* [25] Lina, G., Boutite, F., Tristan, A., Bes, M., Etienne, J. and Vandenesch, F. (2003). Bacterial Competition for Human Nasal Cavity Colonization: Role of Staphylococcal agr Alleles. Applied and Environmental Microbiology, 69 (1), pp. 18-23.
  * [26] Lowy, F., 2003. Antimicrobial resistance: the example of Staphylococcus aureus. Journal of Clinical Investigation, 111 (9), pp. 1265-1273.
  * [27] Macpherson, A. J. & Harris, N. L. Interactions between commensal intestinal bacteria and the immune system. Nat. Rev. Immunol. 4, 478-485 (2004).
  * [28] Manges AR, Steiner TS, Wright AJ. Fecal microbiota transplantation for the intestinal decolonization of extensively antimicrobial resistant opportunistic pathogens: a review. Infect Dis. 2016; 48 (8): 587-592.
  * [29] Mashburn, L., Jett, A., Akins, D. and Whiteley, M., 2005. Staphylococcus aureus Serves as an Iron Source for Pseudomonas aeruginosa during In Vivo Co-culture. Journal of Bacteriology, 187 (2), pp. 554-566.
  * [30] Massey, R., Horsburgh, M., Lina, G., Höök, M. and Recker, M., 2006. The evolution and maintenance of virulence in Staphylococcus aureus: a role for host-to-host transmission?. Nature Reviews Microbiology, 4 (12), pp. 953-958.
  * [31] Milo, R., Phillips, R. and Orme, N. (2016) Cell biology by the numbers. New York, N.Y: Garland Science, Taylor & Francis Group.
  * [32] Moran, J., Crank, E., Ghabban, H. and Horsburgh, M., 2016. Deferred growth inhibition assay to quantify the effect of bacteria-derived antimicrobials on competition. Journal of Visualized Experiments, 115.
  * [33] Moran, J.C. & amp; Horsburgh, M.J., 2016. Whole-genome sequence of Staphylococcus epidermidis TU3298. Genome Announcements, 4 (2).
  * [34] Morin, P. J. (1999). Community Ecology. England: Blackwell Science.
  * [35] Murray, J., 2003. Mathematical biology. New York: Springer.
  * [36] Peschel, A., Schnell, N., Hille, M., Entian, K. and Götz, F., 1997. Secretion of the lantibiotics epidermin and gallidermin: sequence analysis of the genes gdmT and gdmH, their influence on epidermin production and their regulation by EpiQ. Molecular and General Genetics MGG, 254 (3), pp. 312-318.
  * [37] Piewngam, P., Zheng, Y., Nguyen, T., Dickey, S., Joo, H., Villaruz, A., Glose, K., Fisher, E., Hunt, R., Li, B., Chiou, J., Pharkjaksu, S., Khongthong, S., Cheung, G., Kiratisin, P. and Otto, M., 2018. Pathogen elimination by probiotic Bacillus via signalling interference. Nature, 562 (7728), pp. 532-537.
  * [38] Press, M., Brown, N. and Barker, M., 1997. Photosynthetic characteristics of dipterocarp seedlings in three tropical rain forest light environments: a basis for niche partitioning?. Oecologia, 112 (4), pp. 453-463.
  * [39] Prestinaci, F., Pezzotti, P., & Pantosti, A. (2015). Antimicrobial resistance: a global multifaceted phenomenon. Pathogens and global health, 109 (7), 309–318.
  * [40] Rasool, K. and Wimpenny, J., 1982. Mixed continuous culture experiments with an antibiotic-producing streptomycete and Escherichia coli. Microbial Ecology, 8 (3), pp. 267-277.
  * [41] Sassone-Corsi, M. et al. (2016) “Microcins mediate competition among Enterobacteriaceae in the inflamed gut, Nature, 540 (7632), pp. 280–283.
  * [42] Sieuwerts, S., de Bok, F., Mols, E., de Vos, W. and van Hylckama Vlieg, J., 2008. A simple and fast method for determining colony forming units. Letters in Applied Microbiology, 47 (4), pp. 275-278.
  * [43] Tilman, D., 2020. In Resource Competition and Community Structure., Volume 17. Princeton university press.
  * [44] Topsall, J., 1992. Basic Laboratory Procedures in Clinical Bacteriology. Pathology, 24 (4), p. 321.
  * [45] Vandermeer, J. (2010) How Populations Grow: The Exponential and Logistic Equations.Nature Education Knowledge 3 (10): 15
  * [46] Wos Oxley ML, Plumeier I, von Eiff C, Taudien S, Platzer M, Vilchez Vargas R, Becker K, Pieper DH. 2010. A poke into the diversity and associations within human anterior nare microbial communities. ISME J. 4: 839-851.
  * [47] Yan M, Pamp SJ, Fukuyama J, Hwang PH, Cho DY, Holmes S, Relman DA. 2013. Nasal micro-environments and interspecific interactions influence nasal microbiota complexity and S. aureus carriage. Cell Host Microbe 14: 631-640.


# Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and
Twistorial DSR Algebras111Submitted to Special Issue of SYMMETRY “Quantum
Group Symmetry and Quanatum Geometry”, ed. A. Ballesteros, G. Gubitosi, F.J.
Herranz (2021)

Jerzy Lukierski
Institute for Theoretical Physics, University of Wroclaw,
pl. Maxa Borna 9, 50-205 Wroclaw, Poland
<EMAIL_ADDRESS>

Dedicated on 90-th Birthday to Sir Roger Penrose
as a tribute to his ideas and achievements

###### Abstract

We construct recently introduced palatial NC twistors by considering the pair
of conjugated (Born-dual) twist-deformed $D=4$ quantum inhomogeneous conformal
Hopf algebras $\mathcal{U}_{\theta}(su(2,2)\ltimes T^{4}$) and
$\mathcal{U}_{\bar{\theta}}(su(2,2)\ltimes\bar{T}^{4}$), where $T^{4}$
describes complex twistor coordinates and $\bar{T}^{4}$ the conjugated dual
twistor momenta. The palatial twistors are suitably chosen as the quantum-
covariant modules (NC representations) of the introduced Born-dual Hopf
algebras. Subsequently, we introduce the quantum deformations of $D=4$
Heisenberg-conformal algebra (HCA) $su(2,2)\ltimes H^{4,4}_{\hslash}$
($H^{4,4}_{\hslash}=\bar{T}^{4}\ltimes_{\hslash}T_{4}$ is the Heisenberg
algebra of twistorial oscillators) providing in twistorial framework the basic
covariant quantum elementary system. The class of algebras describing
deformation of HCA with dimensionfull deformation parameter, linked with
Planck length $\lambda_{p}$, is called the twistorial DSR (TDSR) algebra,
following the terminology of DSR algebra in space-time framework. We describe
the examples of TDSR algebra linked with Palatial twistors which are
introduced by the Drinfeld twist and the quantization map in
$H_{\hslash}^{4,4}$. We also introduce generalized quantum twistorial phase
space by considering the Heisenberg double of Hopf algebra
$\mathcal{U}_{\theta}(su(2,2)\ltimes T^{4}).$

Keywords— quantum deformations; quantum gravity; classical and quantum twistor
geometry

##  1 Introduction

###  1.1 Towards Quantum Gravity

One can distinguish two basic levels in quantization procedure of physical
models describing contemporary fundamental interactions:

(i) The first level can be called quantum-mechanical with canonically
quantized phase space coordinates and possible presence of classical gravity
only as a static background.

On such a level, we find all familiar relativistic quantum field theories,
e.g., QED and QCD (fields quantized, space-time geometry flat and
Minkowskian).

(ii) The second level also has quantized gravity and noncommutative space-
times (all fields, including gravity and space-time geometry are quantized).

Quantum gravity (QG) remains a subject of rather hypothetical models (see,
e.g., [1, 2, 3, 4]), however it is mostly agreed that QG effects require at
ultra-short distances the replacement of classical Einsteinian space-time by
quantum noncommutative space-time geometry (see [5]). The QG-generated
noncommutativity corrections appear as proportional to the powers of Planck
mass $m_{p}$ or inverse powers of Planck length $\lambda_{p}$.

$\lambda_{p}=\frac{\hslash}{m_{p}c}=\sqrt{\frac{\hslash G}{c^{3}}}$ (1)

where $c$ is the light velocity and $G$ is the gravitational Newton constant.
The QG origin of Planck length can be seen from Formula (1), with simultaneous
presence of $\hslash$ and $G$.

