Quantum Klein Space and Superspace

Quantum Klein Space and Superspace

Rita FIORESI †, Emanuele LATINI †‡ and Alessio MARRANI §

R. Fioresi, E. Latini and A. Marrani

† Dipartimento di Matematica, Università di Bologna,
† Piazza di Porta S. Donato 5, I-40126 Bologna, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>
‡ INFN, Sez. di Bologna, viale Berti Pichat 6/2, 40127 Bologna, Italy

§ Museo Storico della Fisica e Centro Studi e Ricerche “Enrico Fermi”,
§ Via Panisperna 89A, I-00184, Roma, Italy<EMAIL_ADDRESS>
Received February 23, 2018, in final form June 15, 2018; Published online June
28, 2018

We give an algebraic quantization, in the sense of quantum groups, of the
complex Minkowski space, and we examine the real forms corresponding to the
signatures $(3,1)$, $(2,2)$, $(4,0)$, constructing the corresponding quantum
metrics and providing an explicit presentation of the quantized coordinate
algebras. In particular, we focus on the Kleinian signature $(2,2)$. The
quantizations of the complex and real spaces come together with a coaction of
the quantizations of the respective symmetry groups. We also extend such
quantizations to the $\mathcal{N}=1$ supersetting.

quantum groups; supersymmetry

17B37; 16T20; 20G42; 81R50; 17B60

##  1 Introduction

Undergoing crucial advances from Euclid to Newton and then from Minkowski to
Einstein, classical space-time has been one of the pillars in the conceptual
building of physics. The theory of relativity formulated by Einstein stressed
the crucial role played by concept of locality, which lies at the heart of the
formulation of effective field theory, one of the most powerful descriptive
frameworks to capture the features of a variety of physical systems at
relatively low-energies.

Thus, it is surprising that quantum physics had little or no influence at all
on the concepts of space and time, despite the essential nature of non-
locality being one of the striking (and most counter-intuitive, though)
features of quantum mechanics. If one excludes quantum gravity, all theories
characterized by quantum processes and dynamics ultimately relies on a
classical picture of space-time.

A quantum theory of gravity is essentially non-local, as it immediately
follows from the existence of a fundamental length, the Planck length. Indeed,
in both superstring/M-theory and loop quantum gravity, a minimal distance
measure occurs, hinting for the fact that the infinitely-differentiable space-
time manifold may be an illusory picture, breaking down at very small scales.
This suggests that space-time may be quantized; Snyder [62] was the first to
introduce the term “quantized space-time”. Consequently, quantum gravity
should face the challenging aim of understanding the nature of quantum space-
time, because it is reasonable to expect that, at some fundamental level, the
classical notion of space-time should be superseded by some suitable quantum
notion.

As it seems intuitively natural from the perspective of local effective field
theories, the usual approach to such fundamental conceptual issues in
contemporary theoretical physics is that quantum space-time should be related
to phenomena in which a strong gravitational field is involved, and/or that
the quantum modifications of classical space-time are limited to very tiny
regions, namely of the size of the Planck length. It thus follows that the
fundamental properties of quantum space-time would not be pertaining to
ordinary non-gravitational phenomena.

One of the basic ways in which the classical notion of space-time is
generalized is non-commutative geometry (cf., e.g., [19, 21, 26, 63], and
references therein). In this framework, the non-commutative algebra of
‘quantum space-time coordinates’ [19, 20, 39, 51, 59, 63] works as an ultra-
violet regulator, with the Planck length being the minimal resolution length
scale which generates uncertainty relations among non-commuting coordinates.
This would allow to describe the ‘fuzzy’ or ‘discrete’ nature of the space-
time, occurring only at very small distances or high energies [36, 47]. Even
if most formulations of non-commutative space-time suffer from explicit
violations of the Lorentz invariance, many examples of covariant non-
commutative space-times, retaining all global symmetries of the underlying
classical theory, have been considered in literature [45, 46, 62, 67].

Studies of simple dynamical models in quantized gravitational background (see,
e.g., [18, 34, 35]) also indicate that the Lie-algebraic space-time (Lorentz,
Poincaré) symmetries are modified into quantum symmetries, described by non-
cocommutative Hopf algebras; these latter, after Drinfeld, have been named
quantum deformations or quantum groups [22].

It is here worth recalling that most of the ‘quantum’ space-times investigated
so far are non-commutative versions of the Minkowski space-time, thus with
signature $(s,t)=(3,1)$ [4, 27, 45, 49, 53, 68]. In [54], Zumino, Wess,
Ogievetsky and Schmidke adopted an ad hoc approach for the quantization of the
$D=4$ Minkowski space, exploiting the coaction of the Poincaré group on it. In
[13, 14, 15] the quantum deformation of the complex (chiral) Minkowski and
conformal superspaces was investigated by exploiting the formal machinery of
flag varieties developed in [27, 28]. Recently, in [17], another space-time
quantization based on the spinorial description, namely a string theory
inspired ${\rm Spin}(3,1)$ worldsheet action, was introduced. Moreover, some
recent studies considered the construction of non-commutative space-times with
non-vanishing cosmological constant, dealing with the intriguing issue of the
interplay between quantum gravity effects and the non-vanishing curvature of
space-time, with interesting cosmological consequences of Planck scale physics
(cf., e.g., [1] and references therein).

As evident from above, in the context of the study of non-commutative space-
time structures at Planckian distances, the procedure of deformation of the
space-time coordinates and of the space-time symmetries plays a key role. In
this respect, one of the main approaches to the classification of deformations
has been provided by the theory of classical $r$-matrices [6, 16, 48, 61];
concerning the various possible signatures in $D=4$, recently, in [8, 9, 10]
the deformations of the orthogonal Lie algebras $\mathfrak{o}(4-k,k)$ (with
$k=0,1,2$), were investigated (together with the quaternionic real form
$\mathfrak{o}^{\ast}(4)$), thus dealing with all possible real forms of
$\mathfrak{o}(4;\mathbb{C})$.

In the present paper, we exploit a novel procedure which allows us to
investigate the quantization deformations of the $D=4$ space-time in all
possible signatures; in particular, we focus on the less known case with two
timelike and two spacelike dimensions, namely on the Kleinian (also named
ultrahyperbolic) signature $(s,t)=(2,2)$. The corresponding real form of
$\mathfrak{o}(4;\mathbb{C})$ is $\mathfrak{o}(2,2)$, which is also used in
two-dimensional double field theory (see, e.g., [41, 60]) or employed as $D=3$
AdS Lie algebra111The quantum deformation of the $D=3$ AdS group was
constructed in [5].. Geometries in Kleinian signature currently remains a vast
and yet unexplored realm, with a rich mathematical structure, investigated
only in a few papers (cf., e.g., [11, 23, 24, 25, 40, 42]). The study of the
Klein signature should not be considered as a mere mathematical
divertissement, but rather its interest stems from relevant physical
motivations. Just to name a few, the study of symmetries of scattering
amplitudes in super Yang–Mills theories and in supergravity stressed out the
relevance of Kleinian signature, especially in $D=4$; in fact, in [55] Ooguri
and Vafa showed that $\mathcal{N}=2$ superstring is characterized by critical
dimension $D=4$, with bosonic part given by a self-dual metric of signature
$s=t=2$. Moreover, $4$-dimensional Kleinian signature essentially pertains to
twistors [56], which provide a powerful computational tool of scattering
amplitudes [65]. In [31] the $4$-dimensional Klein space $\mathcal{M}_{2,2}$
was studied, through its definition inside the related Klein-conformal space,
along with its supersymmetric extensions, namely the Klein $\mathcal{N}=1$
superspace $\mathcal{M}_{2,2|1}$ and the corresponding Klein-conformal
$\mathcal{N}=1$ superspace. To the best of our knowledge, this, together with
our previous work [31], is the first investigation of the superspace with
Kleinian (bosonic) signature.

A direct approach to quantum deformations stems from [30, 31], and it exhibits
an intrinsic elegance based on split algebras $\mathbb{A}_{s}$’s in particular
the split complex numbers $\mathbb{C}_{s}$. This is the subject of the present
paper, which, as mentioned, will consider the quantum deformation of both real
Klein and Klein-conformal $\mathcal{N}=1$ spaces. We will not pursue the
aforementioned approach based on $r$-matrices, but rather we will deal with
the procedure introduced by Manin [50]; it is here worth pointing out that
Manin’s approach is equivalent to the one based on $r$-matrices, at least for
${\rm SL}(n)$ groups, which for $n=4$ is the case under investigation. In the
1980’s, developing ideas in part due to Penrose a few years earlier [57], he
introduced the approach to $D=4$ Minkowski space as the manifold of the points
of a big cell in the Grassmannian of complex two-dimensional subspaces of a
complex four dimensional space (twistor space), extending this framework to
the supersymmetric ($\mathcal{N}=1$) case [52]. After some general but rather
abstract studies in [44, 64], the study of quantization of flag manifolds and
of complex Minkowski space à la Manin was worked out in [27], in which were
also given the two involutions of the quantum complex Minkowski space,
respectively defining the real Minkowski space-time and the real Euclidean
space in four dimensions.

The plan of the paper is as follows.

In Section 2 we give a brief account of the classical Minkowski and Klein
spaces, both in the complex and real setting, together with their symmetry
groups.

