text{lim}}\frac{f_{1}\star_{h}f_{2}-f_{2}\star_{h}f_{1}}{h}=\\{f_{1},f_{2}\\}.$
(2)

Equipped with such a product, the algebra of functions is related to non-
commutative geometry, as introduced by Connes [25], in the following way. A
theorem by Gelfand and Naimark [23] states that there is a one to one
correspondence of commutative $C^{*}$ algebras and compact Hausdorff spaces,
i.e. there is a duality between the category of certain spaces and algebras of
functions on that space. Extending this concept, one can view quantum spaces
as a dual category of associative but not commutative algebras (see also [26]
for a pedagogical introduction). These spaces are not necessarily manifolds
and to describe the symmetries of them we need to generalize the notion of a
group as well. Therefore it is helpfull to focus on functions on a group
instead of the group itself, much like in the treatment of quantum spaces. As
will be discussed in section Hopf Algebras, this introduces coproduct and
antipode as pullback from the group multiplication and inverse as well as a
counit and thus leads to the notion of a Hopf-algebra.

It should be noted that different authors use varying more or less restrictive
definitions of what a quantum group is. For example [24] takes the stance to
view every Hopf algebra with interesting applications as a quantum group. In
particular the deformations of enveloping Hopf algebras of Lie algebras (see
section Deformations of Universal Enveloping Algebras) serve as a guiding
example. Others have focussed on objects with extra structure that put more
emphasis on the origin as quantized algebras of coordinate functions as e.g.
the compact matrix pseudogroups of [4]. In this thesis we adopt the former
point of view and loosely use the terms quantum group and Hopf algebra
synonymously.

#### Lie Bialgebras and Poisson Lie Groups

The concept of a Lie bialgebra can on the one hand be seen as an infinitesimal
version of a Hopf algebra, on the other hand it arises also naturally in the
context of Poisson Lie groups, as we will show in the following to motivate
the definition [21].

A Poisson Lie group is a Lie group that is also a Poisson manifold. A Poisson
manifold in turn is a manifold $M$ with a bilinear map defined for smooth real
functions on $M$

$\\{,\\}:C^{\infty}(M)\times C^{\infty}(M)\rightarrow C^{\infty}(M),$

which is antisymmetric, satisfies the Jacobi identity

$\\{f_{1},\\{f_{2},f_{3}\\}\\}+\\{f_{3},\\{f_{1},f_{2}\\}\\}+\\{f_{2},\\{f_{3},f_{1}\\}\\}=0$

and is a derivation

$\\{f_{1}f_{2},f_{3}\\}=f_{1}\\{f_{2},f_{3}\\}+\\{f_{1},f_{3}\\}f_{2}.$

Additionally the group multiplication $\mu$ and the Poisson structure of the
Poisson Lie group have to be compatible in the sense that $\mu$ is a Poisson
map, i.e.

$\displaystyle\\{f_{1},f_{2}\\}(x_{0}y_{0})=\\{f_{1}(xy_{0}),f_{2}(xy_{0})\\}(x_{0})+\\{f_{1}(x_{0}y),f_{2}(x_{0}y)\\}(y_{0})$
(3)

Restricted to the maximal ideal $I$ of functions vanishing at the unit $e$ of
$M$, $I^{2}$, generated by functions of second order in local coordinates that
also vanish at $e$, is a Lie-ideal, i.e.

$\displaystyle\\{,\\}:I\otimes I^{2}\rightarrow I^{2}.$ (4)

Thus the bracket can be defined on the coset $I/I^{2}$ consisting of
equivalence classes $\\{[f]=f+g,f\in I,g\in I^{2}\\}$ and the result of taking
a bracket does not depend on the representative of the equivalence class. One
can show that $I/I^{2}$ is isomorphic to the cotangent space $T^{*}_{e}M$ [27]
which can be identified with the dual Lie algebra $\mathfrak{g}^{*}$ of $M$
where $\\{,\\}$ is therefore defining a Lie bracket. Its dual
$\\{,\\}^{*}\equiv\delta:\mathfrak{g}\rightarrow\mathfrak{g}\otimes\mathfrak{g}$
is called the cobracket which satisfies the Co-Jacobi identity

$\displaystyle\text{Cycl}((\delta\otimes\text{id})\delta(x))=0$ (5)

with $\text{Cycl}(a\otimes b\otimes c)=a\otimes b\otimes c+c\otimes a\otimes
b+b\otimes c\otimes a$. As the Jacobi identity for a Lie algebra encodes the
concept of associativity, the co-Jacoby identity is said to express the dual
coassociativity.

Duality here refers to the dual vector space $V^{*}$ which can be seen as the
vector space of linear functionals on $V$. The evaluation of these functionals
on elements of $V$ defines a nondegenerate bilinear form
$\braket{\cdot,\cdot}$ called the dual pairing.

Furthermore, (3) ensures a compatibility relation between the bracket and
cobracket. To see this, we note that expressed in terms of the Poisson
bivector $\Pi:M\rightarrow\bigwedge^{2}TM$, defined by

$\displaystyle\\{f,g\\}=(df\otimes dg)\Pi,$ (6)

(3) reads

$\displaystyle(df\otimes dg)\Pi(x_{0}y_{0})$ $\displaystyle=(d(f\circ
R_{y_{0}})\otimes d(g\circ R_{y_{0}}))\Pi(x_{0})+(d(f\circ L_{x_{0}})\otimes
d(g\circ L_{x_{0}}))\Pi(y_{0})$ $\displaystyle=(df\otimes
dg)(d_{x}(R_{y_{0}})\otimes d_{x}(R_{y_{0}})\Pi(x_{0})+d_{y}(R_{x_{0}})\otimes
d_{y}(R_{x_{0}})\Pi(y_{0})),$ (7)

where $R_{x},L_{x}:M\rightarrow M,y\mapsto yx,xy$ are the left and right
translation and the subscript on the exterior derivative $d_{x}$ means that
the resulting map, e.g. $d_{x}R$, starts from the tangent space at $x$. In
order to turn the bivector into a map to the exterior product of the Lie
algebra $\bigwedge^{2}\mathfrak{g}\simeq\bigwedge^{2}T_{e}M$ one can use the
right translation

$\displaystyle\tilde{\Pi}:M\rightarrow\bigwedge^{2}\mathfrak{g},\tilde{\Pi}(x)=d_{e}(R_{x^{-}1})\Pi(x).$
(8)

Applying this to (7) yields

$\displaystyle\tilde{\Pi}(x_{0}y_{0})=d_{e}(R_{y_{0}^{-1}x_{0}^{-1}}(d_{x}(R_{y_{0}})\otimes
d_{x}(R_{y_{0}})\Pi(x_{0})+d_{y}(R_{x_{0}})\otimes
d_{y}(R_{x_{0}})\Pi(y_{0}))$ $\displaystyle=d_{e}(R_{x_{0}^{-}1})\otimes
d_{e}(R_{x_{0}^{-}1})\Pi(x_{0})+d_{e}(R_{x_{0}^{-1}})d_{e}(L_{x_{0}})d_{y_{0}}(R_{y_{0}^{-1}})\otimes
d_{e}(R_{x_{0}^{-1}})d_{e}(L_{x_{0}})d_{y_{0}}(R_{y_{0}^{-1}})\Pi(y_{0})$
$\displaystyle=\tilde{\Pi}(x_{0})+\text{Ad}_{x_{0}}\otimes\text{Ad}_{x_{0}}\tilde{\Pi}(y_{0})$
(9)

where we introduced the adjoint action111Cf. section Modules, Comodules and
Module Algebras for more details.
$\text{Ad}:\mathfrak{g}\otimes\mathfrak{g}\rightarrow\mathfrak{g}$ that is the
derivative of the map $\Phi_{x}:h\mapsto xhx^{-1}=L_{x}(R_{x^{-}1}(h))$. Since
$M$ is also a Lie algebra we can write
$x_{0}=e^{ta},y_{0}=e^{tb},a,b\in\mathfrak{g}$ and subtracting (9) from the
same equation with $x_{0}\leftrightarrow y_{0}$ gives in second order of $t$

$\displaystyle d\tilde{\Pi}([a,b])=[a\otimes 1+1\otimes
a,d\tilde{\Pi}(b)]-[b\otimes 1+1\otimes b,d\tilde{\Pi}(a)]$ (10)

and we identify the differential of the right-translated bivector
$d\tilde{\Pi}$ with $\delta$. This compatibility relation between bracket and
cobracket is called cocycle condition since it has a cohomological
interpretation which will be explored in the next section.

To summarize, we showed that the Lie algebra $\mathfrak{g}$ of a Poisson Lie
group has the structure of a  Lie bialgebra (LBA), i.e. it is a Lie algebra
with a cobracket
$\delta:\mathfrak{g}\rightarrow\mathfrak{g}\otimes\mathfrak{g}$ satisfying (5)
and (10).

#### Coboundary and Triangular Lie Bialgebras

In the following, we will show that all cobrackets of Lie bialgebras are
cocycles of certain cohomology groups and introduce special coboundaries with
properties that turn out to play an important role in the construction of
quantum groups [28].

Starting with a Lie algebra $\mathfrak{g}$ and a $\mathfrak{g}$-module $V$
(cf. Modules, Comodules and Module Algebras) the vector spaces of cochains are
defined as $C^{n}=\text{Hom}(\Lambda^{n}\mathfrak{g},V)$, i.e. homomorphisms
from the $n$-th exterior product of $\mathfrak{g}$ into $V$. The coboundary
operators $\partial_{n}:C^{n}\rightarrow C^{n+1}$ are given by

$\displaystyle\partial_{n}(f)(x_{1}\wedge...\wedge x_{n}+1)=$
$\displaystyle\sum_{i=1}^{n+1}(-1)^{i}x_{i}\triangleright
f(x_{1}\wedge...\wedge\hat{x_{i}}...\wedge x_{n+1})$
$\displaystyle+\sum_{i<j}(-1)^{i+j}f([x_{i},x_{j}]\wedge
x_{1}...\wedge\hat{x_{i}}...\wedge\hat{x_{j}}...\wedge x_{n+1}),$ (11)

where $\hat{x_{i}}$ means that the $i$-th tensor leg is dropped and
$\triangleright$ denotes the left action on the module. One can check that
$\partial_{n}\partial_{n-1}=0$ so that the image $\text{Im }\partial_{n-1}$,
the space of the $n$-coboundaries, is contained in the Kernel $\text{Ker
}\partial_{n}$ whose elements are called the $n$-cocycles. The quotient
$\text{Ker }\partial_{n}/\text{Im }\partial_{n-1}$ is the $n$-th cohomology
group of $\mathfrak{g}$ with values in $V$, denoted by
$H^{n}(\mathfrak{g},V)$.

Considering the case $V=\bigwedge^{2}\mathfrak{g}$, we see that the condition
that $\delta:\mathfrak{g}\rightarrow\mathfrak{g}\wedge\mathfrak{g}$ is a
1-cocycle, i.e. $\partial_{1}(\delta)(x)=0$ coincides with (10) justifying the
name. The coboundaries can by definition be written in the form

$\displaystyle\delta_{r}(x)=\partial_{0}(r)(x)=[x\otimes 1+1\otimes x,r]$ (12)

for some $r\in\bigwedge^{2}\mathfrak{g}$ and the cocycle condition is
satisfied automatically because $\partial_{n}\partial_{n-1}=0$. We define a
coboundary Lie bialgebra to be the triple $(\mathfrak{g},[,],r)$ such that
$(\mathfrak{g},[,],\delta_{r})$ is a Lie bialgebra. By directly calculating
that

$\displaystyle\text{Cycl}((\delta_{r}\otimes\text{id}))\delta_{r}(x)+[x,[[r,r]]]=0,$
(13)

one can show that $r$ defines a coboundary Lie bialgebra if and only if

$\displaystyle[[r,r]]\equiv[r_{12},r_{13}]+[r_{12},r_{23}]+[r_{13},r_{23}]=\Omega,$
(14)

i.e. if its Schouten bracket
$[[,]]:\bigwedge^{2}\mathfrak{g}\rightarrow\bigwedge^{3}\mathfrak{g}$ is an
ad-invariant (cf. section Modules, Comodules and Module Algebras) element
$\Omega$. This condition is the modified classical Yang Baxter equation
(mCYBE) which can be seen as the infinitesimal version of the quantum YBE
introduced in section Quasitriangular Hopf Algebras and Quantum $R$-Matrices.
Coboundary Lie bialgebras with an r-matrix that satisfies the CYBE with a
vanishing rhs are called triangular, otherwise quasitriangular.

Consider two coboundary Lie bialgebras over $\mathfrak{g}$ with r-matrices
$r_{1},r_{2}=\tilde{\phi}(r_{1}))\equiv(\phi\otimes\phi)(r_{1})$ that are
related by an Lie algebra automorphism, i.e. a linear map
$\phi:\mathfrak{g}\rightarrow\mathfrak{g}$ such that

$\displaystyle[\phi(x_{i}),\phi(x_{j})]=f^{k}_{ij}\phi(x_{k}),$ (15)

where $\\{x_{i}\\}$ is a basis of $\mathfrak{g}$ and the structure constants
$f^{k}_{ij}$ are those of $(\mathfrak{g},[,])$. These two Lie bialgebras are
then equivalent in the sense that the structure constants $g^{ij}_{k}$ of the
cobracket $\delta_{r_{1}}$ (in other words
$\delta_{r_{1}}(x_{i})=g^{kj}_{i}x_{k}\wedge x_{j}$) are identical to those of
$\delta_{r_{2}}$ because we have

$\displaystyle\delta_{r_{2}}(\phi(x))=[\phi(x)\otimes
1+1\otimes\phi(x),r_{2}]=\tilde{\phi}(\delta_{r_{1}}(x)).$ (16)

Therefore we are interested in equivalence classes of coboundary Lie
bialgebras that are the orbits under Lie algebra automorphisms in the space of
solutions to the mCYBE.

An important result about Lie bialgebras on semi-simple Lie algebras is the
Whitehead Lemma [29]. A Lie algebra is semi-simple if it is the direct sum of
simple Lie algebras which have no proper non-trivial ideals, i.e. subalgebras
$\mathfrak{I}\subset\mathfrak{g}$ such that
$[\mathfrak{g},\mathfrak{I}]=\mathfrak{I}$. The Lemma states that all
cohomology groups $H^{n}(\mathfrak{g},V)$ of cocycles on a finite-dimensional
semi-simple Lie algebra $\mathfrak{g}$ with values in any finite-dimensional
module $V$ vanish. In particular, this means that all Lie bialgebras in a
finite-dimensional semi-simple Lie algebra are coboundary.

#### Real Forms of Complex Lie Algebras and $*$-Structures

In the context of symmetry algebras of space-times with cosmological constant
we will encounter algebras that can be written as different real forms of a
single complex Lie algebra which allows for a unified description.

A complex Lie algebra is simply a Lie algebra defined over the field of
complex numbers $\mathbb{C}$. Starting from a real algebra $W$ (defined over
$\mathbb{R}$) one can construct a complex algebra by setting
$W^{\mathbb{C}}=\mathbb{C}\otimes_{\mathbb{R}}W$ as vector spaces.
$W^{\mathbb{C}}$ is called the complexification of $W$. We call any real
algebra whose complexification is a complex algebra $A$ a real form of $A$.
There is a one to one correspondence between real forms and involutive
antiautomorphisms, i.e. bijective maps $*:A\rightarrow A$ such that

$\displaystyle(a^{*})^{*}=a,\quad(\alpha a)^{*}=\bar{\alpha}a^{*},\quad\forall
a\in A,\alpha\in\mathbb{C}.$ (17)

This means that for a given $*$-map one can find a basis of $A$ such that the
structure constants are real and $a^{*}=-a,\forall a\in A$.

A $*$-Lie bialgebra is a complex Lie bialgebra such that all the operations
are $*$-homomorphisms [21]. In particular for the bracket we have

$\displaystyle[a^{*},b^{*}]=(ba)^{*}-(ab)^{*}=-[a,b]^{*}$ (18)

and for the cobracket we demand

$\displaystyle\delta(a^{*})=(\delta(a))^{*\otimes*},\quad(a\otimes
b)^{*\otimes*}\equiv a^{*}\otimes b^{*}.$ (19)

For coboundary Lie bialgebras with r-matrix $r=r_{1}\wedge r_{2}$ this
condition implies

$\displaystyle[a^{*}\otimes 1+1\otimes a^{*},r]=$
$\displaystyle[a^{*},r_{1}]\wedge r_{2}+r_{1}\wedge[a^{*},r_{2}]$
$\displaystyle\overset{!}{=}$ $\displaystyle[a\otimes 1+1\otimes
a,r]^{*}=-[a^{*},r_{1}^{*}]\wedge r_{2}^{*}-r_{1}^{*}\wedge[a^{*},r_{2}^{*}]$
(20) $\displaystyle\Rightarrow r_{1}^{*}\wedge r_{2}^{*}=$ $\displaystyle-
r_{1}\wedge r_{2}$ (21)

and by linearity all r-matrices have to satisfy

$\displaystyle r^{*\otimes*}=-r,$ (22)

i.e. the reality condition constrains the consistent coboundary Lie bialgebras
and different real forms of a complex Lie algebra might have different sets of
allowed r-matrices.

##### Duality

The notion of a finite dimensional Lie bialgebra is self-dual in the sense of
the following proposition due to [28].

######  Proposition 1.

Let $(\mathfrak{g},[,],\delta)$ be a finite dimensional Lie bialgebra, and
$[,]^{*}:\mathfrak{g}^{*}\rightarrow\mathfrak{g}^{*}\otimes\mathfrak{g}^{*},\delta^{*}:\mathfrak{g}^{*}\otimes\mathfrak{g}^{*}\rightarrow\mathfrak{g}^{*}$
the respective dual maps of $[,],\delta$. Then
$(\mathfrak{g}^{*},[,]^{*},\delta^{*})$ is a Lie bialgebra.

As we will see in concrete examples later on, the condition that
$\mathfrak{g}$ is finite dimensional is indeed crucial.