In order to study algebraically the QG modifications of the space-time
geometry in Special Relativity, one can look at the $\lambda$-dependent
deformations $\mathcal{U}_{\lambda}(\mathcal{P}^{3,1})$ of the Poincaré
algebra $\mathcal{P}^{3,1}=o(3,1)\ltimes P^{3,1}$, where $P^{3,1}$ denotes the
four-momenta sector and $\lambda$ describes an elementary length parameter
which can be fixed $\lambda=\lambda_{p}$. Further, we consider the Minkowski
space-time coordinates $x_{\mu}\in M^{3,1}$ together with covariantly acting
Poincaré symmetry and introduce the semi-direct product algebra

$\mathbb{A}=\mathcal{P}^{3,1}\ltimes M^{3,1}\simeq
o(3,1)\ltimes(P^{3,1}\ltimes M^{3,1})$ (2)

where $P^{3,1}\ltimes M^{3,1}$ describes the relativistic phase space
$\mathbb{P}^{3,1}=(M^{3,1};P^{3,1})$, which after the first quantization level
is endowed with relativistic Heisenberg algebra structure. Such algebra
$\mathbb{A}_{\hbar}$, also called Heisenberg–Lorentz algebra, can be further
deformed into quantum algebra $\mathcal{U}_{\lambda}(\mathbb{A}_{\hbar})$,
which, to describe quantum symmetry, should have Hopf algebra or Hopf-
algebroid structure. We stress that only $U_{\lambda}(\mathcal{P}^{3;1})$ can
be introduced as Hopf algebra, i.e., Drinfeld quantum symmetry group [6, 7].
If $\hbar\neq 0$. the relativistic quantum phase space
$\mathbb{P}^{3;1}_{\hbar}$ implies that $U_{\lambda}(\mathbb{A}_{\hbar})$ has
the algebraic structure of Hopf algebroid [8, 9, 10, 11, 12, 13].

Such class of deformations of algebra (2) provides so-called DSR (Doubly
Special Relativity) algebra describing quantum space-times with covariantly
acting quantum symmetry. We use the original name for DSR algebras [14, 15,
16, 17], however some authors use the name DSR for “Deformed Special
Relativity’, which has a vague informative content.

The name “doubly” is due to the dependence on two parameters: $c$ (light
velocity) and $\lambda_{p}$ (Planck length, or $m_{p}\sim(l_{p})^{-1})$. The
first parameter, $c$, appears in the physical basis of the relativistic
classical algebra $\mathbb{A}$ and the second parameter, $\lambda$, determines
the QG-induced modification of the algebraic structure (2).

It was argued already in the 1930s [18] that QG models should at the basic
level depend on three fundamental nonvanishing constants, $c$, $\hslash$ and
$G$, where $G$ can be replaced by $\lambda_{p}$ or $m_{p}$ (see (1)); if the
cosmological constant or de Sitter radius of the Universe is finite, it
introduces additional geometric parameter. The variant of DSR algebra with
additional de Sitter radius as additional geometric parameter was called
Triply Special Relativity (TSR); see [19].

The model of quantum space-time symmetries, which was an inspiration for
introducing DSR algebras, is provided by the $\kappa$-deformed Poincaré–Hopf
algebra [20, 21] with semi-direct product structure presented in [22] in so-
called bicrossproduct basis.

Our aim is to describe some class of quantum-deformed twistors and provide the
counterpart of DSR algebra in the noncommutative framework of quantum
twistors. It should be recognized here that there are already several
interesting papers dealing with quantum deformations of twistors and their
geometries (see, e.g., [23, 24, 25, 26, 27, 28, 29]).

###  1.2 Elements of Twistor Theory

For more than half of a century, Roger Penrose and his collaborators (see,
e.g., [23, 30, 31, 32]) have promoted the idea that the space-time manifold is
a secondary geometric construction, and primary geometric objects are
twistors. In $D=4$ flat space-time, twistors are introduced as four-
dimensional complex conformal spinors $t_{A}\in T^{4}$, endowed with the $D=4$
conformal-invariant pseudo-Hermitian $U(2,2)$ scalar product

$(\bar{t},t)\equiv\bar{t}_{A}\eta^{AB}t_{B},\qquad A,B=1,2,3,4$ (3)

where $\bar{t}_{A}\in\bar{T}^{4}$ are the complex-conjugated dual twistors and
$\eta^{AB}=(1,1,-1,-1).$ If we introduce twistors as the pair of $D=4$ Weyl
spinors $(\alpha=1,2)$

$t_{A}=(\pi_{\alpha},\omega^{\dot{\alpha}})$ (4)

the alternative $U(2,2)$ frame should be used, with the metric

$\eta^{AB}=\left(\begin{array}[]{cc}0&1_{2}\\\ 1_{2}&0\end{array}\right),$ (5)

leading to the formula

$(\bar{t},t)\equiv\bar{t}_{A}\eta^{AB}t_{B}=\bar{\pi}_{\dot{\alpha}}\omega^{\dot{\alpha}}+H.C.$
(6)