In Section 3, we consider quantum groups and their homogenous spaces. We
approach the theory of deformation according to Manin [51], that is, we
quantize the coordinate rings of our algebraic geometric objects, spaces and
groups. This approach is equivalent to the $r$-matrix formulation, proposed
originally by Fadeev et al. in [26] (see also [16] for a complete account, and
references therein). Our approach is slightly more general than the ones
present the literature, since we take our ground ring $A$ to be a $k$-algebra,
where $k$ is $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{C}_{s}$. Observe that one
could also furtherly generalize this approach by substituting the complex and
split complex units $i$ and $j$ with the so called dual or parabolic one
$\epsilon$, with $\epsilon^{2}=0$. The use of the parabolic unit plays a
relevant role within the context of non-Euclidean spaces [66] and it was
applied in the framework of quantum groups and quantum Cayley–Klein algebras
in [2, 3, 38].

As we shall see, this allows a unified treatment for all signatures of the
real forms of the complex Minkowski space, and provides a natural
generalization to the quantum supersetting. Furthermore, in our treatment, the
actions of the symmetry groups get quantized in tandem with the homogeneous
spaces, though we have to interpret them as coactions, since, in the quantum
group setting, as usual, the geometric space is replaced by its algebra of
functions (in this very case, the algebra of polynomial functions).

In Section 4 we examine the real forms of the quantum spaces introduced in the
previous section, thus producing a deformation of Euclidean, Minkowski and
Klein spaces, together with their metrics. We also compute explicitly the
commutation relations among generators for the quantum rings of the Euclidean,
Minkowski and Klein spaces, thus giving a presentation of such quantum rings.
The metrics appear naturally as the quantum determinants of suitable quantum
matrices. Our approach allows to obtain at one stroke also the quantization of
the symmetry groups: we in fact obtain also the quantum Poincaré groups as
well as a coaction of such quantum groups on the corresponding quantum spaces.

Finally, in Section 5 we generalize the constructions of the previous sections
to the $\mathcal{N}=1$ supersetting. In particular, in Section 5.1 we
introduce the notion of quantum supergroup, whereas in Section 5.2 we consider
the quantum chiral Minkowski superspace. Then, for brevity’s sake, in Section
5.3 we focus (working on $\mathbb{C}_{s}$) on the real form associated to the
Klein involution.

##  2 The classical spaces

In this section we describe the classical Minkowski and Klein spaces using an
approach, which turns out particular fruitful for our subsequent treatment of
the quantizations of these spaces together with their symmetry groups and real
forms.

###  2.1 The (split) complex Minkowski space

The complex Minkowski space can be realized inside the conformal space
$\mathrm{SL}_{4}(\mathbb{C})/P$, $P$ being an upper maximal parabolic, as an
open set, called the big cell. It admits a natural action of the Poincaré
group and Weyl dilations, which sit as subgroups inside the conformal group
(we address the reader to [32, 33] for more details) that reads as

$\displaystyle\underbrace{\begin{pmatrix}x&0\\\
tx&y\end{pmatrix}}_{\textrm{Poincar\'{e}$\times$dilations}},\underbrace{\begin{pmatrix}I_{2}\\\
n\end{pmatrix}}_{\textrm{big cell}}\mapsto\underbrace{\begin{pmatrix}I_{2}\\\
ynx^{-1}+t\end{pmatrix}}_{\textrm{big cell}},$ (2.1)

where all of $x$, $y$, $t$ and $n$ are $2\times 2$ matrices with complex
coefficients. Notice that $\mathrm{diag}(x,y)$ parametrizes dilations, while
translations are represented by the matrix $t$. We then identify the complex
Minkowski space with the translational part of the Poincaré group, that is the
space of $2\times 2$ complex matrices. Note that the subgroup

$\displaystyle H=\left\\{\begin{pmatrix}x&0\\\
0&y\end{pmatrix},x,y\in\mathrm{SL}_{2}(\mathbb{C})\right\\}$

preserve the determinant of $n$, that we then identify with the norm of a
vector; this establishes the homomorphism

$\displaystyle\mathrm{SL}_{2}(\mathbb{C})\times\mathrm{SL}_{2}(\mathbb{C})\rightarrow\textrm{SO}_{4}(\mathbb{C}).$

In [31] we have shown how this picture can be consistently generalized by
substituting $\mathbb{C}$ with $\mathbb{C}_{s}$ in order to consider a more
general picture and different real forms; we will refer to it generally as the
complex Minkowski space $\mathcal{M}^{k}_{4}$, and it will be clear from the
context which algebra we use; the determinant of the various real forms will
naturally select different pseudo-Riemannian signatures.

###  2.2 Real forms and involutions

Let $k=\mathbb{C}$ or $\mathbb{C}_{s}$ and $\mathcal{A}$ be a commutative
algebra over $k$. An involution is a map
$*\colon\mathcal{A}\longrightarrow\mathcal{A}$ satisfying the properties

$\displaystyle(\alpha f+\beta
g)^{*}=\bar{\alpha}f^{*}+\bar{\beta}g^{*},\qquad(fg)^{*}=f^{*}g^{*},\qquad(f^{*})^{*}=f,$

where $f,g\in\mathcal{A}$, $\alpha,\beta\in k$ and the bar indicates the usual
conjugation in $k$. In this setting, we do not need to specify whether the
involution is multiplicative or antimultiplicative, since we are working with
commutative algebras, while in the quantum case this choice will be crucial.
We then consider the following three involutions on functions on
$\mathbb{C}[\mathrm{M}_{2}(k)]$, the algebra of polynomials on the matrices:

$\displaystyle*_{M}\colon\ \begin{pmatrix}a&b\\\
c&d\end{pmatrix}\mapsto\begin{pmatrix}a&c\\\ b&d\ \end{pmatrix},$
$\displaystyle*_{E}\colon\ \begin{pmatrix}a&b\\\
c&d\end{pmatrix}\mapsto\begin{pmatrix}d&-c\\\ -b&a\end{pmatrix},$
$\displaystyle*_{K}\colon\ \begin{pmatrix}a&b\\\
c&d\end{pmatrix}\mapsto\begin{pmatrix}a&b\\\ c&d\end{pmatrix}.$

We refer to $*_{M}$ (resp. $*_{E}$, $*_{K}$) as the Minkowski $($resp.
Euclidean, Klein$)$ involution. In fact, that, if $k=\mathbb{C}$, the fixed
points of these involutions are explicitly given as follows.

  * •

The fixed points of $*_{M}$ are Hermitian matrices222This is the quadratic
Jordan algebra over $\mathbb{C}$, and this justifies our notation; we denote
the space of hermitian matrices by $J_{2}^{\mathbb{C}}$ and we write a generic
element $m$ explicitly as

$\displaystyle m=\begin{pmatrix}x_{0}+x_{1}&x_{2}+ix_{3}\\\
x_{2}-ix_{3}&x_{0}-x_{1}\end{pmatrix}.$

Given $m,p\in J_{2}^{\mathbb{C}}$, one can define the bilinear form

$\displaystyle
g(m,p)=\tfrac{1}{2}\mathrm{tr}\big{(}m\left(p-\mathrm{tr}(p)\right)\big{)},\qquad
Q:=g(m,m)=-\det m$

with signature $(3,1)$. One can then identify Hermitian matrices with the real
Minkowski space $\mathcal{M}_{3,1}\sim\mathbb{R}^{3,1}$ that is naturally
equipped with the action of the Lorentz group,
$H=\mathrm{SL}_{2}(\mathbb{C})\times\mathrm{SL}_{2}(\mathbb{C})$, sitting as a
subgroup into the conformal group. $H$ acts on the Minkowski space by

$\displaystyle n\rightarrow ynx^{-1}.$

Requiring that this action maps $J_{2}^{\mathbb{C}}$ into itself forces
$x^{-1}=y^{\dagger}$, thus we recover the canonical action of
$\mathrm{SL}_{2}(\mathbb{C})$ on Hermitian matrices, establishing the
isomorphism $\mathrm{SL}_{2}(\mathbb{C})\sim\textrm{Spin}(3,1)$.

  * •

The fixed points of $*_{E}$ are complex matrices
$e\in\mathrm{M}_{2}(\mathbb{C})$ satisfying the relations
$e^{{\dagger}}=\omega e^{t}\omega^{-1}$, with $\omega$ being the usual
symplectic matrix. We denote this space by $N_{2}^{\mathbb{C}}$, and the
generic element will be decomposed as

$\displaystyle e=\begin{pmatrix}x_{0}+ix_{1}&x_{2}+ix_{3}\\\
-x_{2}+ix_{3}&x_{0}-ix_{1}\end{pmatrix}.$

On the space of those matrices we define the bilinear form

$\displaystyle
g(e,f)=\tfrac{1}{2}\mathrm{tr}\big{(}ef^{{\dagger}}\big{)},\qquad
Q:=g(e,e)=\det e$

that yields the Riemannian metric with signature $(4,0)$. We identify the
fixed points of $*_{E}$ with the real Euclidean space
$\mathcal{M}_{4,0}\sim\mathbb{R}^{4,0}$. The isometry group can be again
obtained by looking at the action of the subgroup $H$ and requiring it maps
${N}_{2}^{\mathbb{C}}$ into itself. A tedious calculation shows that in this
case one gets $x,y\in\mathrm{SU}_{2}$ as one would expect since
$\mathrm{SU}_{2}\times\mathrm{SU}_{2}\sim\textrm{Spin}(4)$.

  * •

The fixed points of $*_{K}$ are real matrices, and the generic element
$s\in\mathrm{M}_{2}(\mathbb{R})$ will be written, introducing a pair of light
cone coordinates, as

$\displaystyle s=\begin{pmatrix}x_{0}+x_{1}&x_{2}+x_{3}\\\
x_{2}-x_{3}&x_{0}-x_{1}\end{pmatrix}.$

We define the bilinear form

$\displaystyle g(s,p)=\tfrac{1}{2}\mathrm{tr}\big{(}s\omega
p^{t}\omega^{-1}\big{)},\qquad Q:=g(s,s)=\det s.$

In this case, we obtain a pseudo-Riemannian metric with signature $(2,2)$. The
isometry group can be again obtained by forcing $H$ to map real matrices into
real matrices. Explicitly, one gets that $x$ and $y$ must be real matrices and
thus elements of $\mathrm{SL}_{2}(\mathbb{R})$.