To explicitly calculate the dual structures of a given Lie bialgebra one can
make use of the dual pairing
$\braket{\cdot,\cdot}:\mathfrak{g}^{*}\otimes\mathfrak{g}\rightarrow\mathbb{C}$
satisfying

$\displaystyle\braket{\delta^{*}(f,g),x}$ $\displaystyle=\braket{f\wedge
g,\delta(x)},$ (23) $\displaystyle\braket{[,]^{*}(f),x\wedge y}$
$\displaystyle=\braket{f,[x,y]}.$ (24)

#### Hopf Algebras

As motivated in the beginning of this chapter it is often useful to study the
(algebra of) functions $\mathcal{F}(G)$ on a group $G$ instead of the group
itself. The multiplication $\mu:G\times G\rightarrow G$ then induces a map
$\mu^{*}\equiv\Delta:\mathcal{F}(G)\rightarrow\mathcal{F}(G\times
G)\simeq\mathcal{F}(G)\otimes\mathcal{F}(G)$ by a pullback

$\displaystyle\Delta(f)(a,b)=f(\mu(a,b)),\quad f\in\mathcal{F}(G);a,b\in G$
(25)

and similarly from the inverse of the group $\iota:G\rightarrow G$ one obtains
$\iota^{*}:\mathcal{F}(G)\rightarrow\mathcal{F}(G)$ by

$\displaystyle\iota^{*}(f)(a)=f(\iota(a)).$ (26)

Evaluating a function at the unit $1_{G}$ of $G$ gives rise to the so-called
counit $\varepsilon:\mathcal{F}(G)\rightarrow\mathbb{C}$

$\displaystyle\varepsilon(f)=f(1_{G}).$ (27)

Thus, we motivated the following definition [24].
A Hopf algebra $(H,\mu,\eta,\Delta,\varepsilon,S)$ over a field $\mathbb{K}$
is a unital algebra $H$ with associative multiplication $\mu:H\otimes
H\rightarrow H$, i.e.

$\displaystyle\mu\circ(\text{id}\otimes\mu)=\mu\circ(\mu\otimes\text{id}),$
(28)

unit $\eta:\mathbb{K}\rightarrow H$

$\displaystyle\eta(\lambda)=\lambda 1_{H},\quad\lambda\in\mathbb{K},$ (29)

coassociative coproduct $\Delta:H\rightarrow H\otimes H$, i.e.

$\displaystyle(\Delta\otimes\text{id})\circ\Delta=(\text{id}\otimes\Delta)\circ\Delta,$
(30)

a counit $\varepsilon:H\rightarrow\mathbb{K}$ satisfying

$\displaystyle\varepsilon(1_{H})=1_{\mathbb{K}}$ (31)

and an antipode $S:H\rightarrow H$ which has to fulfill

$\displaystyle\mu\circ(S\otimes\text{id})\circ\Delta=\mu\circ(\text{id}\otimes
S)\circ\Delta=\eta\circ\varepsilon.$ (32)

Thus, $H$ is an algebra with $\mu$ and $\eta$ as well as a coalgebra with
$\Delta$ and $\varepsilon$. Furthermore, the coproduct and the counit have to
be algebra homomorphisms or, equivalently, the product and unit have to be
coalgebra homomorphisms, i.e.

$\displaystyle\Delta\circ\mu=\mu_{H\otimes
H}\circ(\Delta\otimes\Delta),\varepsilon\circ\mu=\mu_{\mathbb{K}}(\varepsilon\otimes\varepsilon),$
(33)

where $\mu_{H\otimes
H}=(\mu\otimes\mu)\circ(\text{id}\otimes\tau\otimes\text{id})$ with the flip
$\tau:H\otimes H\rightarrow H\otimes H,a\otimes b\mapsto b\otimes a$.

When writing down relations for the coproduct the Sweedler notation will be
useful. Instead of

$\displaystyle\Delta(a)=\sum_{i}a_{1i}\otimes a_{2i},\quad a,a_{1i},a_{2i}\in
H$ (34)

we write

$\displaystyle\Delta(a)=a_{(1)}\otimes a_{(2)},$ (35)

which implies the summation.

#### Modules, Comodules and Module Algebras

A representation of an algebra $A$ with multiplication $\mu$ on a vector space
$V$ is an algebra homomorphism $\varphi$ mapping from $A$ to the algebra of
linear operators $\mathcal{L}(V)$ on $V$ [24]. This entails linearity of
$\varphi$ as well as

$\displaystyle\varphi(ab)=\varphi(a)\varphi(b),\quad\varphi(1_{A})=1_{\mathcal{L}(V)},\quad
a,b\in A.$ (36)

The same concept is formulated with a linear map $\varphi:A\otimes
V\rightarrow V$,called a left action and $V$, called a left $A$-module,
satisfying

$\displaystyle\varphi\circ(\text{id}\otimes\varphi)=\varphi(\mu\otimes\text{id}),\quad\varphi\circ(\eta\otimes\text{id})=\text{id}.$
(37)

Instead of $\varphi(a\otimes v)$ we may also write $a\triangleright v$.
Similarly, a linear map $\varphi:V\otimes A\rightarrow V$ is a right action
and $V$ a right $A$-module if

$\displaystyle\varphi\circ(\text{id}\otimes\varphi)=\varphi(\text{id}\otimes\mu),\quad\varphi\circ(\text{id}\otimes\eta)=\text{id}$
(38)

and a $A$-bimodule $V$ is a left as well as a right module that are compatible
in the following sense

$\displaystyle(a\triangleright v)\triangleleft
b=a\triangleright(v\triangleleft b).$ (39)

Every associative algebra is a bimodule of itself with the algebra
multiplication as the so-called regular action.

A Hopf algebra $H$ with antipode $S$ canonically becomes a left $H$-module by
endowing it with the left adjoint action
$\text{ad}_{L}:H\rightarrow\mathcal{L}(H)$ defined as

$\displaystyle\text{ad}_{L}(a)b=a_{(1)}bS(a_{(2)}),\quad a,b\in H.$ (40)

In the case of a Lie algebra $\mathfrak{g}$ (which can be turned into a Hopf
algebra in a canonical way as we will see in section Deformations of Universal
Enveloping Algebras) the adjoint action is just given by the Lie bracket

$\displaystyle\text{ad}_{L}(a)b=[a,b],\quad a,b\in\mathfrak{g}.$ (41)

We denote the invariant (under the action) elements of a module by

$V_{A}\equiv\\{v\in V:a\triangleright v=0,\forall a\in A\\}.$

The dual notion to the module of an algebra is defined for a coalgebra $B$
with coproduct $\Delta$ and counit $\varepsilon$. A left $B$-comodule is a
vector space $V$ such that there exists a linear map, the left coaction,
$\varphi:V\rightarrow B\otimes V$ satisfying

$\displaystyle(\text{id}\otimes\varphi)\circ\varphi=(\Delta\otimes\text{id})\circ\varphi,\qquad(\varepsilon\otimes\text{id})\circ\varphi=\text{id}.$
(42)

Using the coproduct, $B$ is itself a left $B$-module, i.e.
$\Delta:B\rightarrow B\otimes B$ is a left coaction since (42) is then just
the coassociativity condition (30).

Importantly, if $A$ is a bialgebra one can construct the tensor product of
representations, i.e. a representation on the tensor product of $A$-modules.
Given two such (left) $A$-modules $V$ and $W$ with corresponding (left)
actions $\varphi_{V},\varphi_{W}$ $V\otimes W$ is a (left) $A$-module with
action

$\displaystyle\varphi_{V\otimes
W}(a)=\varphi_{V}(a_{(1)})\otimes\varphi_{W}(a_{(2)}),\quad a\in A.$ (43)

For the dual description of non-commutative spacetime the following concept of
module algebras is also of great importance. Consider a bialgebra $B$ with
counit $\varepsilon$. A left $B$-module $W$ which is also an algebra with
multiplication $\mu$ and unit $\eta$ is a $B$-module algebra if $\mu$ and
$\eta$ are $B$-module homomorphisms, i.e. if

$\displaystyle b\triangleright\mu(w,v)=\mu(b_{(1)}\triangleright
w,b_{(2)}\triangleright v),\quad b\triangleright 1_{W}=\varepsilon(b)1_{W}.$
(44)

#### Deformations of Universal Enveloping Algebras

An important class of Hopf algebras, especially for our purposes, are
deformations of universal enveloping algebras, which will be introduced in the
following [24].

First, we need the notion of a topological extension of a vector space in
order to allow for formal power series in a formal parameter $h$, playing the
role of a deformation parameter. Let $\mathbb{K}$ be a field with
characteristic $0$. Then the set of

$\displaystyle f=\sum_{n=0}^{\infty}a_{n}h^{n},\quad a_{n}\in\mathbb{K}$ (45)

with the addition and multiplication defined by

$\displaystyle f+g=\sum_{n=0}^{\infty}(a_{n}+b_{n})h^{n},\quad
fg=\sum_{n=0}^{\infty}\left(\sum_{r+s=n}a_{r}b_{s}\right)h^{n}$ (46)

is a ring called $\mathbb{K}[[h]]$. For a $\mathbb{K}$ vector space $V$ we
analogously define $V[[h]]$ as the set

$\displaystyle f=\sum_{n=0}^{\infty}a_{n}h^{n},\quad a_{n}\in V$ (47)

and multiplication and addition are given by (46), where the coefficients
$a_{n},b_{n}$ are elements of $V$. On a $\mathbb{K}[[h]]$ vector space $V$ one
can introduce the so-called $h$-adic topology [30], i.e. the topology in which
the sets $\\{h^{n}V+v|n\in\mathbb{N}_{0}\\}$ are a basis of the neighborhood
of $v\in V$. The completion of the tensor product
$V\otimes_{\mathbb{K}[[h]]}W$ of two $\mathbb{K}[[h]]$ vector spaces $V,W$ in
the $h$-adic topology is denoted by $V\hat{\otimes}W$. One can show that
$V[[h]]$ is complete in the $h$-adic topology and that
$V[[h]]\hat{\otimes}W[[h]]=(V\otimes W)[[h]]$.

Now we are ready to define the deformation of a Hopf algebra. For a Hopf
algebra $(H,\mu,\eta,\Delta,\varepsilon,S)$ a deformation is a Hopf algebra
$(H_{h},\mu_{h},\eta_{h},\Delta_{h},\varepsilon,S_{h})$ such that $H_{h}$ is
isomorphic to $H[[h]]$ and

$\displaystyle\mu_{h}=\mu\;\text{ mod }h,\quad\Delta_{h}=\Delta\;\text{ mod
}h.$ (48)

Two deformations $H_{h}$ and $H^{\prime}_{h}$ of $H$ are said to be equivalent
if there exists an isomorphism of Hopf algebras $f_{h}:H_{h}\rightarrow
H^{\prime}_{h}$ that is the identity $\text{mod }h$.

As stated above, of particular interest are deformations of the universal
enveloping algebra (so-called quantum universal envelope QUE)
$U(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$, i.e. the free algebra of
finite powers of elements of $\mathfrak{g}$ with Lie bracket $[,]$ subject
only to the relation $ab-ba-[a,b]=0$. $U(\mathfrak{g})$ is naturally a Hopf
algebra with the coproduct $\Delta$, antipode $S$ and counit $\varepsilon$
defined on the primitive elements by

$\displaystyle\Delta(x)=x\otimes 1+1\otimes x,\quad
S(x)=-x,\quad\varepsilon(x)=0,\quad\forall x\in\mathfrak{g}.$ (49)

We will call this Hopf algebra the undeformed Hopf algebra of $\mathfrak{g}$.
Note that $U(\mathfrak{g})$ also has a unit.

The obstructions and uniqueness of constructing a full (all orders of $h$)
deformation given a first order (i.e. $\text{mod }h$) deformation are
quantified by a cohomology derived from the Hochschild cohomology of algebras
and coalgebras (see [21] and references therein). For deformations of a
universal enveloping algebra of a semi-simple Lie algebra one finds e.g. that
$U(\mathfrak{g})[[h]]$ is isomorphic to $U(\mathfrak{g})$ as an algebra, i.e.
one can set $\mu_{h}=\mu$ for all deformations.

#### Quasitriangular Hopf Algebras and Quantum $R$-Matrices

In quantum mechanics the existence of indistinguishable particles is encoded
by using multi-particle states that are obtained from tensorproducts of one-
particle states by (anti)symmetrization. All permutations of a given
tensorproduct of a vector space $V$(the one-particle states) are related by
(multiple) flips $\tau:v\otimes w\mapsto w\otimes v$ and our goal is to find a
quantum analog of this operation that will allow for covariant statistics on
the multi-particle states (cf. [31]).

As described in section Modules, Comodules and Module Algebras, given a
representation $\varphi:H\otimes V\rightarrow V$ of a Hopf algebra $H$ on $V$,
the map in (43) with $\varphi_{V}=\varphi_{W}\equiv\varphi$ is a
representation $\tilde{\varphi}:H\otimes V\otimes V\rightarrow V\otimes V$.
For an analog of $\tau$ on this tensorproduct to be covariant we mean that it
is an intertwiner of this representation. In general, for two $H$ modules
$V_{1},V_{2}$ an intertwiner is an isomorphism $\phi:V_{1}\rightarrow V_{2}$
such that

$\displaystyle\phi(\varphi_{V_{1}}(x,v))=\varphi_{V_{2}}(x,\phi(v)).$ (50)

With $V_{1}=V_{2}\equiv V\otimes V$ and $\phi=\tau$ this becomes

$\displaystyle a_{(2)}\triangleright w\otimes a_{(1)}\triangleright
v=a_{(1)}\triangleright w\otimes a_{(2)}\triangleright
v\Leftrightarrow\Delta^{\text{op}}=\Delta.$ (51)

It turns out that a general solution to this problem can be formulated by
introducing quasitriangular Hopf algebras, i.e. Hopf algebras
$(H,\mu,\eta,\Delta,S,\varepsilon)$ with an invertible element $R\in H\otimes
H$ such that

$\displaystyle\Delta^{\text{op}}(x)=R\Delta(x)R^{-1},$ (52)
$\displaystyle(\Delta\otimes\text{id})R=R_{13}R_{23},\quad(\text{id}\otimes\Delta)R=R_{13}R_{12}.$
(53)

$R$ is called a universal quantum $R$-matrix and $H$ is called triangular if
$R_{21}=R^{-1}$.

Then, given a representation $T_{V}:H\rightarrow\mathcal{L}(V)$ the map
$\tilde{R}\equiv\tau\circ(T_{V}\otimes T_{V})(R)$ is an intertwiner of the
action $\tilde{T}_{V}:H\rightarrow\mathcal{L}(V\otimes V)$ induced by the
coproduct. Analogously the problem of finding such an intertwiner can be
formulated as finding representations of the Artin braid group, defined by the
relations among the generators $\sigma_{i},i\in\mathbb{N}$

$\displaystyle\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},\quad\text{for
}|i-j|>1$ (54)
$\displaystyle\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1},$
(55)

since $\tilde{R}$ fulfills the cubic relations

$\displaystyle\tilde{R}_{12}\tilde{R}_{23}\tilde{R}_{12}=\tilde{R}_{23}\tilde{R}_{12}\tilde{R}_{23}$
(56)

if $R$ is a universal $R$-matrix and the quadratic relations are satisfied
automatically. Furthermore, these cubic relations are equivalent to the so-
called quantum Yang-Baxter equation (QYBE) [24] for $R$

$\displaystyle R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}.$ (57)

The notation already hints at a connection with Lie bialgebras, the
infinitesimal version or classical limit of QUE. Indeed, if a Hopf algebra $H$
is a QUE on a Lie algebra $\mathfrak{g}$, this Lie algebra is a Lie bialgebra
with cobracket given by

$\displaystyle\delta(x)=\frac{\Delta(x)-\Delta^{\text{op}}}{h}|_{h=0}.$ (58)

If $H$ is quasitriangular, so is the corresponding LBA as the linearized
version of the QYBE for $R\sim 1+r$ implies the CYBE for $r$. Note that $R$ is
not necessarily anti-symmetric and a non anti-symmetric $r$ satisfying the
CYBE does not imply that it defines a triangular LBA but one can obtain an
equivalent one with $r^{\prime}=-r^{\prime}_{21}$ satisfying the mCYBE.
Furthermore, the LBA is triangular iff $H$ is triangular [21].

#### Specialization and Classical Limit of the Standard Deformation

In section Deformations of Universal Enveloping Algebras we encountered Hopf
algebras with an $h$-adic topology with a formal parameter $h$. While this is
a useful concept, it is ultimately problematic for the application to physical
models where we want to interpret $h$ as an observable energy scale e.g. the
Planck scale. The problem of finding a Hopf algebra (the so-called q-analog)
with the same (co)algebra structure where $h$ can be specialized to a complex
parameter $q$ is known as specialization [21], [24], [36]. Given a topological
Hopf algebra $H[[h]]$ the q-analog is a subset $\tilde{H}\subset H[[h]]$ over
the field $\mathbb{C}[q,q^{-1}]$ of finite monomials in $q,q^{-1}$.

An instructive example of such a q-analog can be constructed in the case of
the standard deformation that we will analyze in the following [24]. Consider
the simple Lie algebra $sl(2,\mathbb{C})$ spanned by $\\{H,E_{+},E_{-}\\}$
subject to the relations

$\displaystyle[H,E_{\pm}]=\pm 2E_{\pm},\quad[E_{+},E_{-}]=H.$ (59)

A deformation of the universal envelope of $sl(2,\mathbb{C})$, called
$U_{h}(sl(2,\mathbb{C}))$ is defined on the set $P$ of polynomials in the
generators endowed with the $h$-adic topology divided by the completion
$\hat{I}$ (in the $h$-adic topology) of the ideal $I$ generated by the
deformed algebra relations

$\displaystyle[H,E_{\pm}]=\pm
2E_{\pm},\quad[E_{+},E_{-}]=\frac{e^{hH}-e^{-hH}}{e^{h}-e^{-h}},$ (60)

i.e. $U_{h}(sl(2,\mathbb{C}))=P[[h]]/\hat{I}$. The coalgebra sector is given
by

$\displaystyle\Delta_{h}(E_{+})=E_{+}\otimes e^{hH}+1\otimes
E_{+},\quad\Delta_{h}(E_{-})=E_{-}\otimes 1+e^{-hH}\otimes E_{-},$ (61)
$\displaystyle\Delta_{h}(H)=H\otimes 1+1\otimes H,\quad
S_{h}(E_{+})=-E_{+}e^{-hH},$ (62) $\displaystyle
S_{h}(E_{-})=-e^{hH}E_{-},\quad
S_{h}(H)=-H,\quad\varepsilon_{h}(E_{\pm})=\varepsilon_{h}(H)=0.$ (63)

This construction can be generalized to any finite dimensional semi-simple Lie
algebra and is known as the standard or Drinfeld-Jimbo deformation [37], [38].
It is the first important tool in the construction of quantum groups.