We recall that the points $z_{\mu}$ of complex Minkowski space-time are
specified in $T^{4}$ by two-dimensional planes with twistor coordinates,
$(\pi_{\alpha},\omega^{\dot{\alpha}})$, satisfying the Cartan–Penrose
incidence relation

$\omega^{\dot{\alpha}}=iz^{\dot{\alpha}\beta}\pi_{\beta},\qquad
z^{\dot{\alpha}\beta}=\frac{1}{2}(\sigma^{\mu})^{\dot{\alpha}\beta}z_{\mu}.$
(7)

The quantum-mechanical twistors $\hat{t}_{A}$,$\hat{\bar{t}}_{A}$ on first
basic quantization level are provided by the oscillator-like canonical
commutation relations (CCR) [23, 30].

$\displaystyle[\hat{\bar{t}}_{A},\vskip 12.0pt plus 4.0pt minus
4.0pt\hat{t}_{B}]$ $\displaystyle=$ $\displaystyle\hbar\eta_{AB}$ (8)
$\displaystyle[\vskip 12.0pt plus 4.0pt minus 4.0pt\hat{t}_{A},\vskip 12.0pt
plus 4.0pt minus 4.0pt\hat{t}_{B}]$ $\displaystyle=$
$\displaystyle[\hat{\bar{t}}_{A},\hat{\bar{t}}_{B}]=0.$ (9)

One can call $\hat{t}_{A}$ the twistor coordinates and $\hat{\bar{t}}_{A}$ the
twistor momenta; they introduce the twistorial analog of the relativistic
quantum-mechanical phase-space algebra for conformal-covariant twistorial
models. We recall that one can obtain the twistor realization of $D=4$
conformal algebra $o(4,2)\simeq su(2,2)$ given by the bilinear products of
Quantum-Mechanical (QM) twistors $\hat{t}_{A}$,$\hat{\bar{t}}_{A}$ (see also
Section 3.1).

The twistors $t_{A}\in T^{4}$ satisfying the Cartan–Penrose incidence
relations (7) provide the geometric alternative for the description by complex
Minkowski space-time geometry. The real Minkowski coordinates
$x_{\mu}(z_{\mu}=x_{\mu}+iy_{\mu})$ are obtained if the $2\times 2$ Hermitian
matrix $z^{\dot{\alpha}\beta}$ is parameterized as follows

$z^{\dot{\alpha}\beta}=\left(\begin{array}[]{cc}x_{0}+x_{3}&x_{1}+ix_{2}\\\
x_{1}-ix_{2}&x_{0}-x_{3}\end{array}\right)=(z^{\dot{\beta}\alpha})^{*}$ (10)

In such a case, one gets from (7) that $(t,t)=0$, i.e., the real Minkowski
coordinates have as twistor counterparts the null twistor planes (so-called
$\alpha$-planes), with vanishing norm (6).

For the discussion of QG effects, it is more appropriate and realistic to
consider twistors corresponding to curved space-time. The simplest examples of
nonflat space-times are the ones with constant curvature $R$ or cosmological
constant $\Lambda=\pm\frac{1}{R^{2}}$, where $\Lambda>0$ for de Sitter and
$\Lambda<0$ for anti-de Sitter geometries. In such a case, the standard
quantization relations (8) and (9) for twistors should be modified, with
respective deformation determined by the $\Lambda$-dependent antisymmetric
constant second rank twistor $I_{AB}$, called in twistor theory the infinity
twistor (see, e.g., [30]). In the basis (4), it is given by the following
formula (we choose further $\Lambda>0$)

$I_{AB}=\left(\begin{array}[]{cc}\frac{\Lambda}{6}\epsilon_{\alpha\beta}&0\\\
0&\epsilon^{\dot{\alpha}\dot{\beta}}\end{array}\right),\qquad I_{AB}=-I_{BA}.$
(11)

One can introduce in twistorial phase space $(T^{4},\bar{T}^{4})$ the
deformation of Poisson structure described by the following complex-
holomorphic $(2,0)$ symplectic two-form [27]