We can easily see that the described action preserves the bilinear form
defined above, and this is again not surprising since
$\mathrm{SL}_{2}(\mathbb{R})\times\mathrm{SL}_{2}(\mathbb{R})\sim\textrm{Spin}(2,2)$.
The proof relies on the fact that for $g\in\mathrm{SL}_{2}(\mathbb{R})$ it
holds that $g^{-1}=\omega g^{t}\omega^{-1}$.

Note that there is a fourth real, non-compact form of
$\mathfrak{o}(4;\mathbb{C)}$: the so-called quaternionic one,
$(\mathfrak{s)o}^{\ast}(4)\sim\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{su}(2)$;
it has been recently treated in [8, 9] (see also, e.g., [37] for recent
applications).

If $k=\mathbb{C}_{s}$, the fixed points of all the three given involutions
yield the Klein space, as one can naively check by substituting $j$ to $i$,
where $j$ is the imaginary unit of the split complex numbers $\mathbb{C}_{s}$
($j^{2}=1$). At the classical level, one thus obtains four equivalent
realizations of the $D=4$ Klein space $\mathcal{M}_{2,2}$. Note that, along
the way, one proves also the isomorphisms

$\displaystyle\mathrm{SL}_{2}(\mathbb{C}_{s})=\widetilde{\mathrm{SU}_{2}}\times\widetilde{\mathrm{SU}_{2}}=\textrm{Spin}(2,2),$

with $\widetilde{\mathrm{SU}_{2}}$ being the unitary $2\times 2$ matrices over
$\mathbb{C}_{s}$ with determinant one.

All this is summarized in the following table:

involution | fixed Points over $k=\mathbb{C},\mathbb{C}_{s}$ |  | space | isometry group
---|---|---|---|---
|  | $\mathbb{C}$ | $\mathcal{M}_{3,1}$ | $\mathrm{SL}_{2}(\mathbb{C})$
$*_{M}$ |  $J_{2}^{k}\colon\big{\\{}m\in\mathrm{M}_{2}(k)$ s.t. $m=m^{{\dagger}}\big{\\}}$ |  |  |
|  | $\mathbb{C}_{s}$ | $\mathcal{M}_{2,2}$ | $\mathrm{SL}_{2}(\mathbb{C}_{s})$
|  | $\mathbb{C}$ | $\mathcal{M}_{4,0}$ | $\mathrm{SU}_{2}\times\mathrm{SU}_{2}$
$*_{E}$ |  $N_{2}^{k}\colon\big{\\{}e\in\mathrm{M}_{2}(k)$ s.t. $e^{{\dagger}}=\omega e^{t}\omega^{-1}\big{\\}}$ |  |  |
|  | $\mathbb{C}_{s}$ | $\mathcal{M}_{2,2}$ | $\widetilde{\mathrm{SU}_{2}}\times\widetilde{\mathrm{SU}_{2}}$
|  | $\mathbb{C}$ | $\mathcal{M}_{2,2}$ | $\mathrm{SL}_{2}(\mathbb{R})\times\mathrm{SL}_{2}(\mathbb{R})$
$*_{K}$ | $\mathrm{M}_{2}(\mathbb{R})$ |  |  |
|  | $\mathbb{C}_{s}$ | $\mathcal{M}_{2,2}$ | $\mathrm{SL}_{2}(\mathbb{R})\times\mathrm{SL}_{2}(\mathbb{R})$

##  3 Quantum deformations

In this section we introduce the quantum deformations of the coordinate rings
of the spaces we have studied above. We construct such deformations together
with the coactions of the deformations of their symmetry groups, and we also
give quantizations of the real forms, which are compatible with such
coactions.

###  3.1 Quantum groups

We now introduce the quantum special linear group, as well as the notion of
quantum space.

Let $k=\mathbb{C},\mathbb{C}_{s}$ and $k_{q}=A\big{[}q,q^{-1}\big{]}$, where
$q$ is an indeterminate. In applications to quantum physics and quantum
gravity, the non-zero real parameter $q$ can be thought as related to the
Planck constant as $q\sim e^{h}$; the classical limit is achieved for
$h\longrightarrow 0$ and $q\longrightarrow 1$.

######  Definition 3.1.

We define quantum space $k_{q}^{n}$ the non commutative algebra

$\displaystyle k_{q}^{n}:=k_{q}\langle
x_{1},\dots,x_{n}\rangle/\big{(}x_{i}x_{j}-q^{-1}x_{i}x_{j},\,i<j\big{)},$

where $k_{q}\langle x_{1},\dots,x_{n}\rangle$ is the free algebra over the
ring $k_{q}$ with generators $x_{1},\dots,x_{n}$.

We define the quantum matrix bialgebra as

$\displaystyle\mathrm{M}_{q}(n):=k_{q}\langle a_{ij}\rangle/I_{M},$

where the indeterminates $a_{ij}$ satisfy Manin’s relations [51] generating
the ideal $I_{M}$

$\displaystyle a_{ij}a_{kj}=q^{-1}a_{kj}a_{ij},\quad i<k\qquad$ $\displaystyle
a_{ij}a_{kl}=a_{kl}a_{ij},\quad i<k,j>l\quad\text{or}\quad i>k,j<l,$
$\displaystyle a_{ij}a_{il}=q^{-1}a_{il}a_{ij},\quad j<l,\qquad$
$\displaystyle
a_{ij}a_{kl}-a_{kl}a_{ij}=\big{(}q^{-1}-q\big{)}a_{ik}a_{jl},\quad i<k,j<l,$
(3.1)

with comultiplication and counit given by

$\displaystyle\Delta(a_{ij})=\sum_{k}a_{ik}\otimes
a_{kj},\qquad\epsilon(a_{ij})=\delta_{ij}.$

The definitions of the comultiplication $\Delta$ and the counit $\epsilon$ are
summarized with the following notation

$\displaystyle\Delta\begin{pmatrix}a_{11}&\dots&a_{1n}\\\ \vdots&&\vdots\\\
a_{n1}&\dots&a_{nn}\end{pmatrix}=\begin{pmatrix}a_{11}&\dots&a_{1n}\\\
\vdots&&\vdots\\\
a_{n1}&\dots&a_{nn}\end{pmatrix}\otimes\begin{pmatrix}a_{11}&\dots&a_{1n}\\\
\vdots&&\vdots\\\ a_{n1}&\dots&a_{nn}\end{pmatrix},$
$\displaystyle\epsilon\begin{pmatrix}a_{11}&\dots&a_{1n}\\\ \vdots&&\vdots\\\
a_{n1}&\dots&a_{nn}\end{pmatrix}=\begin{pmatrix}1&\dots&0\\\ \vdots&&\vdots\\\
0&\dots&1\end{pmatrix},$

$k_{q}^{n}$ and $\mathrm{M}_{q}(n)$ are quantizations of the module $k^{n}$
and the matrix algebra $\mathrm{M}_{n}(k)$ respectively, that is, when $q=1$,
$k_{q}^{n}$ becomes the algebra of polynomials with coefficients in $k$ and
similarly holds for $\mathrm{M}_{n}(k)$. Since $\mathrm{M}_{n}(k)$ has a
natural action on $k^{n}$, we have that dually $\mathrm{M}_{q}(n)$ coacts on
$k_{q}^{n}$ (see [52], and [32, Chapter 5] for more details).

######  Proposition 3.2.

We have natural left and right coactions of the bialgebra of quantum matrices
on the quantum space $k_{q}^{n}$

$\displaystyle\lambda\colon\ k_{q}^{n}\longrightarrow\mathrm{M}_{q}(n)\otimes
k_{q}^{n},\qquad$ $\displaystyle\rho\colon\ k_{q}^{n}\longrightarrow
k_{q}^{n}\otimes\mathrm{M}_{q}(n),$ $\displaystyle\hphantom{\lambda\colon}{}\
x_{i}\mapsto\sum_{j}a_{ij}\otimes x_{j},\qquad$
$\displaystyle\hphantom{\rho\colon}{}\ x_{i}\mapsto\sum_{j}x_{j}\otimes
a_{ji}.$

Now we turn to the special linear group $\mathrm{SL}_{n}(k)$. The algebra of
polynomials with coefficients in $k$ on the special linear group is obtained
from the matrix bialgebra by imposing the condition that the determinant is
equal to $1$. We consider the quantum exterior algebra in $n$ variables

$\displaystyle\wedge^{n}_{q}:=k_{q}\langle\chi_{1},\dots,\chi_{n}\rangle/\big{(}\chi_{i}\chi_{j}+q\chi_{j}\chi_{i},\chi_{i}^{2},\,i<j\big{)}.$

Also $\wedge^{n}_{q}$ admits a (right) coaction of $\mathrm{M}_{q}(n)$, using
the transpose as one expects

$\displaystyle r\colon\
\wedge^{n}_{q}\longrightarrow\wedge^{n}_{q}\otimes\mathrm{M}_{q}(n),\qquad\chi_{i}\mapsto\sum_{j}a_{ij}\otimes\chi_{j}.$

######  Definition 3.3.

We define quantum determinant, the element in $\mathrm{M}_{q}(n)$
${\mathrm{det}}_{q}(a_{ij})$ (or ${\mathrm{det}}_{q}$ for short) defined by
the equation

$\displaystyle
r(\chi_{1}\cdots\chi_{n})={\mathrm{det}}_{q}(a_{ij})\otimes\chi_{1}\cdots\chi_{n}.$

A small calculation gives

$\displaystyle{\mathrm{det}}_{q}(a_{ij})=\sum_{\sigma}(-q)^{-\ell(\sigma)}a_{1\sigma(1)}\cdots
a_{n\sigma(n)}=\sum_{\sigma}(-q)^{-\ell(\sigma)}a_{\sigma(1)1}\cdots
a_{\sigma(n)n},$

where $\sigma$ runs through all the permutations of the first $n$ integers.