When trying to naively interpret $h$ as a complex number and use the formulas
(60)-(63) verbatim for the q-analog one encounters two problems. First, the
coproducts and antipodes are infinite series in $H$ and also in $h$ and second
that (60) is not defined for $h=0$. Both of these can be remedied by passing
to a slightly different formulation of $U_{h}(sl(2,\mathbb{C}))$ which we will
call $\tilde{U}_{q}(sl(2,\mathbb{C}))$. Define $q=e^{h},K=e^{hH}$ and

$\displaystyle G=\frac{K-K^{-1}}{q-q^{-1}}.$ (64)

The relations in (60) (and an extra condition for $G$) for the new Hopf
algebra with generators $E_{\pm},K,K^{-1},G$ then read

$\displaystyle[G,E_{+}]=E_{+}(qK+q^{-1}K^{-1}),\quad[G,E_{-}]=-(qK+q^{-1}K^{-1})E_{-},$
(65) $\displaystyle[E_{+},E_{-}]=G,\quad(q-q^{-1})G=K-K^{-1},$ (66)

which contains finitely many terms but is not linear. Similarly (61)-(63) can
be rewritten and we have the additional relations

$\displaystyle\Delta(G)=G\otimes K+K^{-1}\otimes G,\quad\Delta(K)=K\otimes
K,\quad\Delta(K^{-1})=K^{-1}\otimes K^{-1},$ (67) $\displaystyle S(G)=-G,\quad
S(K)=K^{-1},\quad S(K^{-1})=K,$ (68)
$\displaystyle\varepsilon(G)=0,\quad\varepsilon(K)=\varepsilon(K^{-1})=1,$
(69)

which are also finite polynomials in the new generators and defined for all
values of $q$. With these relation $\tilde{U}_{q}(sl(2,\mathbb{C}))$ is a
suitable q-analog of $U_{h}(sl(2,\mathbb{C}))$ and one can now try to find a
classical limit. In the algebra sector for $q=1$ one finds that $K$ is in the
center and $K^{2}=1$. Indeed, $\tilde{U}_{1}(sl(2,\mathbb{C}))$ is just the
central extension of the universal envelope of $sl(2,\mathbb{C})$ by a central
$K$ with $K^{2}=1$.

For the coalgebra sector the classical limit means the corresponding LBA (cf.
previous section) and one easily obtains that it has a cobracket defined with
classical r-matrix

$\displaystyle r=E_{+}\wedge E_{-}.$ (70)

As the Schouten bracket yields the ad-invariant element

$\displaystyle[[r,r]]=4E_{+}\wedge E_{-}\wedge H,$ (71)

the LBA is quasitriangular.

#### Drinfeld Twist Deformation

In section Quasitriangular Hopf Algebras and Quantum $R$-Matrices we saw that
one can easily obtain a LBA from a quantum group and in the previous section
the first method of constructing a quantum group given a LBA was presented.
That method, however, only works for semi-simple Lie algebras. Another method
of obtaining new Hopf algebras that does not have this restriction is the so-
called Drinfeld twist presented in the following [39].

Let $(H,\mu,\eta,\Delta,\varepsilon,S)$ be a Hopf algebra and
$\mathcal{F}=f_{\alpha}\otimes f^{\alpha}$ an invertible element in $H\otimes
H$ satisfying the 2-cocycle condition

$\displaystyle\mathcal{F}_{12}(\Delta_{0}\otimes
1)(\mathcal{F})=\mathcal{F}_{23}(1\otimes\Delta_{0})(\mathcal{F})$ (72)

and the normalization

$\displaystyle\varepsilon(f_{\alpha})f^{\alpha}=1.$ (73)

Then $(H,\mu,\eta,\Delta_{\mathcal{F}},\varepsilon,S_{\mathcal{F}})$ with

$\displaystyle\Delta_{\mathcal{F}}(x)=\mathcal{F}\Delta(x)\mathcal{F}^{-1},\quad
S_{\mathcal{F}}(x)=vS(x)v^{-1},$ $\displaystyle v=\mu\circ(\text{id}\otimes
S)(\mathcal{F}),\quad
v^{-1}=\mu\circ(S\otimes\text{id})(\mathcal{F}^{-1}),\quad\forall x\in H$ (74)

is a Hopf algebra as well. In particular, $\Delta_{\mathcal{F}}$ is
coassociative

$\displaystyle(\Delta_{\mathcal{F}}\otimes\text{id})\circ\Delta_{\mathcal{F}}=$
$\displaystyle\left(\mathcal{F}_{12}(\Delta_{0}\otimes\text{id})\mathcal{F}^{-1}_{12}\right)\circ\left(\mathcal{F}\Delta_{0}\mathcal{F}^{-1}\right)$
$\displaystyle=$
$\displaystyle\mathcal{F}_{12}\left[(\Delta_{0}\otimes\text{id})(\mathcal{F})(\Delta_{0}\otimes\text{id})\circ\Delta_{0}(\Delta_{0}\otimes\text{id})\left(\mathcal{F}^{-1}\right)\right]\mathcal{F}^{-1}_{12}$
$\displaystyle=$
$\displaystyle\mathcal{F}_{23}\left[(\text{id}\otimes\Delta_{0})(\mathcal{F})(\text{id}\otimes\Delta_{0})\circ\Delta_{0}(\text{id}\otimes\Delta_{0})\left(\mathcal{F}^{-1}\right)\right]\mathcal{F}^{-1}_{23}$
$\displaystyle=$
$\displaystyle(\text{id}\otimes\Delta_{\mathcal{F}})\circ\Delta_{\mathcal{F}},$
(75)

where equation (72), the analog relation for the inverse twist element and the
coassociativity of $\Delta_{0}$ was used in the third line. If the original
Hopf algebra was quasitriangular with universal R-matrix $R$ so is the twisted
one with $R_{\mathcal{F}}=\mathcal{F}_{21}R\mathcal{F}^{-1}$.

Starting from the undeformed Hopf algebra of the universal envelope of some
Lie algebra, this construction can be used to construct deformed
quasitriangular QUE with $R=\mathcal{F}_{21}\mathcal{F}^{-1}$. We already know
that the linearized and anti-symmetrized $R$ yields a classical r-matrix $r$
and if it is given by a twist $r$ is even triangular. Conversely, one can show
that given a triangular r-matrix $r$ there is a twist such that the associated
R-matrix has $r$ as the classical limit [21]. A related, even more general
result from [40], [41] states that all first order deformations encoded in a
LBA structure can be extended to a full deformation. In particular, it is
stated that the category of Lie bialgebra over a field with h-adic topology
and the category of QUE are equivalent.

#### Star Product

We saw in (Modules, Comodules and Module Algebras) that the action of a Hopf
algebra $H$ on a product in a module algebra $A$ with product $\mu$ involves
the coproduct. If the coproduct is deformed one can ask how the product in the
module has to change so that the deformed action is compatible with it. In the
case of a twist deformation with twist $\mathcal{F}$ the answer can be
explicitly given in the form of the so-called star product
$\star:A_{\star}\otimes A_{\star}\rightarrow A_{\star}$ where $A=A_{\star}$ as
a vector space. It is defined by [42]

$\displaystyle a\star b=\mu\left(\mathcal{F}^{-1}\triangleright(a\otimes
b)\right)$ (76)

and the compatibility with the action

$\displaystyle h\triangleright(a\star
b)=\left(h_{(1)^{\mathcal{F}}}\triangleright
a\right)\star\left(h_{(2)^{\mathcal{F}}}\triangleright b\right),$ (77)

where the twisted coproduct is written in the Sweedler notation
$\Delta_{\mathcal{F}}(h)=h_{(1)^{\mathcal{F}}}\otimes h_{(2)^{\mathcal{F}}}$,
follows from the compatibility of the undeformed action with the product $\mu$

$\displaystyle
h\triangleright\left(\mu\left(\mathcal{F}^{-1}\triangleright(a\otimes
b)\right)\right)$
$\displaystyle=\mu\left(\Delta(h)\triangleright\left(\mathcal{F}^{-1}\triangleright(a\otimes
b)\right)\right)$
$\displaystyle=\mu\left(\mathcal{F}^{-1}\triangleright\left(\Delta^{\mathcal{F}}(h)\triangleright(a\otimes
b)\right)\right)=\left(h_{(1)^{\mathcal{F}}}\triangleright
a\right)\star\left(h_{(2)^{\mathcal{F}}}\triangleright b\right).$ (78)

Furthermore, the star product is associative.

Two examples of star products on modules will be especially important in this
work. The first is the Hopf algebra itself via the adjoint action which we
discuss in the following. Starting from a universal envelope $U\mathfrak{g}$
of a Lie algebra $\mathfrak{g}$ by a twist deformation one obtains a QUE
$Ug^{\mathcal{F}}$. Using the adjoint action a star product

$\displaystyle\star:U\mathfrak{g}\otimes U\mathfrak{g}\rightarrow
U\mathfrak{g},\quad h\star g=(\bar{f}^{\alpha}\triangleright
h)(\bar{f}_{\alpha}\triangleright g)$ (79)

can be defined on $U\mathfrak{g}_{\star}\equiv(U\mathfrak{g},\star)$. On the
generators of $\mathfrak{g}$ the star product can be used to define a braided
commutator

$\displaystyle[u,v]_{\star}=u\star v-\left(\bar{R}^{\alpha}\triangleright
v\right)\star\left(\bar{R}_{\alpha}\triangleright u\right),$ (80)

where
$R^{-1}=\mathcal{F}\mathcal{F}_{21}^{-1}\equiv\bar{R}^{\alpha}\otimes\bar{R}_{\alpha}$
is the inverse R-matrix. The braided commutator closes in $\mathfrak{g}$ and
satisfies braided antisymmetry and the braided Jacobi identity

$\displaystyle[u,v]_{\star}$
$\displaystyle=-[\bar{R}^{\alpha}(v),\bar{R}_{\alpha}(u)]_{\star},$ (81)
$\displaystyle[u,[v,t]_{\star}]_{\star}$
$\displaystyle=[[u,v]_{\star},t]_{\star}+[\bar{R}^{\alpha}(v),[\bar{R}_{\alpha}(u),t]_{\star}]_{\star},$
(82)

turning $\mathfrak{g}_{\star}\equiv(\mathfrak{g},[,]_{\star})$ into a deformed
Lie algebra. Furthermore, one can show that $\mathfrak{g}_{\star}$, sometimes
referred to as a quantum Lie algebra, generates $U\mathfrak{g}$ .

On $U\mathfrak{g}_{\star}$ one can naturally define a Hopf algebra structure
by

$\displaystyle\Delta_{\star}(u)=u\otimes
1+\bar{f}^{\alpha}\bar{R}^{\beta}S^{-1}(\bar{f}_{\alpha})\otimes\bar{R}_{\beta}(u),\quad
S^{-1}_{\star}(u)=-(\bar{R}^{\beta}\triangleright
u)\star(\bar{f}^{\alpha}\bar{R}_{\beta}S^{-1}(\bar{f}_{\alpha})),$ (83)

which is isomorphic to
$(U\mathfrak{g}^{\mathcal{F}},\mu,\Delta_{\mathcal{F}},S_{\mathcal{F}},\varepsilon)$
with the isomorphism

$\displaystyle D:U\mathfrak{g}_{\star}\rightarrow
U\mathfrak{g}^{\mathcal{F}},\quad D(u\star v)=D(u)D(v),$ (84) $\displaystyle
D^{-1}:a\mapsto D^{-1}(a)=(\bar{f}^{\alpha}\triangleright
a)\bar{f}_{\alpha}\equiv a^{\mathcal{F}}.$ (85)

It will be useful to express the Hopf algebra structures of
$U\mathfrak{g}_{\star}$ in terms of $a^{\mathcal{F}}$, i.e.

$\displaystyle[a^{\mathcal{F}},b^{\mathcal{F}}]_{\star}$
$\displaystyle=a^{\mathcal{F}}\star
b^{\mathcal{F}}-(\bar{R}^{\alpha}\triangleright
b)^{\mathcal{F}}\star(\bar{R}_{\alpha}\triangleright a)^{\mathcal{F}},$ (86)
$\displaystyle\Delta_{\mathcal{F}}(a^{\mathcal{F}})$
$\displaystyle=a^{\mathcal{F}}\otimes
1+\bar{R}^{\alpha}\otimes(\bar{R}_{\alpha}\triangleright a)^{\mathcal{F}}.$
(87)

The deformed bracket can also be expressed as [43]

$\displaystyle[a^{\mathcal{F}},b^{\mathcal{F}}]_{\star}=a^{\mathcal{F}}_{(1^{\mathcal{F}})}b^{\mathcal{F}}S_{\mathcal{F}}(a^{\mathcal{F}}_{(2^{\mathcal{F}})}).$
(88)

An important property of the deformed coproduct (83) is

$\displaystyle\Delta_{\star}(\mathfrak{g}_{\star})\subset\mathfrak{g}_{\star}\otimes
1+U\mathfrak{g}_{\star}\otimes\mathfrak{g}_{\star},$ (89)

which does not hold in general for e.g. $\Delta_{\mathcal{F}}$. This property
allows for the construction of a deformed differential calculus in the sense
of Woronowicz [44] (see also [45] for an explicit construction for
$\kappa$-Minkowski)which in [42] was used to define a theory of gravity in a
non-commutative setting. The deformed infinitesimal diffeomorphisms under
which the theory is invariant are described by $U\mathfrak{g}_{\star}$ giving
an argument that this form is preferred over other, isomorphic forms like
$U\mathfrak{g}_{\mathcal{F}}$. In particular, it is possible to define a
deformed quadratic wave operator on the algebra of the infinitesimal
diffeomorphisms (vector fields) that can be expressed as [43]

$\displaystyle\Box^{\mathcal{F}}\equiv D(\Box),$ (90)

where $\Box$ is the undeformed wave operator.

The second example for making use of the star product is directly related to
non-commutative geometry and will be discussed in the next section.

###  $\kappa$-Poincaré and Non-commutative Geometry

In this section, the $\kappa$-Poincaré quantum group is introduced and used to
illustrate the relation of non-trivial coalgebra structure and non-commutative
geometry (see e.g. [47]).

In the seminal paper by Lukierski et.al. [46] the $\kappa$-Poincaré quantum
group was first introduced. They employed the Drinfeld-Jimbo deformation
discussed in section Specialization and Classical Limit of the Standard
Deformation for the symmetry algebra of anti de Sitter (AdS) spacetime (cf.
chapter Asymptotic Symmetries of Spacetime). That symmetry algebra is the
semi-simple $so(3,2)$ which is a real form of $\mathfrak{o}(5,\mathbb{C})$.
When taking the contraction limit (which will also be discussed in greater
detail in chapter Asymptotic Symmetries of Spacetime and Quantum Groups on
Asymptotic Symmetry Algebras) of $so(3,2)$, the Poincaré algebra in four
dimensions

$\displaystyle[M_{\mu\nu},M_{\rho\lambda}]=i(\eta_{\mu\lambda}M_{\nu\rho}-\eta_{\nu\lambda}M_{\mu\rho}+\eta{\nu\rho}M_{\mu\lambda}-\eta_{\mu\rho}M_{\nu\lambda}),$
(91)
$\displaystyle[M_{\mu\nu},P_{\rho}]=i(\eta_{\nu\rho}P_{\mu}-\eta_{\mu\rho}P_{\nu}),\quad[P_{\mu},P_{\nu}]=0,$
(92)

with the Minkowski metric $\eta_{\mu\nu}=\text{diag}(1,-1,-1,-1)$ is obtained
on the algebraic level.

In the bicrossproduct formulation ([32]) the algebra and coalgebra relations
of the $\kappa$-Poincaré are both deformed, i.e. one has

$\displaystyle[M_{0i},k_{j}]=i\delta_{ij}\left(\frac{\kappa}{2}\left(1-e^{-2k_{0}/\kappa}\right)+\frac{{\bf{k}}^{2}}{2\kappa}\right)-i\frac{1}{\kappa}k_{i}k_{j},$
(93)

for the algebra sector while the rest of the relations are undeformed. The
coproducts take the form

$\displaystyle\Delta(k_{i})=k_{1}\otimes 1+e^{k_{0}/\kappa}\otimes
k_{i},\quad\Delta(k_{0})=k_{0}\otimes 1+1\otimes k_{0}.$ (94)

However, one can choose a basis (the so-called classical basis, also discussed
in [33], [34], [35])

$\displaystyle P_{0}(k)=$
$\displaystyle\kappa\sinh\left(\frac{k_{0}}{\kappa}\right)+\frac{{\bf{k}}^{2}}{2\kappa}e^{\frac{k_{0}}{\kappa}},$
(95) $\displaystyle P_{i}(k)=$ $\displaystyle k_{i}e^{\frac{k_{0}}{\kappa}},$
(96)

so that the algebra relations are undeformed and the coalgebra sector can be
expressed as [36]

$\displaystyle\Delta_{\tau}(P_{\mu})=$ $\displaystyle
P_{\mu}\otimes\Pi_{\tau}+1\otimes
P_{\mu}-\frac{\tau_{\mu}}{\kappa}P^{\rho}\Pi_{\tau}^{-1}\otimes
P_{\rho}-\frac{\tau_{\mu}}{2\kappa^{2}}C_{\tau}\Pi_{\tau}^{-1},$ (97)
$\displaystyle\Delta_{\tau}(M_{\mu\nu})=$ $\displaystyle M_{\mu\nu}\otimes
1+1\otimes
M_{\mu\nu}+\frac{1}{\kappa}P^{\rho}\Pi_{\tau}^{-1}\otimes(\tau_{\nu}M_{\rho\mu}-\tau_{\mu}M_{\rho\nu})$
$\displaystyle-\frac{1}{2\kappa^{2}}C_{\tau}\Pi_{\tau}^{-1}\otimes(\tau_{\mu}\tau^{\rho}M_{\rho\nu}-\tau_{\nu}\tau^{\rho}M_{\rho\mu}),$
(98)

where the symbols

$\displaystyle\Pi_{\tau}$
$\displaystyle=\frac{1}{\kappa}\tau^{\mu}P_{\mu}+\sqrt{1+\frac{\tau^{2}}{\kappa^{2}}C},$
(99) $\displaystyle C_{\tau}$
$\displaystyle=\frac{2\kappa^{2}}{\tau^{2}}\left(\sqrt{1+\frac{\tau^{2}}{\kappa^{2}}C}-1\right),\quad
C=P^{\mu}P_{\mu},$ (100)

have been used222The quadratic element $C_{\tau}$ is argued to play the role
of a deformed Casimir operator in [55] because it satisfies
$[C_{\tau},x_{\mu}]=2P_{\mu}$ when acting on the dual non-commutative
coordinates (cf. below). In chapter Quantum Groups on Asymptotic Symmetry
Algebras we will employ a different construction of deformed Casimir operators
based on the generators discussed in section Star Product. and $C_{\tau}=C$
for $\tau=0$. The original version of Lukierski et.al. corresponds to the
choice $\tau_{\mu}=(1,0,0,0)\rightarrow\tau^{2}=1$ which is also called the
time-like $\kappa$-Poincaré but other choices are possible, i.e. the spacelike
$\tau^{2}=-1$ and light-like (or light-cone) $\tau^{2}=0$ versions. The light-
like form is special since it is triangular and can thus be obtained from a
twist.