$\Omega_{2}=d(I_{AB}t^{A}dt^{B})=I_{AB}dt^{A}\wedge
dt^{B}=\frac{\Lambda}{6}d\pi_{\alpha}\wedge
d\pi^{\alpha}+d\omega_{\dot{\alpha}}\wedge d\omega^{\dot{\alpha}}$ (12)

generated by the holomorphic $(1,0)$ Liouville one-form
$\Omega_{1}=I_{AB}t^{A}dt^{B}$, where $\Omega_{2}=d\Omega_{1}$. In dual
twistor space $\bar{T}^{4}$, the complex-anti-holomorphic $(0,2)$ symplectic
two-form (we denote $\bar{t}_{A}\equiv t_{A}^{\dagger}=\eta_{AB}\bar{t}^{B}$)

$\bar{\Omega}_{2}=d(\bar{I}^{AB}\bar{t}_{A}d\bar{t}_{B})=\bar{I}^{AB}d\bar{t}_{A}\wedge
d\bar{t}_{B}=\frac{\Lambda}{6}d\bar{\pi}^{\dot{\alpha}}\wedge
d\bar{\pi}_{\dot{\alpha}}+d\bar{\omega}^{\alpha}\wedge d\bar{\omega}_{\alpha}$
(13)

is complex-conjugated to (12), which leads to the relation

$\bar{I}^{AB}=\frac{1}{2}\epsilon^{ABCD}I_{CD}=\left(\begin{array}[]{cc}\epsilon_{\alpha\beta}&0\\\
0&\frac{\Lambda}{6}\epsilon^{\dot{\alpha}\dot{\beta}}\end{array}\right).$ (14)

It appears that, within the framework of Hopf-algebraic quantum deformations
of inhomogeneous conformal algebras $su(2,2)$, one gets separately the
deformations of twistors $\hat{t}^{A}\in T^{4}$ and
$\hat{\bar{t}}\in\bar{T}^{4}$, which lead to the quantum-mechanical (first
level) quantization of symplectic structures (12) and (13).

The deformed twistors obtained by the quantization of symplectic Poisson
structures (12)–(14) are called the palatial twistors [27]. After the
quantization procedure, one gets the holomorphic and anti-holomorphic
noncommutativity relations modifying (9) as follows (further, in many
formulae, we put $\hbar=c=1$).

$[\hat{t}^{A},\hat{t}^{B}]=\hbar\bar{I}^{AB},\qquad[\hat{\bar{t}}_{\dot{A}},\hat{\bar{t}}_{\dot{B}}]=\hbar
I_{\dot{A}\dot{B}}.$ (15)

###  1.3 Twist Deformations: From Space-Time
to Twistors

Let us recall the well-known twisting procedure (see, e.g., [33, 34]) of the
Poincaré algebra $\mathcal{U}(o(3,1)\ltimes P^{3,1})$ which is semi-dual to
twisted Poincaré algebra $\mathcal{U}(o(3,1)\ltimes M^{3,1})$.

For these two basic relativistic algebras (in Minkowski space-time and four-
momentum space), one introduces the following basic Abelian twists

$\displaystyle\bar{F}$ $\displaystyle=$
$\displaystyle\exp\left(i\frac{\lambda^{2}}{2}\theta^{\mu\nu}p_{\mu}\otimes
p_{\nu}\right),\qquad p_{\mu}\in P^{3,1}$ (16) $\displaystyle F$
$\displaystyle=$
$\displaystyle\exp\left(i\frac{1}{2\lambda^{2}}\theta_{\mu\nu}x^{\mu}\otimes
x^{\nu}\right),\qquad x^{\mu}\in M^{3,1}$ (17)

where $\theta_{\mu\nu}=-\theta_{\nu\mu}$ are real and dimensionless; the
dimensionfull deformation parameter $\lambda$ representing an elementary
length in QG applications may be chosen as Planck length $\lambda_{p}$. If we
insert the twists (16) and (17) into the formulae for twist-deformed Hopf
algebra modules (see, e.g., [5, 35]), we get the quantization maps

$\hat{x}^{\mu}=m[\bar{F}^{-1}(\vartriangleright\otimes 1)(x^{\mu}\otimes
1)],\qquad\hat{p}_{\mu}=m[F^{-1}(\vartriangleright\otimes 1)(p_{\mu}\otimes
1)],$ (18)

One obtains explicitly the well-known noncommutative $\theta$-deformed
coordinates called also DFR (Dopplicher, Fredenhagen, Roberts). quantum space-
times [36, 37], where standard parameters $\theta^{\mu\nu}_{DSR}$ in DSR
deformation are
dimensionfull—$\theta^{\mu\nu}_{DSR}=\lambda^{2}\theta^{\mu\nu}$, with the
dimensionality $[\theta^{\mu\nu}_{DSR}]=L^{2}$ and $[\theta^{\mu\nu}]=L^{0}$.
The choice of $\theta^{\mu\nu}$ is purely geometric, selects directions in
space-time, which are being deformed. We obtain from (18)