Furthermore $\det_{q}$ is central, i.e., commutes with all the elements in
$\mathrm{M}_{q}(n)$ and it is a group-like element, that is

$\displaystyle\Delta({\mathrm{det}}_{q})={\mathrm{det}}_{q}\otimes{\mathrm{det}}_{q}.$

######  Definition 3.4.

We define quantum special linear group over $k$ the algebra

$\displaystyle\mathrm{SL}_{q}(n)=\mathrm{M}_{q}(n)/({\mathrm{det}}_{q}-1).$

Notice that, most immediately, $\mathrm{SL}_{q}(n)$ is a quantum deformation
of the special linear group $\mathrm{SL}_{n}(k)$.

$\mathrm{SL}_{q}(n)$ is an Hopf algebra, with $\Delta$, $\epsilon$ inherited
by $\mathrm{M}_{q}(n)$ and antipode given by

$\displaystyle S_{q}(a_{ij})=(-q)^{j-i}A_{ji},$

where $A_{ji}$ is the quantum determinant in the quantum matrix bialgebra
generated by the indeterminates $a_{rs}$ with $r,s=1,\dots,n$ and $r\neq j$,
$s\neq i$.

The following definition will be useful later.

######  Definition 3.5.

We say that a certain set of indeterminates $\\{a_{ij}\\}$ is a quantum matrix
if the $a_{ij}$’s satisfy Manin’s relations (3.1). We also speak of a
$q^{-1}$-quantum matrix, meaning that it satisfies Manin’s relations (3.1)
with $q$ replaced by $q^{-1}$.

###  3.2 The quantum spaces

We want to quantize the setting introduced in Sections 2 and 2.1. We proceed
heuristically imposing the following equality (see also [32, Chapter 5])

$\displaystyle\begin{pmatrix}x&0\\\ tx&y\end{pmatrix}\begin{pmatrix}I_{2}&s\\\
0&I_{2}\end{pmatrix}=\begin{pmatrix}a_{11}&\dots&a_{14}\\\ \vdots&&\vdots\\\
a_{41}&\dots&a_{44}\end{pmatrix},$ (3.2)

which will yield the variables $x$, $y$, $t$ in terms of the coordinates
$a_{ij}$ of $\mathrm{SL}_{q}(n)$ (for the physical meaning of $x$, $y$, $t$
refer to Section 2.1). Equation (3.2) holds only when $D_{12}^{12}$ is
invertible. After a small calculation we get the equalities

$\displaystyle x=(x_{ij})=\begin{pmatrix}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{pmatrix},\qquad
t=(\boldsymbol{t}_{kj})=\begin{pmatrix}-q^{-1}D_{23}{D_{12}}^{-1}&D_{13}{D_{12}}^{-1}\\\
-q^{-1}D_{24}{D_{12}}^{-1}&D_{14}{D_{12}}^{-1}\end{pmatrix},$ $\displaystyle
s=\begin{pmatrix}-q{D_{12}}^{-1}D_{12}^{23}&-q{D_{12}}^{-1}D_{12}^{24}\\\
{D_{12}}^{-1}D_{12}^{13}&{D_{12}}^{-1}D_{12}^{14}\end{pmatrix},\qquad
y=(y_{kl})=\begin{pmatrix}D_{123}^{123}{D_{12}}^{-1}&D_{124}^{123}{D_{12}}^{-1}\\\
D_{123}^{124}{D_{12}}^{-1}&D_{124}^{124}{D_{12}}^{-1}\end{pmatrix},$

where $1\leq i,j\leq 2$, $3\leq k,l\leq 4$ and $D^{j_{1}\dots
j_{r}}_{i_{1}\dots i_{r}}$ is the quantum determinant obtained by taking the
rows $i_{1},\dots,i_{r}$ and columns $j_{1},\dots,j_{r}$ (we may omit the
column index, when we take the first $r$ columns).

The entries of the matrix $t$ correspond to the generators of the quantum
Minkowski space: in fact, we identify the Minkowski space with the
translational part of the Poincaré group as it is classically expressed in
formula (2.1). For the time being, we write the elements of $t$, $x$, $y$, $s$
in a matrix form, for convenience. We observe that $t$ is not a quantum
matrix, but it is close to it; we in fact obtain a quantum matrix by
exchanging the two columns, i.e.,
$\left(\begin{smallmatrix}\boldsymbol{t}_{32}&\boldsymbol{t}_{31}\\\
\boldsymbol{t}_{42}&\boldsymbol{t}_{41}\end{smallmatrix}\right)$ is a quantum
matrix. In details we have

$\displaystyle\boldsymbol{t}_{32}\boldsymbol{t}_{31}=q^{-1}\boldsymbol{t}_{31}\boldsymbol{t}_{32},\qquad\boldsymbol{t}_{32}\boldsymbol{t}_{42}=q^{-1}\boldsymbol{t}_{42}\boldsymbol{t}_{32},\qquad\boldsymbol{t}_{31}\boldsymbol{t}_{41}=q^{-1}\boldsymbol{t}_{41}\boldsymbol{t}_{31},\qquad\boldsymbol{t}_{42}\boldsymbol{t}_{41}=q^{-1}\boldsymbol{t}_{41}\boldsymbol{t}_{42},$
$\displaystyle\boldsymbol{t}_{32}\boldsymbol{t}_{41}=\boldsymbol{t}_{41}\boldsymbol{t}_{32}+\big{(}q^{-1}-q\big{)}\boldsymbol{t}_{31}\boldsymbol{t}_{42},\qquad\boldsymbol{t}_{31}\boldsymbol{t}_{42}=\boldsymbol{t}_{42}\boldsymbol{t}_{31}.$

This heuristic reasoning leads to the following definition in analogy with the
classical setting.

######  Definition 3.6.

We define quantum Poincaré group $\mathcal{P}_{q}$ as the algebra generated
inside $\mathrm{SL}_{q}(4)$ by the elements in $x$, $y$, $t$ described above.
This is a quantum deformation of the Poincaré group. We define quantum
Minkowski space the subring $\mathcal{M}_{4}^{k,q}$ in $\mathrm{SL}_{q}(4)$
generated by the elements $\boldsymbol{t}_{kj}$. This is a quantum deformation
of the coordinate algebra of the translations.

We define the quantum Lorentz group as

$\displaystyle\mathcal{L}_{q}=\mathcal{P}_{q}/(t,{\det}_{q}(x)-1,{\det}_{q}(y)-1).$

Our definition is very natural and in fact gives us a coaction of the quantum
Poincaré group on the Minkowski space.

######  Proposition 3.7.

The quantum Minkowski space admits a natural coaction of the quantum Poincaré
group

$\displaystyle\delta\colon\ \mathcal{M}_{4}^{k,q}$
$\displaystyle\longrightarrow\mathcal{P}_{q}\otimes\mathcal{M}_{4}^{k,q},$
$\displaystyle\boldsymbol{t}_{ij}$
$\displaystyle\longmapsto\sum_{s,r}y_{is}S(x_{rj})\otimes\boldsymbol{t}_{sr}+t_{ij}\otimes
1,$

where we rescale all indices $i,j,r,s=1,2$.

###### Proof.

This is a natural consequence of our construction. In fact the coaction
$\delta$ corresponds to the restriction of the comultiplication $\Delta$ of
$\mathrm{SL}_{q}(4)$. In particular we have

$\displaystyle\Delta\begin{pmatrix}x&0\\\
tx&y\end{pmatrix}=\begin{pmatrix}x&0\\\
tx&y\end{pmatrix}\otimes\begin{pmatrix}x&0\\\ tx&y\end{pmatrix},$

implying

$\displaystyle\Delta(tx)=tx\otimes x+y\otimes tx.$

Multiplying then by $S(x)\otimes S(x)$ on both sides one gets the result.
Notice that in our notation we identify the translation matrix $t_{ij}$ with
the coordinates on the Minkowski space $\boldsymbol{t}_{ij}$. ∎

This gives also a coaction of the quantum Lorentz group by setting the
generators $t=0$.

We have then that the quantum Minkowski space is a quantum homogeneous space
for the quantum Poincaré group and the quantum Lorentz group see also [32] for
more details on quantum homogeneous spaces in general.

######  Remark 3.8.

We consider two quantum planes $\mathbb{C}_{q}[\chi_{1},\chi_{2}]$ and
$\mathbb{C}_{q}[\psi_{1},\psi_{2}]$, that is
$\chi_{2}\chi_{1}=q\chi_{1}\chi_{2}$ and $\psi_{2}\psi_{1}=q\psi_{1}\psi_{2}$,
then one has the following morphism

$\displaystyle\mathbb{C}_{q}^{2}\stackrel{{\scriptstyle\ell}}{{\oplus}}\mathbb{C}_{q}^{2}$
$\displaystyle\longrightarrow\mathcal{M}^{\ell,q}_{\mathbb{C}},$
$\displaystyle\begin{pmatrix}\chi_{1}\\\
\chi_{2}\end{pmatrix}\otimes\begin{pmatrix}\psi_{2}&-q\psi_{1}\end{pmatrix}$
$\displaystyle\mapsto\begin{pmatrix}t_{11}&t_{12}\\\
t_{21}&t_{22}\end{pmatrix}.$

Thus, in analogy with the classical case, we can view the quantum plane as
quantum spinors.