#### Non-commutative Geometry

From ordinary quantum mechanics we know that the operators for position and
momenta are dual to each other in the sense that e.g. the momentum operator
can be represented as $P_{\mu}\equiv-i\partial_{\mu}$ and thus acts on the
coordinates via

$\displaystyle\braket{P_{\mu},x_{\nu}}=P_{\mu}\triangleright
x_{\nu}=-i\eta_{\mu\nu},$ (101)

leading to the Heisenberg relation

$\displaystyle[P_{\mu},x_{\nu}]\equiv\braket{P_{\mu},x_{\nu}}=-i\eta_{\mu\nu}.$
(102)

This canonical action on the dual Minkowski algebra can be understood as dual
to the regular action, i.e. the first equality in (101) follows from

$\displaystyle\braket{P_{\nu}\otimes P_{\rho},\Delta(x_{\mu})}$
$\displaystyle=\braket{P_{\nu}P_{\rho},x_{\mu}}=\braket{P_{\nu},P_{\rho}\triangleright^{*}x_{\mu}},$
(103) $\displaystyle P_{\rho}\triangleright^{*}x_{\mu}$
$\displaystyle=\braket{x_{\mu(1)},P_{\nu}}x_{\mu(2)},$ (104)

if we endow the dual algebra with a primitive coproduct $\Delta(x)=x\otimes
1+1\otimes x$.

Now, as we saw in section Modules, Comodules and Module Algebras, if the
algebra of momenta is a Hopf algebra one can look for a representation on the
algebra of the form

$\displaystyle P_{\mu}\triangleright(x^{\nu}\cdot
x^{\rho})=(P_{\mu(1)}\triangleright x^{\nu})\cdot(P_{\mu(2)}\triangleright
x^{\rho}).$ (105)

When comparing to (77) it becomes apparent that the product $\cdot$ that
satisfies this module algebra relation for a deformed coproduct is the star
product which allows us to calculate what we will interpret as deformed
commutation relations for Minkowski space. For example, with the coproduct
from (94) one finds e.g.

$\displaystyle k_{j}\triangleright[x^{0},x^{i}]=(k_{j(1)}\triangleright
x^{0})(k_{j(2)}\triangleright x^{i})-(k_{j(1)}\triangleright
x^{i})(k_{j(2)}\triangleright x^{0})=\frac{1}{\kappa}\delta{ij},$ (106)

indicating that

$\displaystyle[x^{0},x^{i}]=\frac{i}{\kappa}x^{i}.$ (107)

This relation (with the rest of the commutators vanishing) constitutes the
time-like $\kappa$-Minkowski algebra and the other variations can be included
in the description again with the help of $\tau_{\mu}$

$\displaystyle[x^{\mu},x^{\nu}]=\frac{i}{\kappa}\left(\tau^{\mu}x^{\nu}-\tau^{\nu}x^{\mu}\right).$
(108)

Besides commutation relations of Lie algebra type that we just encountered
there is also the so-called canonical or Moyal-Weyl non-commutative spacetime
defined by [22]

$\displaystyle[x^{\mu},x^{\nu}]=\Theta^{\mu\nu},$ (109)

which will be discussed in the next section. Note that in this situation the
direct formulation in terms of the star product (cf. below) is more suitable
because the $k_{j}$ acting on a constant would just be zero.

Another point of view to describe the relation between non-commutativity and
quantum groups is to start from the commutation relations (107) which form
what is known in the mathematical literature as the $an(3)$ algebra. A five
dimensional representation of $an(3)$ is given by the matrices [47]

$\displaystyle x^{0}=-\frac{i}{\kappa}\begin{pmatrix}0&{\bf{0}}&1\\\
{\bf{0}}&{\bf{0}}&{\bf{0}}\\\ 1&{\bf{0}}&0\end{pmatrix},\quad
x^{i}=\frac{i}{\kappa}\begin{pmatrix}0&{\bf{\epsilon}}_{i}^{T}&0\\\
{\bf{\epsilon}}_{i}&{\bf{0}}&{\bf{\epsilon}}_{i}\\\
0&-{\bf{\epsilon}}_{i}^{T}&0\end{pmatrix},$ (110)

where ${\bf{\epsilon}}_{i}$ is the unit vector with a single entry at the
$i$’th position. As usual, the associated Lie group ($AN(3)$) is obtained by
exponentiation and thus a generic group element can be written as

$\displaystyle\hat{e}_{k}=e^{ik_{i}x^{i}}e^{ik_{0}x^{0}},$ (111)

parametrized by the functions $k_{\mu}$ on the group manifold, which becomes

$\displaystyle\hat{e}_{k}=\frac{1}{\kappa}\begin{pmatrix}\bar{P}_{4}&-{\bf{P}}e^{-k_{0}/\kappa}&P_{0}\\\
-{\bf{P}}&\kappa 1&-{\bf{P}}\\\
\bar{P}_{0}&{\bf{P}}e^{-k_{0}/\kappa}&P_{4}\end{pmatrix}$ (112)

in terms of the matrix representation. The momenta $P_{\mu}(k)$ are given by
(95) and

$\displaystyle\bar{P}_{0}(k)=$
$\displaystyle\kappa\sinh\left(\frac{k_{0}}{\kappa}\right)-\frac{{\bf{k}}^{2}}{2\kappa}e^{\frac{k_{0}}{\kappa}},\quad\bar{P}_{4}(k)=\kappa\cosh\left(\frac{k_{0}}{\kappa}\right)+\frac{{\bf{k}}^{2}}{2\kappa}e^{\frac{k_{0}}{\kappa}},$
(113) $\displaystyle P_{4}(k)=$
$\displaystyle\kappa\cosh\left(\frac{k_{0}}{\kappa}\right)-\frac{{\bf{k}}^{2}}{2\kappa}e^{\frac{k_{0}}{\kappa}}.$
(114)

The vector $(0,{\bf{0}},\kappa)\in\mathbb{R}^{5}$, the origin of the momentum
space, has norm $\kappa^{2}$ and acting on it with $\hat{e}_{k}$ gives

$\displaystyle\hat{e}_{k}\begin{pmatrix}0\\\ {\bf{0}}\\\
\kappa\end{pmatrix}=\begin{pmatrix}P_{0}\\\ -{\bf{P}}\\\ P_{4}\end{pmatrix}$
(115)

so that the invariance of the norm implies

$\displaystyle P_{4}^{2}-P_{0}^{2}+{\bf{P}}^{2}=\kappa^{2},\quad
P_{0}+P_{4}=e^{k_{0}/\kappa}>0.$ (116)

But this just means that the momentum space is a submanifold of the four
dimensional de Sitter space with curvature radius $1/\kappa$.

From the group perspective the deformed coproduct acquires the interpretation
of a non-standard addition ($\hat{+}$) of momenta in the group multiplication,
i.e.

$\displaystyle\hat{e}_{k\hat{+}l}\equiv\hat{e}_{k}\hat{e}_{l}=e^{i(k_{i}+e^{-k_{0}/\kappa}l_{i})x^{i}}e^{i(k_{0}+l_{0})x^{0}},$
(117)

with the help of the Baker Campbell Hausdorff formula.

A connection between the non-commutative algebra as in (108) and a star
product between commutative coordinate functions $\text{x}^{\mu}$ is given by
a Weyl map (see e.g. [48]) $\Omega$

$\displaystyle\Omega\left(e^{k_{\mu}\text{x}^{\mu}}\right)$
$\displaystyle\equiv\hat{e}_{k}=e^{ik_{i}x^{i}}e^{ik_{0}x^{0}},$ (118)
$\displaystyle(f\star g)$ $\displaystyle=\Omega^{-1}(\Omega(f)\Omega(g)),$
(119)

for coordinate functions $f,g$ of $\text{x}^{\mu}$. Then the deformed
commutator is written as

$\displaystyle[\text{x}^{\mu},\text{x}^{\nu}]\equiv\text{x}^{\mu}\star\text{x}^{\nu}-\text{x}^{\nu}\star\text{x}^{\mu}.$
(120)

Note that while for the star product we also have the unique prescription from
the previous section, the Weyl map is in general ambiguous, i.e. it depends on
ordering ambiguities. For a further star product construction related to
$\kappa$-Minkowski see also [52].

#### Broken and Deformed Symmetries

Now that we saw how the formalism of Hopf algebras relates to spacetime non-
commutativity let us explore two different types of physical motivation of
this phenomenon [110].

An instructive analogy for this distinction is the treatment of the speed of
light in Galilean relativity, where the laws of motion (Newton’s law) hold in
all frames related by a galilean transformation. Such transformations relate
frames that differ by constant relative motion and velocities simply add as
vectors. When Maxwell introduced his equations for electromagnetism the speed
of light $c$ was introduced and two different ways of incorporating it into
Galilean relativity are conceivable. First, one could treat $c$ just as a
ordinary velocity that transforms like a vector. Such a view breaks the
relativistic symmetry as the Maxwell equation would have a different form in
distinct inertial frames, singling out a frame where $c$ has the value that
one measures in our universe (ether). Alternatively, Einstein postulated that
$c$ has the same value in all inertial frames. This deforms the Galilean into
special relativity, e.g. the addition of velocities is no longer vectorial but
no frame is singled out.

When spacetime non-commutativity of the form [49]

$\displaystyle[x^{\mu},x^{\nu}]=\frac{1}{\kappa^{2}}\Theta^{\mu\nu}(\kappa x)$
(121)

is introduced, where the rhs is expanded as [51]

$\displaystyle\Theta^{\mu\nu}(\kappa
x)=\Theta^{\mu\nu}_{(0)}+\kappa\Theta^{\mu\nu}_{(1)\rho}x^{\rho}+\kappa^{2}\Theta^{\mu\nu}_{(2)\rho\tau}x^{\rho}x^{\tau}+...,$
(122)

one similarly has the choice how to treat the energy scale $\kappa$ or the
coefficients $\Theta$. One possibility is to assume that $\Theta^{\mu\nu}$
transforms as an ordinary tensor. This choice can be motivated by a simple
system of quantum mechanical particles in a strong magnetic field as follows
[54]. Consider the Lagrangian of a particle in the $x,y$ plane with mass $m$,
charge $q$ and velocity $v$ in the magnetic field $B$

$\displaystyle L=\frac{mv^{2}}{2}+qBxv_{y}.$ (123)

The canonical momenta are simply given by

$\displaystyle p_{x}=\frac{\partial L}{\partial\dot{x}}=mv_{x},\quad
p_{y}=mv_{y}+qBx$ (124)

and the canonical commutation relations $[p_{y},y]=i$ yield for large $B$

$\displaystyle\frac{i}{qB}=\frac{1}{qB}[p_{y},y]=\frac{1}{qB}[x,mv_{y}+qBx]\simeq[y,x].$
(125)

The physical interpretation of this emerging effective non-commutativity is
that even though the model allows for states in which both $x$ and $y$ can be
measured sharply simultaneously these states correspond to high energy if $B$
is very large and thus a low-energy observer can not excite them. An analogous
mechanism was discovered in string theory where the Kalb-Ramond field $B^{ij}$
(which generalizes the electromagnetic potential) plays the role of the
magnetic field [53] [54]. This low-energy effective non-commutativity was
mostly studied in the case of constant $B^{ij}$ corresponding to the zeroth
order in (122) and is also known as canonical or Moyal non-commutativity [22].

Another possibility consists in postulating that the scale $\kappa$, often
identified with the Planck mass $m_{\text{Pl}}$, is observer independent. Such
a scenario is called doubly special relativity [50] because of the status as a
fundamental constant of nature both of the speed of light and $\kappa$ and it
has been mostly studied in the context of $\kappa$-Poincaré [47],[117]
introduced above (the first order in (122)) but is not limited to it [57].
Usually the interpretation is that the momentum space described by a Hopf
algebra is more fundamental in the sense that it is immediately accessible by
experiments whereas the spacetime (module algebra) is emergent and properties
like locality are generally relative for different observers. Thus, the
classical notion of a local continuous spacetime is recovered only in a limit.

In this thesis we will adopt the latter point of view and focus mainly on the
Lie algebra type non-commutativity but also the zeroth order (which we will
also call canonical non-commutativity and second order (called quadratic non-
commutativity, cf. [51]) terms in (122) can be consistently described in this
framework.

Let us close with a general remark on the motivation for deformation. If a new
physical theory is discovered the old ones are not fully abandoned, they
usually remain valid as an approximation, i.e. they can be seen as limits of
the new theory. The parameter(s) controlling the limit are often new constants
of nature. Deformations, as we met them in this chapter, are in some sense the
attempt to reverse this procedure, to look for new physics in the “vicinity”
of established theories. Also special relativity and quantum mechanics are, in
hindsight, motivating examples for this strategy.

## Asymptotic Symmetries of Spacetime

In the following chapter, we review the symmetries of spacetimes with certain
asymptotic behavior in the framework of general relativity. We mainly focus on
the algebraic features and investigate four and three dimensional spacetimes
with different values for the cosmological constant. Also the corresponding
conserved charges and their respective algebras are studied.

### 4D Gravity

In the three spatial and one time dimensions of spacetime the phenomenon of
gravity is described by Einsteins theory of general relativity (GR). This
theory is formulated in terms of a differentiable manifold representing
spacetime endowed with a semi-Riemannian metric $g_{\mu\nu}$ that is affected
by the energy-momentum tensor $T_{\mu\nu}$ via the Einstein field equations

$\displaystyle R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi
GT_{\mu\nu}$ (126)

where $R_{\mu\nu}$ and $R$ are the Ricci curvature and scalar determined by
the metric. The theory is, as the name suggests, relativistic in the sense of
special relativity, i.e. physical laws hold equally in all inertial frames and
do not single out a preferred one. This implies that in the absence of sources
and a cosmological constant the symmetry group of the vacuum is the Poincaré
group. GR generalizes the relativity principle even further by postulating
that it holds for non-rotating, freely falling frames of reference which
locally can not be distinguished from a uniformly moving frame in the absence
of a gravitational field. Therefore the metric can be expanded locally around
the Minkwski metric of flat space and it transforms covariant under general
coordinate transformations. In (126) $\Lambda$ denotes the cosmological
constant and can be interpreted as the energy density of spacetime. Its origin
is still not very well understood and observations suggest that it has a small
but positive value in our universe. On shorter scales the effect of $\Lambda$
is negligable but it has important implications for the global structure of
spacetime as we will explore later.

#### Asymptotic Flatness

As already mentioned above, (126) also suggests that in the absence of sources
and vanishing $\Lambda$ the curvature vanishes and it is well-known that the
resulting flat spacetime is invariant under Lorentz transformations and
translations which form the Poincaré group whose Lie algebra we encountered in
the previous chapter. When Bondi, Metzner, Sachs and van der Burgh [58], [59],
[60] studied the symmetries of a spacetime that is arbitrary in a finite
region but approaches flat space in the limit of infinite radial coordinate
$r$, they probably also expected to find the Poincaré group. However, it
turned out that the symmetry group of such an asymptotically flat spacetime is
enhanced to an infinite dimensional group.

In the following we will shortly review this finding. It is useful to first
introduce Penrose diagrams which allow for a visualization of the infinite
distances of spacetime while highlighting the causal structure (see e.g.
[61]).

Figure 1: A Penrose diagram of Minkowski space. Red lines indicate surfaces of
constant time $t$ and blue lines surfaces of constant radius $r$ [61].

As figure 1 shows in the example of an eternal Minkowski spacetime the
infinite region is squished into a finite one by a conformal transformation in
such a way that lightrays move in a $45^{\circ}$ angle. Every pair of points
with the same radial coordinate $r$ and time $t$ is actually a two-sphere and
the lightrays start from the lower boundary called past light-like infinity
$\mathcal{I}^{-}$, where $r=\infty=t$ but the advanced time $v=t+r$ is finite,
and move to the future light-like infinity $\mathcal{I}^{+}$ parametrized by
the retarded time $u=t-r$. Massive particles that start from the time-like
infinity $i^{-}$ on the other hand will always be overtaken on their way to
$r=\infty$ by all lightrays and end up in the future time-like infinity
$i^{+}$. The sphere $i^{0}$ labels the so-called spatial infinity.

##### Bondi Metric

Because of the (gauge) invariance of GR under general coordinate
transformations one has to make a gauge choice fixing some of the metric
components. The choice

$\displaystyle g_{rr}=0,\quad
g_{rA}=0,\partial_{r}\text{det}(g_{AB}/r^{2})=0,$ (127)

defining the Bondi gauge, implies that $u$ is labeling null hypersurfaces as
the normal $n^{\mu}=g^{\mu\nu}\partial_{\nu}u$ is null. Furthermore, the
angular coordinates $x^{A}=(z,\bar{z})$ are defined such that their
directional derivative along $n^{\mu}$ vanishes. Here $z$ and its complex
conjugate $\bar{z}$ are coordinates of a stereographic projection of the two-
sphere into the complex plane. A general metric in the Bondi gauge

$\displaystyle
ds^{2}=-Udu^{2}-2e^{\beta}dudr+g_{AB}\left(dx^{A}+\frac{1}{2}U^{A}du\right)\left(dx^{B}+\frac{1}{2}U^{B}du\right)$
(128)

is parametrized by functions $U,U^{A},\beta$ which can depend on all
coordinates. In order to obtain a metric that describes asymptotically flat
spacetime one thus has to impose fall-off conditions on these functions that
restricts their behavior for large $r$. Even though there is no general
consensus as to what exactly these conditions should be, they are ultimately
tied to observation and should allow for phenoma that are known to exist but
exclude the unphysical ones. We will first adapt some of the loosest
restrictions and discuss possible objections later on (see also [65])333We
will sometimes use the term BMS algebra to loosely refer to asymptotic
symmetry algebras and name specific algebras with calligraphic letters.. After
imposing the fall-off conditions the general metric reads

$\displaystyle ds^{2}=$ $\displaystyle-
du^{2}-2dudr+2r^{2}\gamma_{z\bar{z}}dzd\bar{z}$
$\displaystyle+\frac{2m_{B}}{r}du^{2}+rC_{zz}dz^{2}+rC_{\bar{z}\bar{z}}d\bar{z}^{2}+D^{z}C_{zz}dudz+D^{\bar{z}}C_{\bar{z}\bar{z}}dud\bar{z}$
$\displaystyle+\frac{1}{r}\left(\frac{4}{3}(N_{z}+u\partial_{z}m_{B})-\frac{1}{4}\partial_{z}(C_{zz}C^{zz})\right)dudz+c.c.+\mathcal{O}(1/r^{2}),$
(129)

with the Bondi mass aspect $m_{B}$ describing the mass of a system as measured
at null infinity, the angular momentum aspect $N_{A}$ and the traceless and
symmetric tensor $C_{AB}$. $C_{AB}$ is related to the Bondi news tensor by
$N_{AB}=\partial_{u}C_{zz}$ whose square is proportional to the energy flux of
gravitational waves (propagating disturbances in the curvature) across
$\mathcal{I}^{+}$. On the sphere one has the metric
$\gamma_{z\bar{z}}=\frac{2}{(1+z\bar{z})^{2}}$ and $D_{A}$ denotes the
covariant derivative with respect to $\gamma_{AB}$.