$\left[\hat{x}^{\mu},\hat{x}^{\nu}\right]=i\lambda^{2}\theta^{\mu\nu}.$ (19)

For the noncommutative $\theta$-deformed four-momenta we get

$\left[\hat{p}_{\mu},\hat{p}_{\nu}\right]=\frac{i}{\lambda^{2}}\theta_{\mu\nu}.$
(20)

If we wish to obtain both deformations (19) and (20) inside one algebraic
structure, we observe that due to Jacobi identity the canonical relations
$[x_{\mu},p_{\nu}]=i\hslash g_{\mu\nu}$ should be modified and both
deformations (19) and (20) can be embedded together only in a Hopf algebroid
(see, e.g., [8, 9, 10, 11]).

The paper is organized as follows. In Section 2, we describe the pair of
twisted inhomogeneous $D=4$ conformal algebras $isu(2,2)\equiv su(2,2)\ltimes
T^{4},\bar{i}su(2,2)\equiv su(2,2)\ltimes\bar{T}^{4}$. Using Hopf-algebraic
twisting, we derive the NC relations (15) for holomorphic (chiral) and anti-
holomorphic (anti-chiral) palatial twistors as the twist-generated quantum
deformations. If we twist the primitive coproducts of $su(2,2)$ generators, we
can show that the twistorial NC phase space coordinates
$(\hat{T}^{4},\hat{\bar{T}}^{4})$ are covariant as Hopf algebra module under
the action of respective twist-deformed inhomogeneous $su(2,2)$ algebras.

In Section 3, we introduce the new notion of twistorial DSR (TDSR) algebra, in
particular as $\Theta_{AB}$-deformed twistorial $D=4$ Heisenberg-conformal
algebra $su(2,2)\ltimes H^{4,4}_{\hslash}$, where
$H^{4,4}_{\hbar}=\bar{T}^{4}\ltimes_{\hslash}T^{4}$ containing both sectors
$T^{4}$, $\bar{T}^{4}$ simultaneously $\Theta$-deformed. For such purpose, we
use the quantization map which follows from the quantized versions of
symplectic structures (12) and (13), and suitably modifies the Kronecker delta
in the relation (8). We propose the twist quantization by a Drinfeld twist
$\mathbb{F}$ (see. e.g., [38]), which is related with the twists (29) and (30)
and only in the linear approximation in $\lambda$ of the quantization map
provides the palatial twistors. Further, we consider generalized twistorial
quantum phase spaces defined by twisted Heisenberg doubles of $isu(2,2)$ and
$\bar{i}su(2,2)$ Hopf algebras. In such a framework, following Brain and
Majid, the quantum deformation of twistor geometry considered in Section 1.2
can be obtained by cotwist quantization of inhomogeneous conformal quantum
groups $ISU(2,2)$ and $\bar{I}SU(2,2)$, which, respectively, are Hopf-dual to
twist-deformed $isu(2,2)$ and $\bar{i}su(2,2)$ quantum symmetry algebras.

In the concluding Section 4, we present an outlook, with directions for
possible future research.

##  2 Twisted Inhomogeneous $D=4$ Conformal Algebras, Born Duality and
Palatial Twistors

###  2.1 From Poincaré to Inhomogeneous Conformal Algebras

If we pass from the $D=4$ relativistic space-time description of the Universe
to twistorial geometric framework, the $D=4$ Lorentz algebra $o(3,1)\simeq
SL(2;\mathbb{C})\oplus\bar{SL(2;\mathbb{C}})$ with the four-vectors
$x_{\mu},p_{\nu}$ is replaced by $D=4$ conformal algebra $o(4,2)\simeq
su(2,2)$ with fundamental translational degrees of freedom described by
twistors $(\hat{t}_{A}\in T^{4},{\hat{\bar{t}}}_{A}\in\bar{T}^{4})$ spanning
the canonical twistorial phase space. In the space-time approach, the orbital
part of Lorentz algebra generators $M_{\mu\nu}$ can be realized in terms of
$D=4$ relativistic phase space variables ($x_{\mu}\in M^{3,1},p_{\mu}\in
P^{3,1})(\mu,\nu=0,1,2,3)$ as follows

$o(3,1):\qquad M_{\mu\nu}=\hat{x}_{[\mu}\hat{p}_{\nu]}.$ (21)

where the Formula (21) is applicable only to the spinless systems; for the
extension with spin in the context of quantum deformations, see, e.g., [13].