##  4 Real forms of the quantum Minkowski space

In this section we want to construct real forms of the complex quantum
Minkowski space $\mathcal{M}_{4}^{k,q}$ as quantum homogeneous space for the
quantum Lorentz group. In other words, we want real forms, which are
compatible with the coaction $\delta$ of Proposition 3.7, setting to zero the
translation part in $\mathcal{P}_{q}$. As usual $k=\mathbb{C}$ or
$\mathbb{C}_{s}$ and we shall specify which case, depending on the real form
under consideration.

###  4.1 Quantum real forms and involutions

In the quantum setting, real forms correspond to involution of quantum
algebras. In particular, if $G_{q}$ is a quantum group, a real form of $G_{q}$
is a pair $(G_{q},*_{q})$, where $*_{q}$ is an antilinear, involutive,
antimultiplicative map respecting the comultiplication and the antipode (see,
e.g., [32, Section 5.3] for more details). For a quantum homogeneous space, we
give the following definition of real form.

######  Definition 4.1.

Let $V_{q}$ be a quantum homogeneous space for the quantum group $G_{q}$, with
coaction $\delta_{V_{q}}\colon V_{q}\longrightarrow G_{q}\otimes V_{q}$.
Assume $(G_{q},*_{G_{q}})$ is a real form of $G_{q}$. We say that
$(V_{q},*_{V_{q}})$ is a real form as quantum homogeneous space of $V_{q}$ if

  * •

$*_{V_{q}}\colon V_{q}\longrightarrow V_{q}$ is an involutive antilinear map,

  * •

$*_{V_{q}}$ is antimultiplicative, that is
$(ab)^{*_{V_{q}}}=b^{*_{V_{q}}}a^{*_{V_{q}}}$, or multiplicative
$(ab)^{*_{V_{q}}}=a^{*_{V_{q}}}b^{*_{V_{q}}}$,

  * •

$*_{V_{q}}$ preserves the coaction, that is

$\displaystyle\delta_{V_{q}}(a^{*_{V_{q}}})=\delta_{V_{q}}(a)^{*_{G_{q}}\times*_{V_{q}}}.$

######  Remark 4.2.

In the following, the compatibility condition with the coaction of the real
form of $G_{q}$ discussed above, will force us to consider also involution for
$G_{q}$ and $V_{q}$ that are not antimultiplicative333In literature,
multiplicative involution for quantum groups have been already considered, see
for example [7].; this is not crucial, since we are not interested in studying
particular representations of the real form of $G_{q}$, but we are interested
just on its action on $V_{q}$.

Inspired by the classical case (treated in Section 2.1), we now consider three
different involutions on $\mathcal{M}_{4}^{k,q}$. As we shall see later, they
are compatible with a suitable real form of the complex Poincaré quantum
group, thus consistently realizing real forms of $\mathcal{M}_{4}^{k,q}$ as
quantum homogeneous space for the suitable quantum group of symmetries:

$\displaystyle\bullet\ *_{{}_{Mq}}\colon\ $
$\displaystyle\mathcal{M}_{4}^{k,q}\longrightarrow\mathcal{M}_{4}^{k,q},$
$\displaystyle\begin{pmatrix}\boldsymbol{t}_{31}&\boldsymbol{t}_{32}\\\
\boldsymbol{t}_{41}&\boldsymbol{t}_{42}\end{pmatrix}\longmapsto\begin{pmatrix}\boldsymbol{t}_{31}&\boldsymbol{t}_{41}\\\
\boldsymbol{t}_{32}&\boldsymbol{t}_{42}\end{pmatrix},$ $\displaystyle
q\longmapsto q;$ $\displaystyle\bullet\ *_{{}_{Eq}}\colon\ $
$\displaystyle\mathcal{M}_{4}^{k,q}\longrightarrow\mathcal{M}_{4}^{k,q},$
$\displaystyle\begin{pmatrix}\boldsymbol{t}_{31}&\boldsymbol{t}_{32}\\\
\boldsymbol{t}_{41}&\boldsymbol{t}_{42}\end{pmatrix}\longmapsto\begin{pmatrix}-q^{-1}\boldsymbol{t}_{42}&\boldsymbol{t}_{41}\\\
\boldsymbol{t}_{32}&-q\boldsymbol{t}_{31}\end{pmatrix},$ $\displaystyle
q\longmapsto q;$ $\displaystyle\bullet\ *_{{}_{Kq}}\colon\ $
$\displaystyle\mathcal{M}_{4}^{k,q}\longrightarrow\mathcal{M}_{4}^{k,q},$
$\displaystyle\begin{pmatrix}\boldsymbol{t}_{31}&\boldsymbol{t}_{32}\\\
\boldsymbol{t}_{41}&\boldsymbol{t}_{42}\end{pmatrix}\longmapsto\begin{pmatrix}\boldsymbol{t}_{31}&\boldsymbol{t}_{32}\\\
\boldsymbol{t}_{41}&\boldsymbol{t}_{42}\end{pmatrix},$ $\displaystyle
q\longmapsto q.$

Notice that while $*_{Mq}$ and $*_{Eq}$ are antimultiplicative involutions of
$k_{q}[\boldsymbol{t}_{ij}]/I_{M}$ (with $I_{M}$ denoting the ideal of Manin’s
relations (3.1)) fixing $k_{q}$, $*_{Kq}$ gives a multiplicative involution of
$k_{q}$; the antimultiplicative property could be recovered sending $q$ to
$q^{-1}$. Let us stress once more that
$\left(\begin{smallmatrix}{\boldsymbol{t}}_{31}&{\boldsymbol{t}}_{32}\\\
{\boldsymbol{t}}_{41}&{\boldsymbol{t}}_{42}\end{smallmatrix}\right)$ is not a
quantum matrix (it becomes such by interchanging the first and the second
columns). This has clearly a consequence, when computing the commutation
relations and the quantum determinant, however we shall pay attention and
proceed nevertheless with this notation. We now proceed analyzing case by case
the quantum structure for both $k=\mathbb{C}$ or $\mathbb{C}_{s}$.

###  4.2 The quantum real Minkowski space

We consider now fixed points of $\mathcal{M}_{4}^{\mathbb{C},q}$ with respect
to $*_{{}_{Mq}}$, i.e., the real form
$\big{(}\mathcal{M}_{4}^{\mathbb{C},q},*_{{}_{Mq}}\big{)}$. In analogy with
the classical case, we parametrize it by the following “real variables”

$\displaystyle\widetilde{\boldsymbol{x}}_{0}=\tfrac{1}{2}(\boldsymbol{t}_{31}+\boldsymbol{t}_{42}),\qquad\widetilde{\boldsymbol{x}}_{2}=\tfrac{1}{2}(\boldsymbol{t}_{32}+\boldsymbol{t}_{41}),\qquad\widetilde{\boldsymbol{x}}_{3}=\tfrac{i}{2}(\boldsymbol{t}_{41}-\boldsymbol{t}_{32}),\qquad\widetilde{\boldsymbol{x}}_{1}=\tfrac{1}{2}(\boldsymbol{t}_{31}-\boldsymbol{t}_{42}),$

that are the following fixed points with respect the action of $*_{{}_{Mq}}$;
their commutation relations are listed below

$\displaystyle\widetilde{\boldsymbol{x}}_{2}\widetilde{\boldsymbol{x}}_{0}=q_{+}\widetilde{\boldsymbol{x}}_{0}\widetilde{\boldsymbol{x}}_{2}+iq_{-}\widetilde{\boldsymbol{x}}_{0}\widetilde{\boldsymbol{x}}_{3},$
$\displaystyle\widetilde{\boldsymbol{x}}_{2}\widetilde{\boldsymbol{x}}_{1}=q_{+}\widetilde{\boldsymbol{x}}_{1}\widetilde{\boldsymbol{x}}_{2}+iq_{-}\widetilde{\boldsymbol{x}}_{1}\widetilde{\boldsymbol{x}}_{3},$
$\displaystyle\widetilde{\boldsymbol{x}}_{3}\widetilde{\boldsymbol{x}}_{0}=q_{+}\widetilde{\boldsymbol{x}}_{0}\widetilde{\boldsymbol{x}}_{3}-iq_{-}\widetilde{\boldsymbol{x}}_{0}\widetilde{\boldsymbol{x}}_{2},$
$\displaystyle\widetilde{\boldsymbol{x}}_{3}\widetilde{\boldsymbol{x}}_{1}=q_{+}\widetilde{\boldsymbol{x}}_{1}\widetilde{\boldsymbol{x}}_{3}-iq_{-}\widetilde{\boldsymbol{x}}_{1}\widetilde{\boldsymbol{x}}_{2},$
$\displaystyle\widetilde{\boldsymbol{x}}_{0}\widetilde{\boldsymbol{x}}_{1}=\widetilde{\boldsymbol{x}}_{1}\widetilde{\boldsymbol{x}}_{0},$
$\displaystyle\widetilde{\boldsymbol{x}}_{2}\widetilde{\boldsymbol{x}}_{3}=\widetilde{\boldsymbol{x}}_{3}\widetilde{\boldsymbol{x}}_{2}+iq_{-}\big{(}\widetilde{\boldsymbol{x}}_{0}^{2}-\widetilde{\boldsymbol{x}}_{1}^{2}\big{)},$

with $q_{+}=\frac{1}{2}\big{(}q^{-1}+q\big{)}$ and
$q_{-}=\frac{1}{2}\big{(}q^{-1}-q\big{)}$. The metric is given by

$\displaystyle
Q_{q}=\det\nolimits_{q}\begin{pmatrix}\boldsymbol{t}_{32}&\boldsymbol{t}_{31}\\\
\boldsymbol{t}_{42}&\boldsymbol{t}_{41}\end{pmatrix}=\widetilde{\boldsymbol{x}}_{2}^{2}+\widetilde{\boldsymbol{x}}_{3}^{2}+q_{+}\big{(}\widetilde{\boldsymbol{x}}_{1}^{2}-\widetilde{\boldsymbol{x}}^{2}_{0}\big{)}.$

###  4.3 The quantum Klein space

We want to replicate this construction for the Klein space. This is the most
interesting case since, as we already observe, we have 4 different ways to
realize it.