##### Supertranslations and Superrotations

Gauge transformations of (128) that preserve the Bondi gauge are infinitesimal
diffeomorphisms $\xi$ satisfying [61]

$\displaystyle\mathcal{L}_{\xi}g_{rr}=0,\quad\mathcal{L}_{\xi}g_{rA}=0,\quad\mathcal{L}_{\xi}\partial_{r}\text{det}\left(g_{AB}/r^{2}\right)=0,$
(130)

where $\mathcal{L}_{\xi}$ denotes the Lie derivative with respect to $\xi$ and
the asymptotic requirements

$\displaystyle\mathcal{L}_{\xi}g_{uu}=\mathcal{O}\left(r^{-1}\right),\quad\mathcal{L}_{\xi}g_{ur}=\mathcal{O}\left(r^{-2}\right),\quad\mathcal{L}_{\xi}g_{uA}=\mathcal{O}\left(r^{0}\right),\quad\mathcal{L}_{\xi}g_{AB}=\mathcal{O}\left(r^{1}\right).$
(131)

We have

$\displaystyle\xi(f,R^{A})=$
$\displaystyle\left(f+\frac{u}{2}D_{A}R^{A}+\mathcal{O}(r^{0})\right)\partial_{u}+\left(R^{A}-\frac{1}{r}D^{A}f+\mathcal{O}(r^{-1})\right)\partial_{A}$
$\displaystyle+\left(-\frac{r+u}{2}D^{A}R^{A}+\frac{1}{2}D_{A}D^{A}f+\mathcal{O}(r^{0})\right)\partial_{r}$
(132)

and we call these solutions of (130) and (131) asymptotic Killing vectors as
they leave the form of (128) invariant. The asymptotic Killing vectors are
parametrized by functions $f$ and $R^{A}$ on the sphere. While $f$ is
unconstrained, the last equation in (131) enforces the conformal Killing
equation on the two-sphere for $R^{A}$

$\displaystyle D_{A}R_{B}+D_{B}R_{A}=\gamma_{AB}D_{C}R^{C}$ (133)

which in turn implies that $R^{z}$ (respective $R^{\bar{z}}$) is holomorphic
(antiholomorphic),i.e $\partial_{z}R^{\bar{z}}=0(\partial_{\bar{z}}R^{z}=0)$.
The Lie derivative of vector fields naturally defines a Lie bracket via
$\mathcal{L}_{\xi}(\xi^{\prime})\equiv[\xi,\xi^{\prime}]$ that reads in first
order

$\displaystyle[\xi(f,R^{A}),\xi(f^{\prime},{R^{\prime}}^{A})]=\xi(\hat{f},\hat{R^{A}}),$
(134)
$\displaystyle\hat{f}=R^{A}D_{A}f^{\prime}+\frac{1}{2}fD_{A}{R^{\prime}}^{A}-{R^{\prime}}^{A}D_{A}f-\frac{1}{2}f^{\prime}D_{A}R^{A},$
(135)
$\displaystyle\hat{R^{A}}=R^{B}D_{B}{R^{\prime}}^{A}-{R^{\prime}}^{B}D_{B}R^{A}$
(136)

for the asymptotic Killing vectors. The vector fields $\xi(f,0)$ form an ideal
of the Lie algebra and are called supertranslations. Acting on the Minkowski
vacuum defined by $m_{B}=N_{AB}=C_{AB}=0$ with a supertranslation does not
change the mass or angular momentum aspect but it transforms

$\displaystyle\mathcal{L}_{f}C_{zz}=-2D^{2}_{z}f.$ (137)

The vanishing of the curvature requires that

$\displaystyle C_{zz}=-2D^{2}_{z}C$ (138)

for some function $C(z,\bar{z})$ and because one has $\mathcal{L}_{f}C=f$,
supertranslations preserve this condition and lead to a degeneracy of the
gravitational vacuum already at the classical level. Only the modes that are
annihilated by $D^{2}_{z}$ are not spontaneously broken in this sense and are
just the usual spacetime translations.

Expanding into monomials in $z,\bar{z}$ according to
$f_{mn}\equiv\frac{z^{m}\bar{z}^{n}}{1+z\bar{z}}\equiv T_{mn}$ and similar for
the superrotations $R^{z}_{n}\equiv z^{n+1}\equiv
l_{n},R^{\bar{z}}_{n}\equiv\bar{z}^{n+1}\equiv\bar{l}_{n}$ [66], one can write
(134) as

$\displaystyle[l_{m},l_{n}]$ $\displaystyle=(m-n)l_{m+n},$
$\displaystyle[l_{m},T_{pq}]$
$\displaystyle=\left(\frac{m+1}{2}-p\right)T_{m+p,q},$ (139)
$\displaystyle[\bar{l}_{m},T_{pq}]$
$\displaystyle=\left(\frac{m+1}{2}-q\right)T_{p,q+m},$
$\displaystyle[T_{mn},T_{pq}]$ $\displaystyle=[l_{m},\bar{l}_{n}]=0,$ (140)

the infinite dimensional BMS algebra in four dimensions denoted by
$\mathfrak{B}_{4}$ in the following. It is the semi-direct product of two
copies of the algebra of vector fields on the sphere, the Witt algebra
(denoted by $\mathfrak{W}$) defined by the first equation in (139) and the
abelian ideal $V_{4}$ generated by the $T_{pq}$. Restricting to
$m,n,p,q\in\\{0,\pm 1\\}$ the algebra (139) is indeed just the ordinary
Poincaré algebra which is therefore a subalgebra of $\mathfrak{B}_{4}$.
However, there exist infinitely many subalgebras generated by

$\\{l_{0},\bar{l}_{0},l_{\pm(1-2m)},\bar{l}_{\pm(1-2m)},T_{mm},T_{1-m,m},T_{m,1-m},T_{1-m,1-m}|m\in\mathbb{Z}\\}$

that are isomorphic to the Poincaré after the rescaling

$\displaystyle
l_{i}\rightarrow\frac{l_{i}}{n},\quad\bar{l}_{i}\rightarrow\frac{\bar{l}_{i}}{n},$
(141)

where $n=1-2m$. These so-called embeddings correspond to classical vacua which
are left invariant by a unique subgroup of the BMS group [61]. An observer
describes the symmetry of the vacuum he/she is in with the Poincaré algebra
($n=1$) and transitions to another vacuum by supertranslations/-rotations.

A useful basis for $\mathfrak{B}_{4}$ is obtained by the transformation

$\displaystyle k_{n}$ $\displaystyle=l_{n}+\bar{l}_{n},$
$\displaystyle\bar{k}_{n}$ $\displaystyle=-i(l_{n}-\bar{l}_{n}),$ (142)
$\displaystyle S_{mn}$ $\displaystyle=\frac{1}{2}(T_{mn}+T_{nm}),$
$\displaystyle A_{mn}$ $\displaystyle=-\frac{i}{2}(T_{mn}-T_{nm}),$ (143)

and the algebra relations then read

$\displaystyle[k_{n},k_{m}]$
$\displaystyle=(n-m)k_{n+m},\quad[\bar{k}_{n},\bar{k}_{m}]=-(n-m)k_{n+m},$
(144) $\displaystyle[k_{n},\bar{k}_{m}]$ $\displaystyle=(n-m)\bar{k}_{n+m},$
(145) $\displaystyle[k_{n},S_{pq}]$
$\displaystyle=\left(\frac{n+1}{2}-p\right)S_{p+n,q}+\left(\frac{n+1}{2}-q\right)S_{p,q+n},$
(146) $\displaystyle[\bar{k}_{n},S_{pq}]$
$\displaystyle=\left(\frac{n+1}{2}-p\right)A_{p+n,q}-\left(\frac{n+1}{2}-q\right)A_{p,q+n},$
(147) $\displaystyle[k_{n},A_{pq}]$
$\displaystyle=\left(\frac{n+1}{2}-p\right)A_{p+n,q}+\left(\frac{n+1}{2}-q\right)A_{p,q+n},$
(148) $\displaystyle[\bar{k}_{n},A_{pq}]$
$\displaystyle=-\left(\frac{n+1}{2}-p\right)S_{p+n,q}+\left(\frac{n+1}{2}-q\right)S_{p,q+n}.$
(149)

In this basis the Poincaré algebra in light-cone coordinates

$\displaystyle P_{\pm}=\frac{P_{0}\pm P_{3}}{\sqrt{2}},\quad
M_{\pm\,1/2}=\frac{M_{0\,1/2}\pm M_{3\,1/2}}{\sqrt{2}},\quad M_{+-}=M_{03}$
(150)

is embedded according to

$\displaystyle k_{0}$
$\displaystyle=\frac{i}{n}M_{+-},\quad\bar{k}_{0}=-\frac{i}{n}M_{12},\quad
k_{n}=\frac{i}{n}\sqrt{2}M_{+1},$ (151) $\displaystyle k_{-n}$
$\displaystyle=\frac{i}{n}\sqrt{2}M_{-2},\quad\bar{k}_{n}=\frac{i}{n}\sqrt{2}M_{+2},\quad\bar{k}_{-n}=\frac{i}{n}\sqrt{2}M_{-1},$
(152) $\displaystyle S_{mm}$ $\displaystyle=\frac{i}{n}\sqrt{2}P_{-},\quad
S_{1-m,1-m}=-\frac{i}{n}\sqrt{2}P_{+},$ (153) $\displaystyle S_{m,1-m}$
$\displaystyle=-\frac{i}{n}P_{1},\quad A_{m,1-m}=-\frac{i}{n}P_{2}.$ (154)

##### Phenomenology

One might ask now whether the enlargement of the Poncaré algebra has any
observable consequences. This is indeed the case and the easiest way to
confirm it is by considering the degenerate vacua related by supertranslations
[67]. An experiment might consist of two inertial detectors near
$\mathcal{I}^{+}$ such that the Bondi news vanishes at early and late times
but in the intermediate time the passing of a gravitational wave results in
non-zero Bondi news. From the geodesic equation one finds that their relative
separations $(s^{z},s^{\bar{z}})$ are altered by the gravitational wave and
after it has passed the spacetime is flat but characterized by a different
$C_{zz}$, i.e. it was supertranslated. The residual change in the separations
is proportional to

$\displaystyle\Delta s^{\bar{z}}=\frac{\gamma^{z\bar{z}}}{2r}\Delta
C_{zz}s^{z}$ (155)

and yields a finite result. This effect is called the gravitational
displacement memory effect (cf. also chapter Applications and Phenomenology).

In the case of the superrotations the situation is a bit more subtle. With the
general parametrization where $m\in\mathbb{Z}$ there are singularities at
$z=0,\infty$, i.e. the functions are only meromorphic. Only the ordinary
Lorentz rotations and boosts with $m=0,1,2$ are holomorphic on the entire
sphere and thus globally well defined as the vector fields $R_{m}\equiv
z^{m}\partial_{z},m<0$ and, after redefining
$\omega=z^{-1}$,$R_{m}=\omega^{2-m}\partial_{\omega},m>2$, have a singularity
at the origin [68]. A general superrotation will thus map a flat spacetime to
one that is flat everywhere except at the singularities which is then called
asymptotically locally flat (ALF) [70]. While some authors argued that
superrotations have to be excluded because of the singularities, others
suggested that they might have a physical interpretation related to cosmic
strings. Cosmic strings are one-dimensional topological defects which were
linked to the following construction due to Penrose [71], [70]. Consider a
Minkowski spacetime that is cut along the light-cone $u=0$. After performing a
diffeomorphism on the patch with $u>0$ it is glued together such that the
metric is continuous across the cut. This procedure introduces singularities,
e.g. at $z=0,\infty$ which describe a cosmic string. In our universe such
strings might exist if at some early time the universe was not simply
connected yet (cf. also section Decay of Cosmic Strings). Vacua with strings
would then be ALF spacetimes and the snapping of a string can therefore be
seen as a transition from an ALF spacetime to an inequivalent asymptotically
flat one which is indeed just a superrotation. In analogy to the
supertranslations a further consequence of the existence of superrotations was
suggested to be the (sub-leading) spin-memory effect [135].

#### Surface Charges

A cornerstone of theoretical physics is the principle that continuous
symmetries lead to conserved quantities as expressed by the Noether theorem.
In the case of a “rigid” or global symmetry of a field theory on the
$D$-dimensional manifold $\mathcal{M}$ with a parameter $\epsilon$ [72]

$\displaystyle x\mapsto
x+\delta_{\epsilon}x,\quad\phi\mapsto\phi+\delta_{\epsilon}\phi,$ (156)

one can obtain the associated Noether current $j^{\mu}$ by replacing
$\epsilon$ with an arbitrary function $\epsilon(x)$ and reading it off from
the variation of the action

$\displaystyle\delta
S=-\int_{\mathcal{M}}d^{D}xj^{\mu}\partial_{\mu}\epsilon.$ (157)

If the transformation (156) is indeed a symmetry of the theory the variation
vanishes on-shell, i.e. if the fields satisfy the equations of motion, which
in turn implies that $j^{\mu}$ is conserved. For a space-like slice $\Sigma$
of $\mathcal{M}$ the associated conserved Noether charge then reads

$\displaystyle Q=\int_{\Sigma}d^{D-1}xn_{\mu}j^{\mu},$ (158)

where $n_{\mu}$ is the time-like normal vector field of $\Sigma$. If, however,
we are dealing with a gauge symmetry then $\epsilon$ will be depending on the
spacetime coordinates from the start and thus the rhs of (157) vanishes
identically leaving us with no possibility to determine $j^{\mu}$. The
situation can be rescued by noting that we can modify the current
$j^{\mu}\rightarrow j^{\mu}+\partial_{\nu}k^{\mu\nu}$ and (157) is unaffected
if $k^{\mu\nu}$ is antisymmetric. The Noether charge then receives a
contribution

$\displaystyle
Q_{\text{surface}}=\int_{\Sigma}d^{D-1}xn_{\mu}\partial_{\nu}k^{\mu\nu}=\int_{\partial\Sigma}d^{D-2}n_{\mu}n^{\prime}_{\nu}k^{\mu\nu},$
(159)

where $n^{\prime}_{\nu}$ is normal to $\partial\Sigma$. We can now associate a
two-form $k^{\mu\nu}$ to the symmetry by using (157) with
$\partial_{\nu}k^{\mu\nu}$ and take (159) as a definition for the charge, it
is conserved if $\nabla_{\mu}k^{\mu\nu}=0$ on-shell, where $\nabla_{\mu}$ is
the covariant derivative.

Conserved charges $Q$ of a symmetry described by a group $G$ generate
infinitesimal symmetry transformations in the following way [72]. Let
$\mathcal{M}^{\prime}$ be a Poisson manifold with Poisson bracket $\\{,\\}$,
the classical phase space of a theory and $F$ an observable, i.e. a function
on $\mathcal{M}^{\prime}$. One can identify the infinitesimal generator
$\xi_{X}$ of the action of $G$ on $\mathcal{M}^{\prime}$, which is a vector
field on $\mathcal{M}^{\prime}$ related to an element $X$ of the Lie algebra
$\mathfrak{g}$ associated to $G$ by

$\displaystyle[\xi_{X},\xi_{Y}]=-\xi_{[X,Y]},$ (160)

with the charge via

$\displaystyle-\\{Q[\xi_{X}],\\}\equiv\xi_{X},\rightarrow\\{Q[\xi_{X}],F\\}=\xi_{X}F\equiv\delta_{X}F.$
(161)

One says that $Q$444Or rather the map
$\tilde{Q}:\mathcal{M}^{\prime}\rightarrow\mathfrak{g}^{*}$,
$\braket{\tilde{Q}(),X}\equiv Q[\xi_{X}]$. is a momentum map of the action of
$G$ on $\mathcal{M}^{\prime}$.

For the symmetry transformation on the fields $\phi$ we can write

$\displaystyle\\{Q[\xi],\phi\\}=-\delta_{\xi}\phi$ (162)

and the Jacobi identity for the Poisson bracket implies (using (160))

$\displaystyle\\{\phi,\\{Q[\xi],Q[\zeta]\\}\\}=-\\{Q[\zeta],\\{Q[\xi],\phi\\}\\}-\\{Q[\xi],\\{\phi,Q[\zeta]\\}\\}=[\delta_{\xi},\delta_{\zeta}]\phi=\\{\phi,Q[[\xi,\zeta]]\\}.$
(163)

As this has to hold for arbitrary fields, one can conclude that the terms in
the bracket have to coincide up to a constant term

$\displaystyle\\{Q[\xi],Q[\zeta]\\}=Q[[\xi,\zeta]]+c(\xi,\zeta)$ (164)

such that $\\{c,\phi\\}=0$. The Jacobi identity then enforces

$\displaystyle c(\\{Q[\xi_{1}],Q[\xi_{2}]\\},Q[\xi_{3}])+\text{cycl.}=0,$
(165)

which is the condition that $c$ is a 2-cocycle of the Chevalley-Eilenberg
cohomology with values in the field $\mathbb{C}$ 555The first line in (11)
vanishes because the action of the field is trivial and thus we obtain (165).
Also, if $c$ were a coboundary derived from a 1-cocycle $c^{\prime}$ we could
redefine

$\displaystyle Q^{\prime}[\cdot]=Q[\cdot]+c^{\prime}(\cdot)$ (166)
$\displaystyle\rightarrow\\{Q^{\prime}[\xi_{1}],Q^{\prime}[\xi_{2}]\\}=Q[[\xi_{1},\xi_{2}]]+c(\xi_{1},\xi_{2})=Q[[\xi_{1},\xi_{2}]]+c^{\prime}([\xi_{1},\xi_{2}])=Q^{\prime}[[\xi_{1},\xi_{2}]]$
(167)

so we find that the algebra of the surface charges is in general a central
extension of the algebra of the symmetry generators classified by the
cohomology group $H^{2}(\mathfrak{g},\mathbb{C})$.