Using the twistorial canonical oscillator algebra (8) and (9), one can
analogously express in terms of twistorial phase space coordinates
$(\hat{t}_{A},{\hat{\bar{t}}}_{A})$ the conformal generators $S_{A\bar{B}}\in
su(2,2)(A,{B}=1,2,3,4)$ (Using the basis (4) the physical 15 generators of
$D=4$ conformal algebra can be expressed by the following bilinear formulas
(see also Section 3.1)).

$\displaystyle p_{\alpha\dot{\beta}}=\pi_{\alpha}\bar{\pi}_{\dot{\beta}}\qquad
M_{\alpha\beta}=\pi_{\alpha}\omega_{\beta}\qquad
M_{\dot{\alpha}\dot{\beta}}=\bar{\pi}_{\dot{\alpha}}\bar{\pi}_{\dot{\beta}}$
$\displaystyle D=\pi_{\alpha}\omega^{\dot{\alpha}}+H.C.\qquad
K_{\alpha\dot{\beta}}=\omega_{\alpha}\bar{\omega}_{\dot{\beta}}$
$S_{A{B}}=\hat{t}_{A}\hat{\bar{t}}_{{B}}-\frac{1}{4}(t,t)\eta_{A{B}}\qquad\eta^{AB}S_{AB}=0$
(22)

where $S_{A{B}}$ are the $4\times 4$ pseudo-Hermitian complex matrix $u(2,2)$
generators

$S_{A{B}}^{\dagger}=\eta_{AC}S_{C{D}}\eta_{{D}{B}}.$ (23)

One can introduce the pair of Poincaré groups
$(\mathcal{P}_{x}^{3,1}=O(3,1)\ltimes M^{3,1}$,
$\mathcal{P}_{p}^{3,1}=O(3,1)\ltimes P^{3,1})$, with $M^{3,1},P^{3,1}$
describing Minkowski coordinates and four-momenta, as two cosets related by
the Fourier transform

$M^{3,1}=\frac{\mathcal{P}_{x}^{3,1}}{o(3,1)}\ni\\{x_{\mu}\\}\qquad
P^{3,1}=\frac{\mathcal{P}_{p}^{3,1}}{o(3,1)}\ni\\{p_{\mu}\\}.$ (24)

The twistorial counterparts of relations (24) appear as follows

$T^{4}=\frac{ISU(2,2)}{SU(2,2)}\qquad\bar{T}^{4}=\frac{\bar{I}SU(2,2)}{SU(2,2)}$
(25)

where $ISU(2,2)=SU(2,2)\ltimes T^{4}$ and
$\bar{I}SU(2,2)=SU(2,2)\ltimes\bar{T}^{4}$ describe the semi-dual pair of
twistorial inhomogeneous $D=4$ conformal groups. The twistors $t_{A}\in T^{4}$
describe the fundamental holomorphic conformal spinors (holomorphic twistors)
and $\bar{t}_{A}\in\bar{T}^{4}$ the complex-conjugated anti-holomorphic ones.
Alternatively, one can use the projective twistors $CP(3)$, described by the
equivalence classes $t_{A}\sim ct_{A}(c\in\mathbb{C})$ in $T^{4}$, which
parameterize the following symmetric coset of $SU(2,2)$ (see, e.g., [26]).

$CP(3)=\frac{SU(2,2)}{S(U(2,1)\otimes U(1))}.$ (26)

The complex Minkowski coordinates $z^{\dot{\alpha}\beta}\in CM(4)$ (see (10))
can be introduced as parameterizing the following complex Grassmanian

$M(4)=\frac{SU(2,2)}{S(U(2)\otimes U(2))}.$ (27)

The coset (27) parameterizes complex 2-planes in $T^{4}$ which are determined
by non-parallel pairs of intersecting twistors $t_{A}^{i}(i=1,2;A=1,\ldots
4;(t^{1},t^{2})\neq 0)$ and satisfy the pair (7) of Cartan–Penrose incidence
relations. The complex Minkowski coordinates
$z^{\mu}=\frac{1}{2}(\sigma^{\mu})_{\dot{\alpha}\beta}z^{\dot{\alpha}\beta}$
are expressed by the pair of intersecting twistor coordinates
$t_{A}^{i}=(\pi^{i}_{\alpha},{\omega}^{\dot{\alpha}i})$ as follows