  * •

We start with the real form $\big{(}\mathcal{M}_{4}^{k,q},*_{{}_{Kq}}\big{)}$;
in this case we can work with $\mathbb{C}$ and $\mathbb{C}_{s}$ at the same
time. We introduce the following real variables

$\displaystyle\boldsymbol{x}_{0}=\tfrac{1}{2}(\boldsymbol{t}_{31}+\boldsymbol{t}_{42}),\\!\\!\qquad\boldsymbol{x}_{2}=\tfrac{1}{2}(\boldsymbol{t}_{32}+\boldsymbol{t}_{41}),\\!\\!\qquad\boldsymbol{x}_{3}=\tfrac{1}{2}(\boldsymbol{t}_{41}-\boldsymbol{t}_{32}),\\!\\!\qquad\boldsymbol{x}_{1}=\tfrac{1}{2}(\boldsymbol{t}_{31}-\boldsymbol{t}_{42}),\\!$

with commutation relations:

$\displaystyle\boldsymbol{x}_{2}\boldsymbol{x}_{0}=q_{+}\boldsymbol{x}_{0}\boldsymbol{x}_{2}-q_{-}\boldsymbol{x}_{0}\boldsymbol{x}_{3},$
$\displaystyle\boldsymbol{x}_{2}\boldsymbol{x}_{1}=q_{+}\boldsymbol{x}_{1}\boldsymbol{x}_{2}-q_{-}\boldsymbol{x}_{1}\boldsymbol{x}_{3},$
$\displaystyle\boldsymbol{x}_{3}\boldsymbol{x}_{0}=q_{+}\boldsymbol{x}_{0}\boldsymbol{x}_{3}-q_{-}\boldsymbol{x}_{0}\boldsymbol{x}_{2},$
$\displaystyle\boldsymbol{x}_{3}\boldsymbol{x}_{1}=q_{+}\boldsymbol{x}_{1}\boldsymbol{x}_{3}-q_{-}\boldsymbol{x}_{1}\boldsymbol{x}_{2},$
$\displaystyle\boldsymbol{x}_{0}\boldsymbol{x}_{1}=\boldsymbol{x}_{1}\boldsymbol{x}_{0},$
$\displaystyle\boldsymbol{x}_{2}\boldsymbol{x}_{3}=\boldsymbol{x}_{3}\boldsymbol{x}_{2}+q_{-}\big{(}\boldsymbol{x}_{0}^{2}-\boldsymbol{x}_{1}^{2}\big{)},$

The quantum determinant now yields

$\displaystyle
Q_{q}=\det\nolimits_{q}\begin{pmatrix}\boldsymbol{t}_{32}&\boldsymbol{t}_{31}\\\
\boldsymbol{t}_{42}&\boldsymbol{t}_{41}\end{pmatrix}=\boldsymbol{x}_{2}^{2}-\boldsymbol{x}_{3}^{2}+q_{+}\big{(}\boldsymbol{x}_{1}^{2}-\boldsymbol{x}^{2}_{0}\big{)}.$

As expected, we get the deformation of a split signature norm.

We analyze now the other 2 options one has working within the algebra
$\mathbb{C}_{s}$ (refer to [31]). In this case we speak of a $\mathbb{C}_{s}$
quantum deformation.

  * •

Fixed points of $*_{Mq}$ on $\mathcal{M}_{4}^{\mathbb{\mathbb{C}}_{s},q}$ are
given by the fixed points

$\displaystyle\boldsymbol{y}_{0}=\tfrac{1}{2}(\boldsymbol{t}_{31}+\boldsymbol{t}_{42}),\\!\\!\qquad\boldsymbol{y}_{2}=\tfrac{1}{2}(\boldsymbol{t}_{32}+\boldsymbol{t}_{41}),\\!\\!\qquad\boldsymbol{y}_{3}=\tfrac{j}{2}(\boldsymbol{t}_{41}-\boldsymbol{t}_{32}),\\!\\!\qquad\boldsymbol{y}_{1}=\tfrac{1}{2}(\boldsymbol{t}_{31}-\boldsymbol{t}_{42}),$

whose commutations relations can be easily obtained by the previous one and
the norm turns out to be

$\displaystyle
Q_{q}=\det\nolimits_{q}\begin{pmatrix}\boldsymbol{t}_{32}&\boldsymbol{t}_{31}\\\
\boldsymbol{t}_{42}&\boldsymbol{t}_{41}\end{pmatrix}=\boldsymbol{y}_{2}^{2}-\boldsymbol{y}_{3}^{2}+q_{+}\big{(}\boldsymbol{y}_{1}^{2}-\boldsymbol{y}^{2}_{0}\big{)}.$

Note that this real form of the complex Minkowski space is isomorphic to the
previous one by the simple replacement $\boldsymbol{y}_{3}\rightarrow
j\boldsymbol{y}_{3}$.

  * •

In the end we can also consider the real form
$\big{(}\mathcal{M}_{4}^{\mathbb{C}_{s},q},*_{{}_{Eq}}\big{)}$; in this case a
convenient choice of fixed points is given by

$\displaystyle\tilde{\boldsymbol{y}}_{0}=\tfrac{j}{2}\big{(}q^{\frac{1}{2}}\boldsymbol{t}_{31}+q^{-\frac{1}{2}}\boldsymbol{t}_{42}\big{)},\qquad\tilde{\boldsymbol{y}}_{2}=\tfrac{1}{2}(\boldsymbol{t}_{32}+\boldsymbol{t}_{41}),\qquad\tilde{\boldsymbol{y}}_{3}=\tfrac{j}{2}(\boldsymbol{t}_{41}-\boldsymbol{t}_{32}),$
$\displaystyle\tilde{\boldsymbol{y}}_{1}=\tfrac{1}{2}\big{(}q^{\frac{1}{2}}\boldsymbol{t}_{31}-q^{-\frac{1}{2}}\boldsymbol{t}_{42}\big{)},$

and the metric, as well as the commutation relations coincide, formally, with
the previous one

We have thus obtained four isomorphic constructions of the $D=4$ quantum Klein
space $\mathcal{M}_{2,2}^{q}$. These correspond to the fixed points of the
involutions $\ast_{Mq}$, $\ast_{Eq}$ on $\mathcal{M}_{4}^{\mathbb{C}_{s},q}$,
and of $\ast_{Kq}$ on $\mathcal{M}_{4}^{k,q}$.

###  4.4 The quantum Klein group

In the classical setting we have that the metric $Q$ is preserved by the
action of the Klein group; it is natural then to expect that its quantum
deformation $Q_{q}$, introduced in the previous section is preserved under the
action of the quantum Klein group, viewed as a suitable real form of the
quantum Lorentz group $\mathcal{L}_{q}$. We now work for convenience with
$\mathbb{C}_{s}$ only and we suppress the subscript $k$. We can define three
antilinear involutions (i.e., real forms) on the quantum Lorentz group (for
the first one, see [32], as for the other two, [43, pp. 102 and 316]):

$\displaystyle\bullet\ *_{PM}\colon\ $
$\displaystyle\mathcal{L}_{q}\longrightarrow\mathcal{L}_{q},$ $\displaystyle
x_{ij}\longmapsto S(y_{ji}),$ $\displaystyle y_{ij}\longmapsto S(x_{ji}),$
$\displaystyle q\longmapsto q;$ $\displaystyle\bullet\ *_{PE}\colon\ $
$\displaystyle\mathcal{L}_{q}\longrightarrow\mathcal{L}_{q},$ $\displaystyle
x_{ij}\longmapsto S(x_{ji}),$ $\displaystyle y_{ij}\longmapsto S(y_{ji}),$
$\displaystyle q\longmapsto q;$ $\displaystyle\bullet\ *_{PK}\colon\ $
$\displaystyle\mathcal{L}_{q}\longrightarrow\mathcal{L}_{q},$ $\displaystyle
x_{ij}\longmapsto x_{ij},$ $\displaystyle y_{ij}\longmapsto y_{ij},$
$\displaystyle q\longmapsto q,$

where $S(x_{ij})$ denotes the antipode for the quantum matrix $x_{ij}$, that
is

$\displaystyle S\begin{pmatrix}x_{11}&x_{12}\\\
x_{21}&x_{22}\end{pmatrix}=\begin{pmatrix}x_{22}&-qx_{12}\\\
-q^{-1}x_{21}&x_{11}\end{pmatrix}$

and similarly for $S(y_{ij})$. Observe that $*_{PK}$ is multiplicative and not
antimultiplicative; this property could be again restored by sending $q$ to
$q^{-1}$. As one can readily check, these maps are well defined and give a
$*$-structure on $\mathcal{L}_{q}$.

We have the following proposition, whose proof consists of a tedious direct
calculation.