In the flat $(\Lambda=0)$ case one finds for the surface charges associated to
the supertranslations [61]

$\displaystyle Q[f]=\frac{1}{4\pi G}\int d^{2}z\gamma_{z\bar{z}}fm_{B},$ (168)

where the integration is over the boundary of a space-like slice, e.g. a
sphere on $\mathcal{I}^{\pm}$ which is a point on the boundary of the Penrose
diagram. Defining $Q^{+}[f]$ and $Q^{-}[f]$ as the charges from integration
over $\mathcal{I}^{+}_{-}$ and $\mathcal{I}^{-}_{+}$, one has a matching
condition implying the conservation of the charges, i.e.

$\displaystyle Q^{+}[f]=Q^{-}[f].$ (169)

Similarly, for the superrotations we have

$\displaystyle Q^{+}[R^{A}]=\frac{1}{8\pi
G}\int_{\mathcal{I}^{+}_{-}}d^{2}z(R_{\bar{z}}N_{z}+R_{z}N_{\bar{z}})=\frac{1}{8\pi
G}\int_{\mathcal{I}^{-}_{+}}d^{2}z(R_{\bar{z}}N_{z}+R_{z}N_{\bar{z}})=Q^{-}[R^{A}],$
(170)

where the conservation follows from a matching condition for the angular
momentum aspect $N_{A}$.

For the $\mathfrak{B}_{4}$(eq. (139)-(140)) it was shown in [73] that the only
consistent central extensions are those of the Witt subalgebras. There, the
second cohomology group with values in the trivial module is one-dimensional
and given by the Gelfand-Fuks cocycle leading to the Virasoro algebra
($\mathfrak{Vir}$), i.e. the centrally extended Witt algebra

$\displaystyle[l_{m},l_{n}]=(m-n)l_{m+n}+\frac{c_{l}}{12}(m^{3}-m)\delta_{m+n,0}$
(171)

with central charge $c_{l}$. We denote this surface charge algebra by
$\mathfrak{B}_{4,c}\equiv V_{4}\rtimes(\mathfrak{Vir}\oplus\mathfrak{Vir})$.
In four dimensions the calculation of these charges $(c_{l},c_{\bar{l}})$ in
general is quite involved as they are non-integrable in the presence of
gravitational waves [74].

#### Asymptotically (Anti) de Sitter Spacetimes

As the solutions to the Einstein equations for a globally source-free
configuration are, besides the Minkowski spacetime, also (Anti) de Sitter
((A)dS) spacetimes depending on the cosmological constant it is natural to
investigate the concept of asymptotic symmetries for these solutions.
Recently, this has been done in [75], where the symmetry group of a spacetime
that is asymptotically Anti de Sitter was investigated. Even though the de
Sitter case is algebraically very similar there are several unresolved
conceptual difficulties in the gravitational description of its asymptotics as
described for example in [76]. The main issue appears to be that the null
infinities $\mathcal{I}^{\pm}$ are space-like for $\Lambda>0$ which makes it
difficult to even formulate boundary conditions.

In [75], just as in the asymptotically flat case, a generic spacetime metric
is constrained by fall-off conditions in the radial coordinate $r$ such that
in the limit $r\rightarrow\infty$ the AdS metric is recovered. The Killing
vectors $\xi_{f,R}=f\partial_{u}+R^{A}\partial_{A}$ of this asymptotically AdS
metric satisfy in first order of $1/r$

$\displaystyle[\xi_{1},\xi_{2}]=\hat{\xi}=\hat{f}\partial_{u}+\hat{R}^{A}\partial_{A},$
(172)
$\displaystyle\hat{f}=R_{1}^{A}\partial_{A}f_{2}+\frac{1}{2}f_{1}\partial_{A}R^{A}_{2}-(1\leftrightarrow
2),\quad\hat{R}^{A}=R^{B}_{1}\partial_{B}R_{2}^{A}+\frac{\Lambda}{3}f_{1}\gamma^{AB}\partial_{B}f_{2}-(1\leftrightarrow
2).$ (173)

Parametrizing the Killing vector fields by whole numbers

$\displaystyle\xi_{f_{pq},0}\equiv
T_{pq}\equiv\frac{z^{p}{\bar{z}}^{q}}{1+z\bar{z}}\partial_{u},\quad\xi_{0,R^{z}_{n}}\equiv
l_{n}\equiv-z^{n+1}\partial_{z},\quad\xi_{0,R^{\bar{z}}_{n}}\equiv\bar{l}_{n}\equiv-\bar{z}^{n+1}\partial_{\bar{z}},$
(174)

yields the following structure

$\displaystyle[T_{p,q},T_{m,n}]$
$\displaystyle=\Lambda\frac{z^{p}\bar{z}^{q}}{1+z\bar{z}}\frac{(1+z\bar{z})^{2}}{2}\left(\partial_{\bar{z}}\frac{z^{m}\bar{z}^{n}}{1+z\bar{z}}\partial_{z}+\partial_{z}\frac{z^{m}\bar{z}^{n}}{1+z\bar{z}}\partial_{\bar{z}}\right)-(mn\leftrightarrow
pq)$
$\displaystyle=\Lambda((n-q)z^{p+m}\bar{z}^{q+n-1}\partial_{z}+(m-p)z^{p+m-1}\bar{z}^{q+n}\partial_{\bar{z}})$
(175) $\displaystyle[l_{m},l_{n}]$
$\displaystyle=(m-n)l_{m+n},\quad[\bar{l}_{m},\bar{l}_{n}]=(m-n)\bar{l}_{m+n}$
(176) $\displaystyle[l_{m},T_{p,q}]$
$\displaystyle=\left(\frac{m+1}{2}-p\right)T_{p+m,q},\quad[\bar{l}_{m},T_{p,q}]=\left(\frac{m+1}{2}-q\right)T_{p,q+m}.$
(177)

Thus, the result (175) can not be written as the sum of basis elements
$\xi_{f_{mn},R^{A}_{p}}$ but is a general vector field on the sphere, i.e. the
four dimensional $\Lambda$-BMS is a Lie algebroid. Technically this is a
consequence of the appearance of the coordinate dependent metric on the sphere
in (173). However, it is still true that the ordinary symmetry algebra of Anti
de Sitter spacetime (denoted by $AdS_{4}$)

$\displaystyle[K_{+},K_{-}]$ $\displaystyle=-i\Lambda
M_{+-},\quad[K_{+},K_{a}]=-i\Lambda M_{+a},$ (178)
$\displaystyle[K_{-},K_{a}]$ $\displaystyle=-i\Lambda
M_{-a},\quad[K_{a},K_{b}]=-i\Lambda M_{ab},$ (179)
$\displaystyle[M_{\mu\nu},M_{\rho\sigma}]$
$\displaystyle=i(\eta_{\mu\sigma}M_{\nu\rho}-\eta_{\nu\sigma}M_{\mu\rho}+\eta_{\nu\rho}M_{\mu\sigma}-\eta_{\mu\rho}M_{\nu\sigma}),$
(180) $\displaystyle[M_{\mu\nu},K_{\rho}]$
$\displaystyle=i(\eta_{\nu\rho}K_{\mu}-\eta_{\mu\rho}K_{\nu}),$ (181)

is embedded into (175) which can be seen by considering the subalgebras
spanned by

$\displaystyle
l_{0},l_{1-2m},l_{-1+2m},\bar{l}_{0},\bar{l}_{1-2m},\bar{l}_{-1+2m},T_{mm},T_{1-m,m},T_{m,1-m},T_{1-m,1-m},$

with the rescaling

$\displaystyle
l_{n}\rightarrow\frac{l_{n}}{1-2m},\quad\bar{l}_{n}\rightarrow\frac{\bar{l}_{n}}{1-2m},\quad\Lambda\rightarrow(1-2m)^{2}\Lambda.$
(182)

We have e.g.

$\displaystyle[K_{-},K_{+}]$
$\displaystyle=[T_{mm},T_{1-m,1-m}]=\Lambda(1-2m)(z\partial_{z}+\bar{z}\partial_{\bar{z}})$
$\displaystyle=(1-2m)^{2}\Lambda(\xi_{l_{0}}+\xi_{\bar{l}_{0}})=i(1-2m)^{2}\Lambda
M_{+-},$ (183)

and similar for the rest of the $AdS_{4}$.

Although there are attempts to define generalizations of bialgebras on some
versions of algebroids, e.g. in [77], [78], there is no established notion of
quantum groups on a Lie algebroid and therefore deformations of asymptotically
(A)dS spacetime symmetries in four dimensions will not be studied directly in
this work. Nonetheless, one can hope to infer valuable insights from the three
dimensional theory of gravity.

### 3D Gravity

As we will see, Einsteins theory of GR in two spatial and one time dimension
is a useful toy model for studying asymptotic symmetries as well as other
problems that are harder to address in four dimensional gravity. On the first
glance, the theory seems to be rather trivial and completely different from
what one would associate with gravity because it does not have a Newtonian
limit and massive particles do not attract each other [79], [80], [81]. In
fact, there are no local gravitational degrees of freedom at all in three
dimensions! This means in particular that no gravitational waves exist and
gravitational energy can not be radiated away. Formally this is a consequence
of the fact that the three dimensional Riemann tensor, which can be expressed
in terms of the Ricci and the Weyl tensor, has the same number of degrees of
freedom as the Ricci tensor. Thus, the Weyl tensor vanishes leading to the
peculiar features mentioned above.

Nonetheless, because of topological and asymptotic properties, there are still
interesting physical phenomena. For example particles can be introduced in the
form of topological defects with the geometry of a cone [82]. The fact that
topological but no local gravitational degrees of freedom exist is also
exemplified by the equivalence to Chern-Simons theory which is a topological
field theory [83]. Because Chern-Simons theory can be quantized, it is hoped
that the reformulation can shed light on the problem of quantum gravity [84].
It was also shown that black hole solutions with an event horizon exist [85].
These so-called BTZ black holes admit (if they are non-extremal) a finite
temperature implying that they decay like their four dimensional counterparts.

#### Asymptotic Symmetries

Depending on the value of the cosmological constant, the three dimensional
spacetime is locally isometric to either Minkowski, de Sitter or Anti de
Sitter space. We will start the discussion of the asymptotic symmetries in the
case of curved spacetime (following [72]) and then view the asymptotically
flat spacetime as a contraction limit $\Lambda\rightarrow 0$.

Analogously to the four dimensional case one can start from a general metric

$\displaystyle
ds^{2}=-\left(1-r^{2}\Lambda\right)dt^{2}+\left(1-r^{2}\Lambda\right)^{-1}dr^{2}+r^{2}d\phi^{2}$
(184)

and impose fall-off conditions so that it asymptotically approaches (A)dS. The
Killing vectors are $\xi_{f,R}=f\partial_{u}+R\partial_{z}$ where
$R=R(z),f=T(z)+u\partial_{z}R$ and we have

$\displaystyle[\xi_{1},\xi_{2}]=\hat{\xi}\equiv\hat{f}\partial_{u}+\hat{R}\partial_{z}$
(185)
$\displaystyle\hat{f}=R_{1}\partial_{z}f_{2}+f_{1}\partial_{z}R_{2}-(1\leftrightarrow
2),\quad\hat{R}=R_{1}\partial_{z}R_{2}-\Lambda
f_{1}\partial_{z}f_{2}-(1\leftrightarrow 2).$ (186)

Here, $u=t-r$ is again the retarded time and $z=e^{i\phi}$ with the angular
coordinate $\phi$. Parametrizing $f_{m}\equiv T_{m}=z^{m+1}$ and $R_{m}\equiv
l_{m}=z^{m+1}$, we find

$\displaystyle[l_{m},l_{n}]$
$\displaystyle=(m-n)l_{m+n},\quad[l_{m},T_{n}]=(m-n)T_{m+n},$ (187)
$\displaystyle[T_{m},T_{n}]$ $\displaystyle=-\Lambda(m-n)l_{m+n}.$ (188)

As in the four dimensional case the symmetry algebra of the globally AdS
spacetime ($AdS_{3}$)

$\displaystyle[\tilde{P}_{+},\tilde{P}_{-}]=-i\eta_{+-}\Lambda
M_{+-},\quad[\tilde{P}_{\pm},\tilde{P}_{2}]=-i\eta_{+-}\Lambda M_{\pm 2},$
(189) $\displaystyle[M_{+2},M_{-2}]=i(-\eta_{22})M_{+-},\quad[M_{+-},M_{\pm
2}]=\pm i\eta_{+-}M_{\pm 2},\quad[M_{+-},\tilde{P}_{\pm}]=\pm
i\eta_{+-}\tilde{P}_{\pm},$ (190) $\displaystyle[M_{\pm
2},\tilde{P}_{2}]=i\eta_{22}\tilde{P}_{\pm},\quad[M_{\pm
2},\tilde{P}_{\mp}]=-i\eta_{+-}P_{2},$ (191)

where we choose the metric to be $\eta_{+-}=1,\eta_{22}=-1$, is a subalgebra
of (187) and there are infinitely many embeddings of the form

$\displaystyle\tilde{P}_{2}=-i\frac{T_{0}}{n},\quad\tilde{P}_{\pm}=i\frac{T_{\pm
n}}{\sqrt{2}n},\quad M_{+-}=-i\frac{l_{0}}{n},\quad M_{\pm
2}=\pm\frac{i}{\sqrt{2}n}l_{\pm n}.$ (192)

The embeddings are indeed isomorphic to (189) with the rescaling
$\Lambda\rightarrow n^{2}\Lambda$. Alternatively, one could set $n=1$ in (192)
and rescale the ambient metric
$\eta_{\mu\nu}\rightarrow\frac{\eta_{\mu\nu}}{n^{2}}$ instead of $\Lambda$.

(189) can also be obtained from the four dimensional Lorentz algebra

$\displaystyle[M_{\mu\nu},M_{\rho\lambda}]=i(\eta_{\mu\lambda}M_{\nu\rho}-\eta_{\nu\lambda}M_{\mu\rho}+\eta_{\nu\rho}M_{\mu\lambda}-\eta_{\mu\rho}M_{\nu\lambda})$
(193)

if $M_{\pm 3},M_{23}$ is identified with $K_{\pm},K_{2}$. Here the metric is
given by $\eta_{+-}=1,\eta_{22}=-1,\eta_{33}=-1$. $K_{i}$ has to be rescaled
according to

$\displaystyle\tilde{P}_{i}=R^{-1}K_{i},$ (194)

where $R^{-1}=\sqrt{|\Lambda|}$ is the inverse AdS radius. Thus, the
generators $\tilde{P}_{i}$ are actually boosts in a higher dimensional
spacetime and acquire the interpretation as momenta only in the contraction
limit $\Lambda\rightarrow 0$ when they commute among themselves and form an
ideal.

#####  $(A)dS_{3}$ as real forms

On the algebraic level one can also obtain the de Sitter case by simply
setting $\Lambda>0$. Depending on the choice of $\Lambda$, (187) is a
different real algebra with one reality condition666We warn the reader that
keeping it real might also go wrong, as exemplified in [86].

$\displaystyle l^{\dagger}_{m}=-l_{m},\quad T_{m}^{\dagger}=-T_{m}.$ (195)

These two algebras can also be expressed as two different real forms of the
complex Lie algebra $\mathfrak{W}\oplus\mathfrak{W}(\mathbb{C})$ 777In the
following, if no real form is specified, we often refer to the complex algebra
without making it explicit in the notation. defined by

$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n},\quad[\bar{L}_{m},\bar{L}_{n}]=(m-n)\bar{L}_{m+n},\quad[L_{m},\bar{L}_{n}]=0.$
(196)

This is established by the isomorphism

$\displaystyle
L_{m}=\frac{1}{2}\left(l_{m}+\frac{1}{\sqrt{-\Lambda}}T_{m}\right),\quad\bar{L}_{m}=\frac{1}{2}\left(l_{m}-\frac{1}{\sqrt{-\Lambda}}T_{m}\right).$
(197)

The embeddings now take the form

$\displaystyle L_{0},\bar{L}_{\pm n},\bar{L}_{0},\bar{L}_{\pm n},$ (198)
$\displaystyle
L_{m}\rightarrow\frac{L_{m}}{n},\quad\bar{L}_{m}\rightarrow\frac{\bar{L}_{m}}{n},$
(199)

they are isomorphic to the algebra $\mathfrak{o}(4,\mathbb{C})$

$\displaystyle[L_{0},L_{\pm 1}]=\mp L_{\pm 1},\qquad[L_{+1},L_{-1}]=2L_{0},$
(200) $\displaystyle[\bar{L}_{0},\bar{L}_{\pm 1}]=\mp\bar{L}_{\pm
1},\qquad[\bar{L}_{+1},\bar{L}_{-1}]=2\bar{L}_{0}.$ (201)

For negative $\Lambda$, i.e. the AdS case, we have from (197) and (195) that

$\displaystyle
L_{m}^{\dagger}=-L_{m},\quad\bar{L}_{m}^{\dagger}=-\bar{L}_{m},$ (202)

whereas for the de Sitter case one finds

$\displaystyle
L_{m}^{\ddagger}=-\bar{L}_{m},\quad\bar{L}_{m}^{\ddagger}=-L_{m}.$ (203)

Restricted to (200), which is related to the standard Cartan-Weyl form

$\displaystyle[H,E_{\pm}]=\pm E_{\pm},\qquad[E_{+},E_{-}]=2H,$ (204)
$\displaystyle[\bar{H},\bar{E}_{\pm}]=\pm\bar{E}_{\pm},\qquad[\bar{E}_{+},\bar{E}_{-}]=2\bar{H},$
(205)

via

$\displaystyle H=-L_{0},\quad E_{\pm}=iL_{\pm
1},\quad\bar{H}=-\bar{L}_{0},\quad\bar{E}_{\pm}=i\bar{L}_{\pm 1},$ (206)

the real forms correspond to two copies of
$\mathfrak{sl}(2,\mathbb{R})\simeq\mathfrak{o}(2,1)$ for AdS and the Kleinian
algebra
$\mathfrak{o}(2,2)\simeq\mathfrak{o}(2,1)\oplus\bar{\mathfrak{o}}(2,1)$ for dS
expressed by

$\displaystyle H^{\dagger}$ $\displaystyle=-H,\quad
E_{\pm}^{\dagger}=E_{\pm},\quad\bar{H}^{\dagger}=-\bar{H},\quad\bar{E}_{\pm}^{\dagger}=\bar{E}_{\pm},$
(207) $\displaystyle H^{\ddagger}$ $\displaystyle=-\bar{H},\quad
E^{\ddagger}_{\pm}=\bar{E}_{\pm}.$ (208)

This can be identified with the real structures listed in [87] with the
automorphism $E_{\pm}\rightarrow-
E_{\pm},\bar{E}_{\pm}\rightarrow-\bar{E}_{\pm}$. Note that this automorphism
of the $\mathfrak{sl}(2)$

$\displaystyle\Phi(E_{\pm})=-E_{\pm},\quad\Phi(H)=H,\rightarrow\Phi(L_{\pm
1})=-L_{\pm 1},\quad\Phi(L_{0})=L_{0},$ (209)

can be extended uniquely to an automorphism of the Witt algebra. On the
algebraic level we thus can describe both dS and AdS spacetime symmetries
simultaneously by switching between the two real forms.

#### Algebra of $AdS_{3}$ Surface Charges

Following [72], one can also introduce light-cone coordinates
$x^{\pm}=\frac{t}{\sqrt{-\Lambda}}\pm\phi$ leading to the metric (in
Fefferman-Graham gauge)

$\displaystyle
ds^{2}=-\frac{1}{r^{2}\Lambda}dr^{2}-\left(rdx^{+}-\frac{4G}{\sqrt{-\Lambda}r}\bar{p}(x^{-})dx^{-}\right)\left(rdx^{-}-\frac{4G}{\sqrt{-\Lambda}r}p(x^{+})dx^{+}\right)$
(210)

and instead of $f$ and $R$ the Killing vectors are then parametrized by
$X(x^{+})$ and $\bar{X}(x^{-})$. In analogy to (170) one has

$\displaystyle
Q[X,\bar{X}]=\frac{1}{2\pi}\int_{0}^{2\pi}d\phi\left(X(x^{+})p(x^{+})+\bar{X}(x^{-})\bar{p}(x^{-})\right).$
(211)

The algebra of these charges is a central extension of the algebra of the
Killing vectors, i.e. $\mathfrak{W}\oplus\mathfrak{W}$. As noted at the end of
section Surface Charges, the Witt algebras only have one possible central
extension and so the general form of the central charge algebra is

$\displaystyle[L_{m},L_{n}]=$
$\displaystyle(m-n)L_{m+n}+\frac{c_{L}}{12}(m^{3}-m)\delta_{m+n,0},$ (212)
$\displaystyle[\bar{L}_{m},\bar{L}_{n}]=$
$\displaystyle(m-n)\bar{L}_{m+n}+\frac{c_{\bar{L}}}{12}(m^{3}-m)\delta_{m+n,0},$
(213) $\displaystyle[L_{m},\bar{L}_{n}]=$ $\displaystyle 0.$ (214)

Brown and Henneaux calculated explicitly that [56]

$\displaystyle c_{L}=c_{\bar{L}}=\frac{3}{2G\sqrt{-\Lambda}}.$ (215)

#### Asymptotic Flatness

One possibility of obtaining the symmetry algebra of asymptotically flat
spacetime in three dimensions would be to consider a general metric, impose
fall-off conditions and analyze the algebra of the Killing vectors just as it
was done for the four dimensional case. However, since we already know the
symmetry algebra of asymptotically (A)dS spacetime and it has the form of a
Lie algebra, we can simply calculate the contraction limit888On the algebraic
level such a procedure is also called Inonu-Wigner contraction [69].
$\Lambda\rightarrow 0$ [68]. Starting from the algebra of the surface charges
(212)-(214) and using (197) with the automorphism
$\bar{L}_{m}\rightarrow-\bar{L}_{-m}$, i.e.

$\displaystyle T_{m}=\sqrt{-\Lambda}(L_{m}+\bar{L}_{-m}),\quad
l_{m}=L_{m}-\bar{L}_{-m},$ (216)

we obtain

$\displaystyle[T_{m},T_{n}]=$
$\displaystyle-\Lambda([L_{m},L_{n}]+[\bar{L}_{-m},\bar{L}_{-n}])=-\Lambda(m-n)l_{m+n}$
$\displaystyle+\delta_{m+n,0}(m^{3}-m)\frac{c_{L}}{12}+\delta_{-m-n,0}((-m)^{3}+m)\frac{\bar{c}_{L}}{12},$
(217) $\displaystyle[l_{m},l_{n}]=$
$\displaystyle(m-n)l_{m+n}+\delta_{m+n,0}(m^{3}-m)\frac{c_{L}}{12}+\delta_{-m-n,0}((-m)^{3}+m)\frac{\bar{c}_{L}}{12},$
(218) $\displaystyle[l_{m},T_{n}]=$
$\displaystyle(m-n)T_{m+n}+\sqrt{-\Lambda}\left(\delta_{m+n,0}(m^{3}-m)\frac{c_{L}}{12}-\delta_{-m-n,0}((-m)^{3}+m)\frac{\bar{c}_{L}}{12}\right).$
(219)

Now the contraction limit and (215) yields the algebra of the surface charges
in three dimensions

$\displaystyle[T_{m},T_{n}]$
$\displaystyle=0,\qquad[l_{m},l_{n}]=(m-n)l_{m+n},$ (220)
$\displaystyle[l_{m},T_{n}]$
$\displaystyle=(m-n)T_{m+n}+\delta_{m+n,0}(m^{3}-m)c^{\prime}_{L},$ (221)

with $c^{\prime}_{L}=\frac{1}{4G}$, which we denote $\mathfrak{B}_{3,c}$ in
the following. Without the central extension terms, (220)-(221) defines the
three dimensional BMS algebra $\mathfrak{B}_{3}$.

The three dimensional Poincaré algebra ($\mathfrak{P}_{3}$)

$\displaystyle[M_{+2},M_{-2}]$
$\displaystyle=i\eta_{11}M_{+-}\,,\quad[M_{+-},M_{\pm 2}]=\mp i\eta_{+-}M_{\pm
1}\,,$ (222) $\displaystyle[M_{+-},P_{\pm}]$ $\displaystyle=\mp
i\eta_{+-}P_{\pm}\,,\quad[M_{\pm 2},P_{2}]=-i\eta_{22}P_{\pm}\,,$ (223)
$\displaystyle[M_{\pm 2},P_{\mp}]$ $\displaystyle=i\eta_{+-}P_{2}\,,$ (224)

is infinitely often embedded in $\mathfrak{B}_{3}$ with

$\displaystyle M_{+-}=-il_{0},\quad M_{\pm 2}=\pm\frac{i}{\sqrt{2}}l_{\pm
n},\quad P_{2}=-iT_{0},\quad P_{\pm}=\frac{i}{\sqrt{2}}T_{\pm n},$ (225)

if the metric is rescaled as $\eta_{+-}=\eta_{-+}=-\eta_{22}=n$, otherwise the
Lorentz generators have to be rescaled by $1/n$.

Note that while the symmetry algebras without central extension in three
dimensions are subalgebras of their four dimensional counterparts, this is not
true for the centrally extended algebras, i.e.
$\mathfrak{B}_{3,c}\not\subseteq\mathfrak{B}_{4,c}$. Thus, arguments that
results from the three dimensional theory can be transferred to four
dimensions as one is a subset of the other, have to be particularly wary of
the influence of contributions from the central extension.

An overview over all the relevant algebras related to the three dimensional
symmetry algebras is given in figure 2.

$\mathfrak{W}\oplus\mathfrak{W}(\mathbb{C})$$\mathfrak{W}\oplus\mathfrak{W},\;\dagger$$\mathfrak{W}\oplus\mathfrak{W},\;\ddagger$$\mathfrak{B}_{3}$$\mathfrak{B}_{3,c}$$\rtimes$$\oplus^{2}$$\oplus^{2}$$\mathfrak{Vir}\oplus\mathfrak{Vir}$$\mathfrak{Vir}$Supertranslations$\mathfrak{W}$$\mathfrak{o}(4,\mathbb{C})$$\mathfrak{sl}(2,\mathbb{C})$$AdS_{3}$$dS_{3}$3D
Poincaré$\rtimes$Translations$so(2,1)$ Figure 2: Schematic overview of
algebras relevant to asymptotic symmetries in three dimensions. The following
properties/relations are encoded in the boundaries of the vertices: round
shape $\hat{=}$ semi-simple, red $\hat{=}$ infinite dimensional, blue
$\hat{=}$ finite dimensional, and the arrows: violet $\hat{=}$ real forms,
green $\hat{=}$ finite dimensional embeddings, yellow $\hat{=}$ contraction
limit, light-blue $\hat{=}$ central extension.

## Quantum Groups on Asymptotic Symmetry Algebras

In this chapter, the consistent quantum groups on the symmetry algebras of
three dimensional asymptotically (anti) de Sitter spacetimes are analyzed. To
this end, we first classify Lie bialgebra structures and for several examples
we will explicitly construct the associated Hopf algebras and investigate
their properties and also restrictions they might impose on the symmetries
themselves based on both physical and mathematical constraints. Subsequently
the asymptotically flat limit and the four dimensional case are considered and
compared.

### Deformations of Asymptotically (Anti) de Sitter Symmetries in 3D Gravity

#### Lie Bialgebra Structures on $\mathfrak{W}\oplus\mathfrak{W}$

Recall from section Coboundary and Triangular Lie Bialgebras that all Lie
bialgebra structures on an algebra $\mathfrak{g}$ are either coboundary or
have a cobracket from the cohomology group
$H^{1}(\mathfrak{g},\bigwedge^{2}\mathfrak{g})$. In Appendix Proof of the
Cohomology Theorems, the following theorems are proven999In [89] the authors
independently prove a slightly more general result than our first theorem at
the cost of a longer proof. The proofs presented here conceptually follow the
proof of $H^{1}(\mathfrak{W},\mathfrak{W})=\\{0\\}$ in [90].

######  Theorem 1.

The first cohomology group
$H^{1}\left(\mathfrak{W},\bigwedge^{2}\mathfrak{W}\right)$ vanishes.

######  Theorem 2.

The first cohomology group
$H^{1}\left(\mathfrak{W}\oplus\mathfrak{W},\bigwedge^{2}\mathfrak{W}\oplus\mathfrak{W}\right)$
vanishes.

Therefore all LBA can be obtained from a classical r-matrix satisfying the
mCYBE (14). This is also true for the LBA on the symmetry algebra of globally
(A)dS space, the real forms of $\mathfrak{o}(4,\mathbb{C})$, as a consequence
of the Whitehead lemma. What is different, though, is that in the infinite
dimensional algebras $\mathfrak{W}$ and $\mathfrak{W}\oplus\mathfrak{W}$ there
is no ad-invariant element
$\Omega\in\left(\bigwedge^{3}\mathfrak{W}\right)_{\mathfrak{W}}$, resp.
$\left(\bigwedge^{3}(\mathfrak{W}\oplus\mathfrak{W})\right)_{\mathfrak{W}\oplus\mathfrak{W}}$
anymore. This follows easily from

$\displaystyle 0\overset{!}{=}$ $\displaystyle
L_{m}\triangleright\sum_{i}\alpha_{i}L_{p_{i}}\wedge L_{q_{i}}\wedge
L_{s_{i}}$ $\displaystyle=$
$\displaystyle\sum_{i}\alpha_{i}((m-p_{i})L_{p_{i}+m}\wedge L_{q_{i}}\wedge
L_{s_{i}}+(m-q_{i})L_{p_{i}}\wedge L_{q_{i}+m}\wedge L_{s_{i}}$
$\displaystyle+(m-s_{i})L_{p_{i}}\wedge L_{q_{i}}\wedge L_{s_{i}+m})$ (226)

if we choose $m>0$ and focus on the term with
$s_{i^{\prime}}>s_{i},p_{i},q_{i}$ without loss of generality for the Witt
algebra. Similarly, one shows that

$\displaystyle\left(\bigwedge^{3}(\mathfrak{W}\oplus\mathfrak{W})\right)_{\mathfrak{W}\oplus\mathfrak{W}}=\\{0\\},\qquad\left(\bigwedge^{2}\mathfrak{W}\right)_{\mathfrak{W}}=\left(\bigwedge^{2}(\mathfrak{W}\oplus\mathfrak{W})\right)_{\mathfrak{W}\oplus\mathfrak{W}}=\\{0\\}.$
(227)

The latter implies that the map from an r-matrix to the cobracket derived from
it is injective. Otherwise, i.e. if there was an ad-invariant element $a\wedge
b\in\bigwedge^{2}\mathfrak{W}$, the cobrackets from $r$ and $r+a\wedge b$
would be identical. Furthermore, as noted in section Coboundary and Triangular
Lie Bialgebras, only r-matrices belonging to different orbits under
automorphisms of the algebra lead to inequivalent LBA.

Thus, one can conclude that all LBA on $\mathfrak{W}\oplus\mathfrak{W}$ are
uniquely given by triangular r-matrices that are classified up to
automorphism. The simple observation that, in contrast to the situation for
the embedded $\mathfrak{o}(4,\mathbb{C})$, only triangular LBA are
coassociative will exclude a wide range of deformations.

#### Classification of Triangular r-matrices

Since the Witt algebra and its central extension, the Virasoro algebra, play a
vital role in various parts of math and physics, including string theory [91],
there were attempts to classify all LBA structures on them, most notably by
[92]. These results remained incomplete though except for some subalgebras,
including the one-sided Witt algebra $\mathfrak{W_{+}}$ (generated by
$\\{L_{m},m\geq-1\\}$), where all r-matrices were shown to be r-matrices of
one of the embeddings. For the two copies of the Witt algebra that are studied
here it is also easy to see that e.g. any element of the form

$\displaystyle\left(\sum_{i}\alpha_{i}L_{i}\right)\wedge\left(\sum_{j}\beta_{j}\bar{L}_{j}\right)$
(228)

is a triangular r-matrix.

However, one has to take into account that the asymptotic symmetry is
spontaneously broken in the bulk in the sense that the vacua related by
supertranslations and superrotations are physically distinguishable (cf.
section Asymptotic Flatness and [61]). There is a correspondence between these
vacua and the embeddings of Poincaré or (A)dS subalgebras which leave the
associated vacuum invariant. Therefore, we require that the restriction of the
Hopf algebra deformed by the twist associated with a given r-matrix to the
(A)dS algebra (which is the $n=1$ embedding for an observer) is a sub-Hopf
algebra and we are interested in r-matrices from (real forms of)
$\mathfrak{o}(4,\mathbb{C})$, where $\\{H,E_{\pm},\bar{H},\bar{E}_{\pm}\\}$ is
replaced with the embedding101010In the following, we often leave the
embedding characterized by the index $n$ unspecified. Also note that from a
mathematical point of view there is no reason the generators that appear in
the r-matrix have to be part of a specific embedding and we will comment on
this possibility later on.. Note that while in the case of positive $\Lambda$
the involution mixes left and right-handed elements, this is not happening for
negative $\Lambda$, leaving the potential possibility to use different
embeddings for them.

The classification of r-matrices of (real forms of)
$\mathfrak{o}(4,\mathbb{C})$ was done in [95] and has been discussed as well
in [94] and [87]. Filtering out the quasi-triangular r-matrices we are left
with

$\displaystyle r_{I}$
$\displaystyle=\chi(E_{+}-\bar{E}_{+})\wedge(H+\bar{H}),$ (229) $\displaystyle
r_{II}$ $\displaystyle=\chi E_{+}\wedge
H+\bar{\chi}\bar{E}_{+}\wedge\bar{H}+\zeta E_{+}\wedge\bar{E}_{+},$ (230)
$\displaystyle r_{III}$ $\displaystyle=\eta H\wedge\bar{H}.$ (231)
$\displaystyle r_{V}$ $\displaystyle=\bar{\chi}\bar{E}_{+}\wedge\bar{H}+\rho
H\wedge\bar{E}_{+}.$ (232)

An important caveat remains; the classification (229)-(232) is up to
automorphisms of the $\mathfrak{o}(4,\mathbb{C})$ but not all of these
automorphisms are also automorphisms of $\mathfrak{W}\oplus\mathfrak{W}$, only
the inverse statement holds in general111111Here we understand the
automorphisms of $\mathfrak{W}\oplus\mathfrak{W}$ to be restricted to the
subalgebra $\mathfrak{o}(4)$ and write
$\text{Aut}^{\prime}(\mathfrak{W}\oplus\mathfrak{W})$ for this (and similar)
restriction(s) so that quotients are well defined.. Therefore, in Appendix
Classification of $r$-matrices the classification of triangular r-matrices on
$\mathfrak{o}(4)$ is revised along the lines of [95] but using only the
$\mathfrak{W}\oplus\mathfrak{W}$ automorphisms

$\displaystyle\varphi_{(\gamma,\bar{\gamma},\epsilon,\bar{\epsilon})}(L_{m})=\gamma^{m}\epsilon
L_{\epsilon
m},\quad\varphi_{(\gamma,\bar{\gamma},\epsilon,\bar{\epsilon})}(\bar{L}_{m})=\bar{\gamma}^{m}\bar{\epsilon}\bar{L}_{\bar{\epsilon}m},$
(233)
$\displaystyle\varphi^{\prime}(L_{m})=\bar{L}_{m},\quad\varphi^{\prime}(\bar{L}_{m})=L_{m},$
(234)

where $0\neq\gamma,\bar{\gamma}\in\mathbb{C},\epsilon,\bar{\epsilon}=\pm 1$.
As a result we obtain the following classes of r-matrices