######  Proposition 4.3.

  1. $1.$

The quantum Lorentz group acts on the quantum Minkowski space as follows

$\displaystyle\lambda_{L}\colon\ \mathcal{M}_{4}^{q}$
$\displaystyle\longrightarrow\mathcal{L}_{q}\otimes\mathcal{M}_{4}^{q},$
$\displaystyle\boldsymbol{t}_{ij}$
$\displaystyle\longmapsto\sum_{s,r}y_{is}S(x_{rj})\otimes\boldsymbol{t}_{sr}.$

  2. $2.$

The maps $*_{PA}$, with $A=M,E,K$ define a $*$-structure and a the real form
of $\mathcal{L}_{q}$. Furthermore, the Klein space is quantum homogeneous with
respect to the coaction of the quantum Klein group. In other words we have

$\displaystyle(*_{PA}\times*_{Aq})\circ\lambda_{L}=\lambda_{L}\circ*_{Aq}.$

  3. $3.$

The quantum Klein quadratic form is a coinvariant for the action of the
quantum Lorentz group

$\displaystyle\lambda_{L}(Q_{q})=1\otimes Q_{q}.$

###### Proof.

The first point is a direct consequence of Proposition (3.7).

The relation $(*_{PA}\times*_{Aq})\circ\lambda_{L}=\lambda_{L}\circ*_{Aq}$
arises from a direct calculation; consider for example the element
$\boldsymbol{t}_{11}$ and the Minkowskian involutions, then one has

$\displaystyle\lambda_{L}(\boldsymbol{t}_{11})=y_{11}S(x_{11})\otimes\boldsymbol{t}_{11}+y_{12}S(x_{21})\otimes\boldsymbol{t}_{22}+y_{12}S(x_{11})\otimes\boldsymbol{t}_{21}+y_{11}S(x_{21})\otimes\boldsymbol{t}_{12}$
$\displaystyle\hphantom{\lambda_{L}(\boldsymbol{t}_{11})}{}=y_{11}x_{22}\otimes\boldsymbol{t}_{11}-q^{-1}y_{12}x_{21}\otimes\boldsymbol{t}_{22}+y_{12}x_{22}\otimes\boldsymbol{t}_{21}-q^{-1}y_{11}x_{21}\otimes\boldsymbol{t}_{12}.$

Applying then $(*_{PM}\times*_{Mq})$ and keeping in mind that the Minkowskian
involution is antimultiplicative one gets

$\displaystyle(*_{PM}\times*_{Mq})\circ\lambda_{L}(\boldsymbol{t}_{11})=y_{11}x_{22}\otimes\boldsymbol{t}_{11}-q^{-1}y_{12}x_{21}\otimes\boldsymbol{t}_{22}-q^{-1}y_{11}x_{21}\otimes\boldsymbol{t}_{12}+y_{12}x_{22}\otimes\boldsymbol{t}_{21}$
$\displaystyle\hphantom{(*_{PM}\times*_{Mq})\circ\lambda_{L}(\boldsymbol{t}_{11})}{}=\lambda_{L}(\boldsymbol{t}_{11}),$

and since $*_{Mq}(\boldsymbol{t}_{11})=\boldsymbol{t}_{11}$ the result follows
naturally. Similarly one proves the same for the other elements and
involutions.

To prove the point 3 it is enough to observe that, due to the Manin relation
and the parabolic structure we consider, one has $x_{ij}y_{km}=y_{km}x_{ij}$
and after some elementary reordering and algebras, one gets

$\displaystyle\lambda_{L}(Q_{q})=\det(y)\det(x)\otimes Q_{q}.$

Considering that ${\det}_{q}(x)=1={\det}_{q}(y)$, we obtain that the quantum
metric is coinvariant as claimed. ∎

Observe that this proposition tells us in fact that we have a well defined
coaction of $(\mathcal{L}_{q},*_{PA})$ on
$\big{(}\mathcal{M}_{4}^{q},*_{A}\big{)}$ preserving the quantum metric.

###  4.5 Formulation with the algebraic star product

In this section, we want to present the results obtained in the previous
section in terms of classical objects. In particular we want to take advantage
of the isomorphism between the algebra $\textrm{M}_{q}(2)$ and
$\big{(}\mathcal{O}(\mathcal{M}_{2,2})\big{[}q,q^{-1}\big{]},\star\big{)}\equiv
k_{q}[\tau_{41},\tau_{42},\tau_{31},\tau_{32}]$, where $\star$ is certain non-
commutative product naturally inherited from the underlying quantum group
structure. This approach and language is probably more familiar to physicists
because of its versatility. As mentioned before, the starting point for this
construction is the observation that the map

$\displaystyle\Phi_{q}\colon\ k_{q}[\tau_{41},\tau_{42},\tau_{31},\tau_{32}]$
$\displaystyle\longrightarrow\textrm{M}_{q}(2),$
$\displaystyle\tau_{41}^{m}\tau_{42}^{n}\tau_{31}^{p}\tau_{32}^{r}$
$\displaystyle\mapsto\boldsymbol{t}_{41}^{m}\boldsymbol{t}_{42}^{n}\boldsymbol{t}_{31}^{p}\boldsymbol{t}_{32}^{r}$

is a module isomorphism and thus has an inverse; $\Phi_{q}:$ is called
quantization map, and essentially it encodes a choice of ordering. It is here
worth stressing that with this choice of ordering one gets a basis for
$\textrm{M}_{q}(2)$ as proved in [15], but in principle one could make other
compatible choices. Next, we define the following non commutative product

$\displaystyle f\star g=\Phi_{q}^{-1}(\Phi_{q}(f)\Phi_{q}(g)),\qquad
f\in\big{(}\mathcal{O}(\mathcal{M}_{2,2})\big{[}q,q^{-1}\big{]}\big{)}.$

Using Manin’s relations (3.1), then one can easily construct the following

$\displaystyle\big{(}\tau_{41}^{a}\tau_{42}^{b}\tau_{31}^{c}\tau_{32}^{d}\big{)}\star\big{(}\tau_{41}^{m}\tau_{42}^{n}\tau_{31}^{p}\tau_{32}^{r}\big{)}=q^{-m(c+b)-d(n+p)}\big{(}\tau_{41}^{a+m}\tau_{42}^{b+n}\tau_{31}^{c+p}\tau_{32}^{d+r}\big{)}$
$\displaystyle\qquad{}+\sum\limits_{k=1}^{\min(d,m)}q^{(k-m)(c+b)+(k-d)(n+p)}F(k,q,d,m)\tau_{41}^{a+m-k}\tau_{42}^{b+k+n}\tau_{31}^{c+k+p}\tau_{32}^{d-k+r},$

where $F(k,q,d,m)$ is a function defined recursively (see [15] for more
detail). We comment that there exists a (unique) differential star product,
thus acting on $C^{\infty}$ functions, that coincides with the one given above
on polynomials.

We can now pullback the metric to the star product algebra, obtaining

$\displaystyle
Q=\Phi_{q}^{-1}(Q_{q})=x_{2}^{2}+x_{3}^{2}-qx_{0}^{2}-qx_{1}^{2},$

where the star product among polynomials on the real variables $x$ and is
naturally inherited from the $\boldsymbol{x}$ commutation relations.

##  5 The $\boldsymbol{\mathcal{N}=1}$ quantum Klein superspace
$\boldsymbol{\mathcal{M}_{2,2|1}^{q}}$

The extension of the natural construction that we have done in the previous
sections to the supersetting offers no difficulty, hence we will summarize
quickly the relevant definitions and the results. For all of the supergeometry
terminology we refer the reader to [12, 32].

###  5.1 Quantum supergroups

We start with the definition of quantum matrix superalgebra and quantum
special linear supergroup. Let $k=\mathbb{C}$ or $\mathbb{C}_{s}$, as in
beginning of Section 3 and $k_{q}=k\big{[}q,q^{-1}\big{]}$.

######  Definition 5.1.

We define quantum matrix superalgebra

$\displaystyle\mathrm{M}_{q}(m|n)\overset{\rm def}{=}k_{q}\langle
a_{ij}\rangle/I_{M},$

where $I_{M}$ is generated by the relations [51]

$\displaystyle
a_{ij}a_{il}=(-1)^{\pi(a_{ij})\pi(a_{il})}q^{(-1)^{p(i)+1}}a_{il}a_{ij},\qquad
j<l,$ $\displaystyle
a_{ij}a_{kj}=(-1)^{\pi(a_{ij})\pi(a_{kj})}q^{(-1)^{p(j)+1}}a_{kj}a_{ij},\qquad
i<k,$ $\displaystyle
a_{ij}a_{kl}=(-1)^{\pi(a_{ij})\pi(a_{kl})}a_{kl}a_{ij},\qquad
i<k,j>l\quad\text{or}\quad i>k,j<l,$ $\displaystyle
a_{ij}a_{kl}-(-1)^{\pi(a_{ij})\pi(a_{kl})}a_{kl}a_{ij}=\eta\big{(}q^{-1}-q\big{)}a_{kj}a_{il},\qquad
i<k,j<l,$

where

$\displaystyle\eta=(-1)^{p(k)p(l)+p(j)p(l)+p(k)p(j)}$

with $p(i)=0$ if $1\leq i\leq m$, $p(i)=1$ otherwise and
$\pi(a_{ij})=p(i)+p(j)$ denotes the parity of $a_{ij}$. $\mathrm{M}_{q}(m|n)$
is a bialgebra with the usual comultiplication and counit

$\displaystyle\Delta(a_{ij})=\sum a_{ik}\otimes
a_{kj},\qquad\epsilon(a_{ij})=\delta_{ij}.$

We define the special linear quantum supergroup $\mathrm{SL}_{q}(m|n)$ as
$\mathrm{M}_{q}(m|n)/(B_{q}-1)$, where $B_{q}$ is the quantum Berezinian,
which is a central element in $\mathrm{M}_{q}(m|n)$ (see [29, 58, 69] and [32,
Chapter 5, Section 5.4] for more details).