$\displaystyle r_{1^{\prime}}$
$\displaystyle=\beta(L_{1}+L_{-1}+2L_{0})\wedge(\bar{L}_{1}+\bar{L}_{-1}+2\bar{L}_{0})+a_{1}+\bar{a}_{1},$
(235) $\displaystyle r_{2^{\prime}}$ $\displaystyle=\beta
L_{1}\wedge\bar{L}_{0}+a_{2},$ (236) $\displaystyle r_{3^{\prime}}$
$\displaystyle=\beta(L_{1}+\epsilon
L_{0}+\epsilon^{\prime}L_{-1})\wedge(\bar{L}_{1}+\bar{L}_{-1}+2\bar{L}_{0})+\bar{a}_{1}+(1-\epsilon)(1-\epsilon^{\prime})a_{2},$
(237) $\displaystyle r_{4^{\prime}}$
$\displaystyle=\beta_{1}L_{1}\wedge\bar{L}_{1}+\beta_{2}(L_{1}+\bar{L}_{1})\wedge(L_{0}+\bar{L}_{0}),$
(238) $\displaystyle r_{5^{\prime}}$
$\displaystyle=\beta(L_{1}+L_{-1})\wedge(\bar{L}_{1}+\bar{L}_{0})+a_{1},$
(239) $\displaystyle r_{6^{\prime}}$ $\displaystyle=(\beta
L_{1}+\beta_{0}L_{0}+\epsilon\beta
L_{-1})\wedge(\bar{\beta}\bar{L}_{1}+\bar{\beta}_{0}\bar{L}_{0}+\bar{\epsilon}\bar{\beta}\bar{L}_{-1}),$
(240) $\displaystyle r_{7^{\prime}}$
$\displaystyle=L_{1}\wedge(\bar{\beta}\bar{L}_{1}+\bar{\beta}\bar{L}_{0}+\epsilon\bar{\beta}\bar{L}_{-1})+a_{2},$
(241) $\displaystyle r_{8^{\prime}}$ $\displaystyle=\beta
L_{1}\wedge\bar{L}_{1}+a_{2}+\bar{a}_{2},$ (242)

where $\epsilon,\epsilon^{\prime},\bar{\epsilon}\in\\{0,1\\}$ and

$\displaystyle a_{1}$ $\displaystyle=\alpha(L_{1}\wedge L_{0}+L_{1}\wedge
L_{-1}-L_{-1}\wedge L_{0}),\quad a_{2}=\alpha L_{1}\wedge L_{0},$ (243)
$\displaystyle\bar{a}_{1}$
$\displaystyle=\bar{\alpha}(\bar{L}_{1}\wedge\bar{L}_{0}+\bar{L}_{1}\wedge\bar{L}_{-1}-\bar{L}_{-1}\wedge\bar{L}_{0}),\quad\bar{a}_{2}=\bar{\alpha}\bar{L}_{1}\wedge\bar{L}_{0}.$
(244)

In the complex $\mathfrak{W}\oplus\mathfrak{W}$ all the parameters in 235-242
can take values in $\mathbb{C}$ but for the real forms associated with the
involutions $\dagger,\ddagger$ (202), (203) constrains the choice of
parameters. For $\dagger$ in all classes except $r_{6^{\prime}}$ this enforces
$\beta,\beta_{1},\beta_{2},\alpha,\bar{\alpha}\in i\mathbb{R}$ and in
$r_{6^{\prime}}$ one can restrict
$\beta,\beta_{0},\bar{\beta},\bar{\beta}_{0}\in\sqrt{i}\mathbb{R}$ without
loss of generality. In the case of positive $\Lambda$, i.e. the involution
$\ddagger$, the reality condition is more restrictive. In particular,

$\displaystyle r_{1^{\prime}}:$ $\displaystyle\quad\alpha=\bar{\alpha}\in
i\mathbb{R},\beta\in\mathbb{R},\qquad r_{2^{\prime}}:\quad\beta=0,\alpha\in
i\mathbb{R},$ $\displaystyle r_{3^{\prime}}:$
$\displaystyle\quad\bar{\alpha}=0,\epsilon=\epsilon^{\prime}=1,\beta\in\mathbb{R},\qquad
r_{4^{\prime}}:\quad\beta_{1}\in\mathbb{R},\beta_{2}\in i\mathbb{R},$
$\displaystyle r_{5^{\prime}}:$ $\displaystyle\quad\text{excluded},\qquad
r_{6^{\prime}}:\quad\epsilon=\bar{\epsilon},\beta_{0}\bar{\beta}=-\frac{\beta^{*}}{\beta_{0}^{*}}\text{
or }(\beta_{0}=\bar{\beta}_{0}=0,\beta,\bar{\beta}\in\mathbb{R}),$
$\displaystyle r_{7^{\prime}}:$ $\displaystyle\quad\text{excluded},\qquad
r_{8^{\prime}}:\quad\beta\in\mathbb{R},\alpha=\bar{\alpha}\in i\mathbb{R}.$

As the r-matrices from (235)-(242) that are not included in (229)-(232) are at
least in $\mathfrak{o}(4)$ automorphism orbits containing them, one can always
use the inverse of the automorphisms to obtain the full twists. For example,
$r=(L_{1}-L_{-1})\wedge(\bar{L}_{1}-\bar{L}_{-1})$ (being automorphic to
$r_{6^{\prime}}$ with $\beta_{0}=\bar{\beta}_{0}=0,\epsilon=\bar{\epsilon}=1$
) is automorphic to $L_{0}\wedge\bar{L}_{0}\hat{=}r_{III}$ by

$\displaystyle\varphi(L_{1})=-\frac{1}{2}(L_{1}+L_{-1})+L_{0},\quad\varphi(L_{-1})=-\frac{1}{2}(L_{1}+L_{-1})-L_{0},\quad\varphi(L_{0})=\frac{1}{2}(L_{1}-L_{-1}).$
(245)

From the abelian twist for $r_{III}$

$\displaystyle\mathcal{F}_{III}=\exp(\eta L_{0}\wedge\bar{L}_{0})$ (246)

one then gets the twist

$\displaystyle\mathcal{F}=\left(\varphi^{-1}\otimes\varphi^{-1}\right)\mathcal{F}_{III}=\exp(\eta(L_{1}-L_{-1})\wedge(\bar{L}_{1}-\bar{L}_{-1})).$
(247)

#### Full Deformations via Twist

From section Drinfeld Twist Deformation, it is known that all triangular LBA
can be extended to a deformation in all orders by a twist satisfying the
2-cocycle condition (72). This does not necessarily entail an explicit method
of constructing the twists but for (the real forms of)
$\mathfrak{o}(4,\mathbb{C})$ all of them are given in [87]. By the final
remark in the previous section it follows that the twists corresponding to the
r-matrices (235)-(242) can be obtained.

##### Abelian Twist

The first example we construct in detail is the abelian twist (see [93])
corresponding to the r-matrix $r_{III}$ from (232). Its name is motivated by
the fact that the two elements used for its construction are commuting. While
there are several other r-matrices with this feature (like (228)), $r_{III}$
is special in the sense that it is the only r-matrix from elements that are
contained in all embeddings. Thus, the abelian twist deformation does not
single out any particular embedding.

The full abelian twist can be expressed as

$\displaystyle\mathcal{F}_{A}=\exp\left(\eta\bar{L}_{0}\wedge
L_{0}\right)\exp\left(\eta(L_{0}\otimes
L_{0}-\bar{L}_{0}\otimes\bar{L}_{0})\right),$ (248)

which factorizes into the twist $\mathcal{F}_{3^{\prime\prime}}$ from [87] and
a factor that only contributes symmetric deformations of the coproduct. This
particular construction is chosen to make contact with the deformations in the
flat limit $(\Lambda\rightarrow 0)$. Explicitly, by making use of the Hadamard
formula

$\displaystyle e^{A}Be^{-A}=\sum_{n=0}^{\infty}\frac{1}{n!}\underset{\text{n
times}}{[\underbrace{A,[A,...[A,}}B]...]$ (249)

for general elements of an algebra $A$ and $B$ one finds the deformed
coproducts

$\displaystyle\Delta_{\mathcal{F}_{3^{\prime\prime}}}(L_{m})=e^{-m\eta\bar{L}_{0}}\otimes
L_{m}+L_{m}\otimes e^{m\eta\bar{L}_{0}},$ (250)
$\displaystyle\Delta_{\mathcal{F}_{3^{\prime\prime}}}(\bar{L}_{m})=e^{m\eta
L_{0}}\otimes\bar{L}_{m}+\bar{L}_{m}\otimes e^{-m\eta L_{0}}$ (251)

and

$\displaystyle\Delta_{\mathcal{F}_{A}}(L_{m})=e^{-m\eta(\bar{L}_{0}+L_{0})}\otimes
L_{m}+L_{m}\otimes e^{m\eta(\bar{L}_{0}-L_{0})}$ (252)
$\displaystyle\Delta_{\mathcal{F}_{A}}(\bar{L}_{m})=e^{m\eta(\bar{L}_{0}+L_{0})}\otimes\bar{L}_{m}+\bar{L}_{m}\otimes
e^{-m\eta(L_{0}-\bar{L}_{0})}$ (253)
$\displaystyle\Delta_{\mathcal{F}_{A}}(L_{0})=L_{0}\otimes 1+1\otimes
L_{0},\quad\Delta_{\mathcal{F}_{A}}(\bar{L}_{0})=\bar{L}_{0}\otimes
1+1\otimes\bar{L}_{0}.$ (254)

The antipodes can also be inferred easily from

$\displaystyle m\circ(S\otimes\text{id})\circ\Delta=1\circ\epsilon$ (255)

and turn out to be

$\displaystyle S_{\mathcal{F}_{3^{\prime\prime}}}(L_{m})=$ $\displaystyle-
L_{m},$ $\displaystyle S_{\mathcal{F}_{3^{\prime\prime}}}(\bar{L}_{m})$
$\displaystyle=-\bar{L}_{m}$ (256) $\displaystyle S_{\mathcal{F}_{A}}(L_{m})=$
$\displaystyle-L_{m}e^{-2m\eta L_{0}},$ $\displaystyle
S_{\mathcal{F}_{A}}(\bar{L}_{m})$
$\displaystyle=-\bar{L}_{m}e^{2m\eta\bar{L}_{0}}.$ (257)

The abelian twist thus induces non-cocommutative coproducts, and consequently
the antipodes are non-trivial, except on $L_{0}$ and $\bar{L}_{0}$. Before
discussing further properties let us consider another example of a twist.

##### Jordanian Twist

One of the first known examples of twists besides of the abelian twist was the
Jordanian twist. In a two dimensional Lie algebra with elements $X,Y$ and (the
only non-abelian) bracket $[X,Y]=\alpha Y$ one has

$\displaystyle\mathcal{F}=\exp\left(X\otimes\log\left(1+\alpha
Y\right)\right).$ (258)

Because the Lorentz sector ( containing (super)rotations and boosts) of all
the relevant asymptotic symmetry algebras is semi-simple, there are always
Cartan elements, e.g. $L_{0},\bar{L}_{0}$ for
$\mathfrak{W}\oplus\mathfrak{W}$, that diagonalize the adjoint action and thus
have the structure of the two dimensional example. As a result, many of the
consistent twists are of this prototypical Jordanian form or have it as a
building block.

For the r-matrix $r_{I}$ one has

$\displaystyle\mathcal{F}_{J,n}=\exp\left(-\frac{1}{n}(L_{0}+\bar{L}_{0})\otimes\log\left(1+\tilde{a}(L_{n}-\bar{L}_{n})\right)\right),$
(259)

so that

$\displaystyle\Delta_{\mathcal{F}_{J,n}}(L_{0})$ $\displaystyle=L_{0}\otimes
1+1\otimes
L_{0}-\frac{1}{n}(L_{0}+\bar{L}_{0})\otimes\frac{d\sigma_{n}}{d(L_{n}-\bar{L}_{n})}nL_{n}$
$\displaystyle=L_{0}\otimes 1+1\otimes
L_{0}-\tilde{a}(L_{0}+\bar{L}_{0})\otimes L_{n}\Pi_{+n}^{-1},$ (260)
$\displaystyle\Delta_{\mathcal{F}_{J,n}}(\bar{L}_{0})$
$\displaystyle=\bar{L}_{0}\otimes
1+1\otimes\bar{L}_{0}+\tilde{a}(L_{0}+\bar{L}_{0})\otimes\bar{L}_{n}\Pi_{+n}^{-1},$
(261)

where

$\displaystyle\sigma_{n}\equiv\log\left(1+\tilde{a}(L_{n}-\bar{L}_{n})\right),\quad\Pi_{+n}=e^{\sigma_{n}}.$
(262)

Using (255) and the previous equations we find

$\displaystyle S_{\mathcal{F}_{J,n}}(L_{0})=$
$\displaystyle-(L_{0}+\tilde{a}(\bar{L}_{0}L_{n}+L_{0}L_{n}))\frac{\Pi_{+n}^{-1}}{1-\Pi_{+n}^{-1}(L_{n}-\bar{L}_{n})}$
$\displaystyle=$
$\displaystyle-(L_{0}+\tilde{a}(\bar{L}_{0}L_{n}+L_{0}L_{n})),$ (263)
$\displaystyle S_{\mathcal{F}_{J,n}}(\bar{L}_{0})=$
$\displaystyle-(\bar{L}_{0}-\tilde{a}(\bar{L}_{0}\bar{L}_{n}+L_{0}\bar{L}_{n})).$
(264)

For general generators we find

$\displaystyle\Delta_{\mathcal{F}_{J,n}}(L_{m})$
$\displaystyle=\mathcal{F}_{J,n}(L_{m}\otimes 1+1\otimes
L_{m})\mathcal{F}^{-1}_{J,n}$
$\displaystyle=L_{m}\otimes\Pi_{+n}^{\frac{m}{n}}+\sum_{l=0}^{\infty}\frac{1}{l!}\left(\frac{L_{0}+\bar{L}_{0}}{n}\right)^{l}\otimes\big{[}\sigma_{n},\big{[}...,\big{[}\sigma_{n},L_{m}\big{]}...\big{]},$
(265)

with

$\displaystyle[\sigma_{n},L_{m}]=$
$\displaystyle\bigg{[}-\sum_{j=0}^{\infty}\frac{(-\tilde{a})^{j}}{j}(L_{n}-\bar{L}_{n})^{j},L_{m}\bigg{]}$
(266) $\displaystyle\big{[}\left(L_{n}-\bar{L}_{n}\right)^{j},L_{m}\big{]}=$
$\displaystyle\sum_{s=0}^{j-1}(L_{n}-\bar{L}_{n})^{s}(n-m)L_{m+n}(L_{n}-\bar{L}_{n})^{j-s}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{j}\bigg{(}\sum_{s_{1}=k-1}^{j-1}...\sum_{s_{k}=0}^{s_{k-1}-1}L_{m+kn}(L_{n}-\bar{L}_{n})^{j-k}\prod_{p=0}^{k-1}(n-(m+pn))\bigg{)},$
(267)

where in the last line we iteratively commuted the $s$ terms in front of
$L_{m+kn}$ to the right. From this formula, it can be seen that in general
there is an infinite number of terms in the coproduct involving a tower of
infinitely many different generators. In the h-adic topology this is well
defined but in the light of the comments in section Specialization and
Classical Limit of the Standard Deformation one could ask in which cases
finite expressions are obtained. We find that this can be achieved when
restricting to (two copies of) the one-sided Witt algebra ($\mathfrak{W}_{-}$)
spanned by $\\{L_{m},\bar{L}_{m};m\leq 1\\}$. In that case there are only two
possible embeddings with $n=\pm 1$ and by choosing $n=1$ the sum over $k$ in
(266) terminates after min$\\{1-m,j\\}$ terms, instead of when $k=j$ which is
not finite as we sum $j$ to infinity. As a consequence also the sum over $l$
terminates after $(1-m)$ terms, only a finite number of generators appear and
as we will see also only a finite number of terms.

In that case we proceed from (265) with the identity

$\displaystyle\sum_{s_{1}=k-1}^{j-1}...\sum_{s_{k}=0}^{s_{k-1}-1}=\frac{j!}{(j-k)!k!}={j\choose
k}.$ (268)

It is straightforward to see that it holds for $k=1$. Now assume that it holds
for $k=k^{\prime},1\leq k^{\prime}<j$ and it follows that the lhs for
$k=k^{\prime}+1$ is

$\displaystyle\sum_{s^{\prime}_{1}=k^{\prime}}^{j-1}\sum_{s^{\prime}_{2}=k^{\prime}-1}^{s^{\prime}_{1}-1}...\sum_{s^{\prime}_{k^{\prime}}=0}^{s^{\prime}_{k^{\prime}-1}-1}=\sum_{s^{\prime}_{1}=k^{\prime}}^{j-1}{s^{\prime}_{1}\choose
k^{\prime}}={j\choose k^{\prime}+1},$ (269)

where in the last step the hockey stick identity was used. Thus, we proved
(268) by induction. Therefore,

$\displaystyle[\sigma_{1},L_{m}]=$
$\displaystyle-\sum_{j=0}^{\infty}\frac{(-\tilde{a})^{j}}{j}\big{[}\left(L_{n}-\bar{L}_{n}\right)^{j},L_{m}\big{]}$
$\displaystyle=$ $\displaystyle
L_{m}-\sum_{j=1}^{\infty}\sum_{k=1}^{\text{min}\\{j,1-m\\}}\frac{(-\tilde{a})^{j}}{j}{j\choose
k}L_{m+k}(L_{1}-\bar{L}_{1})^{j-k}\prod_{p=0}^{k-1}(1-(m+p))$ (270)

and comparing with

$\displaystyle\frac{d^{k}\sigma_{1}}{dL_{1}^{k}}=-\sum_{j=k}^{\infty}\frac{(-\tilde{a})^{j}}{j}\frac{j!}{(j-k)!}(L_{1}-\bar{L}_{1})^{j-k},$
(271)

one finds that the summand in (270) and

$\displaystyle
L_{m}+\sum_{k=1}^{1-m}L_{m+k}\frac{d^{k}\sigma_{1}}{dL_{1}^{k}}\frac{1}{k!}\prod_{p=0}^{k-1}(1-(m+p))$
(272)

are identical. Splitting the sums in (270) and (272) according to

$\displaystyle\left(\sum_{j=1}^{1-m}\sum_{k=1}^{j}+\sum_{j=2-m}^{\infty}\sum_{k=1}^{1-m}\right)...$
$\displaystyle\left(\sum_{k=1}^{1-m}\sum_{j=k+1}^{1-m}+\sum_{k=1}^{1-m}\sum_{j=2-m}^{\infty}\right)...$

and using

$\displaystyle\sum_{j=1}^{1-m}\sum_{k=1}^{j}...=\sum_{k=1}^{1-m}\sum_{j=k}^{1-m}...,$
(273)

it follows that (266) is indeed given by (272) if the one-sided Witt algebra
with $n=1$ is considered. Plugging the result into (265) yields

$\displaystyle\Delta_{\mathcal{F}_{J,1}}(L_{m})=$ $\displaystyle
L_{m}\otimes\Pi_{+}^{m}+\sum_{l=1}^{1-m}\frac{1}{l!}\bigg{(}\sum_{k_{1}=0}^{1-m}\frac{1}{k!}\prod_{p_{1}=0}^{k_{1}-1}(1-(m+p_{1}))$