We now turn to the more relevant definitions for us, namely the quantum
Poincaré supergroup $\mathcal{SP}_{q}$ and the $4|1$ dimensional quantum
Minkowski superspace $\mathcal{M}^{k,q}$ as quantum homogeneous space, that is
together with a coaction of $\mathcal{SP}_{q}$ on it.

######  Definition 5.2.

We define quantum Poincaré supergroup $\mathcal{SP}_{q}$ as the quotient of
$\mathrm{SL}_{q}(m|n)$ by the generators $a_{ij}$, $\alpha_{k5}$,
$\alpha_{5l}$, $i,j=1,2$ or $i,j=3,4$ or $i=3,4$, $j=1,2$ and $k=1,2$,
$l=3,4$. In quantum matrix form

$\displaystyle\mathcal{SP}_{q}=\left(\begin{matrix}L&0&0\\\ M&R&\chi\\\
\phi&0&d\end{matrix}\right),$

where

$\displaystyle L=\left(\begin{matrix}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{matrix}\right),\qquad M=\left(\begin{matrix}a_{31}&a_{32}\\\
a_{41}&a_{42}\end{matrix}\right),\qquad R=\left(\begin{matrix}a_{33}&a_{34}\\\
a_{43}&a_{44}\end{matrix}\right),$
$\displaystyle\chi=\left(\begin{matrix}\alpha_{35}\\\
\alpha_{45}\end{matrix}\right),\qquad\phi=\left(\begin{matrix}\alpha_{53}&\alpha_{54}\end{matrix}\right),\qquad
d=a_{55}$

in terms of the generators $a_{ij}$, $\alpha_{kl}$ of $\mathrm{SL}_{q}(m|n)$,
the quantum special linear supergroup.

###  5.2 Quantum chiral Minkowski superspace

We now define the quantum Minkowski superspace $\mathcal{SM}^{k,q}$ as
generated by the matrices $t$ and $\tau$, in analogy with our heuristic
derivation in (3.2), obtained through the equality

$\displaystyle\begin{pmatrix}L&0&0\\\ tL&R&0\\\ \tau
L&\nu&d\end{pmatrix}\begin{pmatrix}I_{2}&s&\sigma\\\ 0&I_{2}&\rho\\\
0&0&1\end{pmatrix}=\begin{pmatrix}a_{11}&\dots&a_{14}&\alpha_{15}\\\
\vdots&&\vdots&\vdots\\\ a_{41}&\dots&a_{44}&\alpha_{45}\\\
\alpha_{51}&\dots&\alpha_{54}&a_{55}\end{pmatrix}.$

After a small calculation, the following result is achieved:

$\displaystyle
t=(\boldsymbol{t}_{kj})=\begin{pmatrix}-q^{-1}D_{23}{D_{12}}^{-1}&D_{13}{D_{12}}^{-1}\\\
-q^{-1}D_{24}{D_{12}}^{-1}&D_{14}{D_{12}}^{-1}\end{pmatrix},$
$\displaystyle\tau=(\tau_{5j})=\begin{pmatrix}-q^{-1}D_{25}{D_{12}}^{-1}&D_{15}{D_{12}}^{-1}\end{pmatrix}.$

As above, both $t$ and $\tau$ are quantum matrices, once the first and second
columns are interchanged.

######  Remark 5.3.

Notice that our definition is over $k$ and so works for $k=\mathbb{C}$, giving
the quantum complex Minkowski superspace, but also over $k=\mathbb{C}_{s}$,
hence giving the quantum $\mathbb{C}_{s}$ Minkowski superspace. The advantage
of our unified treatment is that, when looking at real forms, we will obtain
at once both the real Minkowski and real Klein quantum superspaces.

Our definition corresponds to identify the Minkowski superspace with the
translation supergroup inside the Poincaré supergroup as we did in Section
3.2.

######  Proposition 5.4.

The $\mathcal{N}=1$ quantum Minkowski superspace $\mathcal{{S}M}^{k,q}$ is a
quantum homogeneous superspace for the quantum Poincaré supergroup. The
coaction is explicitly given as

$\displaystyle\delta^{(s)}\colon\ \mathcal{M}^{k,q}$
$\displaystyle\longrightarrow\mathcal{SP}_{q}\otimes\mathcal{M}^{k,q},$
$\displaystyle\boldsymbol{t}_{ij}$ $\displaystyle\mapsto t_{ij}\otimes
1+\sum_{u,v}r_{iu}S(\ell_{vj})\otimes\boldsymbol{t}_{uv}+\sum_{v}\chi_{i5}S(\ell_{vj})\otimes\tau_{5v},$
$\displaystyle\tau_{5j}$ $\displaystyle\mapsto\psi_{5j}\otimes
1+\sum_{v}dS(\ell_{vj})\otimes\tau_{5v},$

where $R=(r_{ij})$, $L=(\ell_{kl})$. $($To ease the notation, we replace the
undetermined in $M$ with $NL$, for a suitable $N$ and similarly $\phi$ with
$\psi L$ for a suitable $\psi.)$

###### Proof.

This again follows naturally from our construction since the coaction of
$\delta^{(s)}$ corresponds to the restriction of the $\mathrm{SL}_{q}(4|1)$
coaction to the blocks $\boldsymbol{t}$ and $\tau$. ∎

###  5.3 Real forms

The chiral complex Minkowski superspace does not admit a physically
interesting real form, because the odd part is spinorial and in this case, the
(Weyl) semispinors are complex. In the unquantized setting, this defect is
fixed by considering extra odd coordinates and pairing them to obtain such a
real form (see [32, Chapter 4] for an exhaustive treatment and the references
within). A quantization of the quantum real Minkowski superspace presents then
difficulties, since the extra odd coordinates must also be quantized and
calculations become intricated.

On the other hand, the quantization of the chiral Klein superspace follows
very naturally in our construction. It should be here recalled that in the non
supersymmetric case we have defined three different involution yielding the
same geometrical structure; in the following, we will focus for simplicity on
the $\mathcal{N}=1$ superextension of the case associated to the Klein
involution.

We define the quantum real chiral Klein superspace as the pair
$\big{(}\mathcal{SM}^{k,q},{*_{SKq}}\big{)}$, where $*_{SKq}$ is the
antilinear involution

$\displaystyle*_{SKq}\colon\qquad\mathcal{SM}^{k,q}$
$\displaystyle\longrightarrow\mathcal{{S}M}^{k,q},$
$\displaystyle\begin{pmatrix}\boldsymbol{t}_{31}&\boldsymbol{t}_{32}\\\
\boldsymbol{t}_{41}&\boldsymbol{t}_{42}\end{pmatrix}$
$\displaystyle\longmapsto\begin{pmatrix}\boldsymbol{t}_{31}&\boldsymbol{t}_{32}\\\
\boldsymbol{t}_{41}&\boldsymbol{t}_{42}\end{pmatrix},$
$\displaystyle\begin{pmatrix}\tau_{51}&\tau_{52}\end{pmatrix}$
$\displaystyle\longmapsto\begin{pmatrix}\tau_{51}&\tau_{52}\end{pmatrix},$
$\displaystyle{q}$ $\displaystyle\longmapsto{q},$

which is the identity on the coordinates. Note that since we want it to be
multiplicative as in the even case, we need to send $q$ to $q$.

Our construction is compatible to what we have discussed in our previous
sections. The metrics are the ones discussed in Section 4.2, since the odd
variables do not modify them (see also [32, Chapter 4]).

$\big{(}\mathcal{SM}^{kq},{*_{SKq}}\big{)}$ is a real homogeneous quantum
superspace, when the antilinear multiplicative involution on the quantum
supergroup $\mathcal{SP}_{q}$ (which is the identity on the generators and
send $q$ to itself) is considered. One can in fact immediately see that
preserves the coaction as given in Proposition 5.4.

##  6 Conclusions

In this paper we propose an algebraic quantization, in the sense of quantum
groups, of the complex and split complex Minkowski spaces
$\mathcal{M}_{4}^{\mathbb{C},q}$ and $\mathcal{M}_{4}^{\mathbb{C}_{s},q}$
viewed as quantum homogenous spaces; we focus our attention on their real
forms, yielding the Lorentzian, Kleinian and Euclidean signatures $(3,1)$,
$(2,2)$, $(4,0)$. All this is summarized in the following table:

real quantum space | signature
---|---
$(*_{{}_{Kq}},\mathcal{M}_{4}^{\mathbb{C},q}$) | $(2,2)$
$(*_{{}_{Kq}}\mathcal{M}_{4}^{\mathbb{C}_{s},q})$ | $(2,2)$
$(*_{{}_{Mq}},\mathcal{M}_{4}^{\mathbb{C},q}$) | $(3,1)$
$(*_{{}_{Mq}},\mathcal{M}_{4}^{\mathbb{C}_{s},q})$ | $(2,2)$
($*_{{}_{Eq}},\mathcal{M}_{4}^{\mathbb{C},q})$ | (4,0)
$(*_{{}_{Eq}},\mathcal{M}_{4}^{\mathbb{C}_{s},q}$) | (2,2)

The beauty of our approach is that those spaces are naturally endowed with the
(co)action of the corresponding isometry quantum group, as we explicitly show;
moreover, for all these spaces, we give an explicit representation of the
deformed coordinates algebra. In this setting we also extend such analysis to
the supercase by constructing the chiral Klein superspace.

### Acknowledgments

We would like to thank Professors Francesco Bonechi, Meng-Kiat Chuah and Fabio
Gavarini for useful discussions and helpful comments. We also wish to thank
our anonymous referees for helpful comments, which have helped us to improve
the clarity of our paper. A.M. wishes to thank the Department of Mathematics
at the University of Bologna, for the kind hospitality during the realization
of this work.

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