$z^{\dot{\alpha}\beta}=-\frac{i}{\pi^{1\alpha}\pi^{2}_{\alpha}}(\omega^{1\dot{\alpha}}\pi^{2\beta}-\omega^{2\dot{\alpha}}\pi^{1\beta}).$
(28)

The primary aim of the Penrose program during the last fifty years was to
encode any curved Einsteinian space-time structure in geometric twistorial
framework; in particular, it was important to find the vocabulary permitting
to translate any general relativity solution in space-time into the twistorial
language. This goal was however achieved only partially, with modest hopes
that the program of finding the twistor formulation of general relativity
theory will be fully successful. However, in last decade, Roger Penrose became
inspired by the idea that perhaps it is an easier task to construct the
twistorial noncommutative version of quantum gravity. Such a view,
conceptually attractive, however still faces the basic question of how the
appropriate formulation of quantum gravity in the space-time picture would
look. On the twistorial side, some first steps towards the construction of
twistorial quantum gravity model were provided by Penrose (see also [29]).

###  2.2 Twist-Deformed Inhomogeneous Conformal
Hopf Algebras and Holomorphic/Anti-Holomorphic
Quantum Twistors

Our first task is to show how the relations (15) can be obtained in the
framework of quantum deformations of inhomogeneous $D=4$ conformal algebras,
with the respective holomorphic twistor coordinates $t_{A}\in T^{4}$ or anti-
holomorphic twistorial complex momenta $\bar{t}_{A}\in\bar{T}^{4}$. For such a
purpose, we consider the pair of semi-dual Hopf algebras
$\mathbb{H}_{0}\equiv\mathcal{U}(isu(2,2))=\mathcal{U}(su(2,2)\ltimes T^{4})$
and
$\bar{\mathbb{H}}_{0}\equiv\mathcal{U}(\bar{i}su(2,2))=\mathcal{U}(su(2,2)\ltimes\bar{T}^{4})$,
with Hermitian-conjugated generators in $T^{4}$ and $\bar{T}^{4}$, but with
the same twistorial realization of $D=4$ conformal subalgebra $su(2,2)$.
Subsequently, one gets the holomorphic and anti-holomorphic palatial twistors
if the Hopf algebras $\mathbb{H}_{0}$ and $\bar{\mathbb{H}}_{0}$ are twisted,
respectively by the following pair of twists,

$\displaystyle\mathcal{\bar{F}}$ $\displaystyle=$
$\displaystyle\exp\left(-\frac{1}{2\lambda}\bar{\Theta}^{AB}{\hat{\bar{t}}}_{A}\wedge{\hat{\bar{t}}}_{B}\right),$
(29) $\displaystyle\mathcal{F}$ $\displaystyle=$
$\displaystyle\exp\left(-\frac{1}{2\lambda}\Theta_{AB}{\hat{t}}^{A}\wedge\hat{t}^{B}\right).$
(30)

where further, in Section 3.1 we justify the same dependence of twists (29)
and (30) from the elementary length parameter $\lambda$.

In the general case, the antisymmetric numerical tensor $\Theta_{AB}$ can be
chosen as complex, but, in the case of Palatial twistors, because
$\Theta_{AB}=I_{AB}$, they are real. The pair of twists (29) and (30) are dual
under the twistorial Born map

${\hat{t}}_{A}\longleftrightarrow{\hat{\bar{t}}}_{A},\qquad\Theta_{AB}\rightarrow\bar{\Theta}_{AB}$
(31)

which, as the twistorial counterpart of the Born duality map
$x_{\mu}\leftrightarrow p_{\mu}$ [39, 40], interchanges the twistorial momenta
${\hat{\bar{t}}}_{A}$ and the twistorial coordinates $\hat{t}_{A}$.

The conformal twists (29) and (30) deforming, respectively, $isu(2,2)$ and
$\bar{i}su(2,2)$ Hopf–Lie algebras are the twistorial counterpart of the
Poincaré twists (16) and (17)—the second one employed quite often in the
space-time approach (see [33, 34]). Further, the pair of Formulae (18) has the
following counterpart in twistorial description

$\hat{t}_{A}^{\bar{\mathcal{F}}}=m[\bar{\mathcal{F}}^{-1}(\triangleright\otimes
1)(\hat{t}_{A}\otimes
1)]=(\bar{\mathcal{F}}_{(1)}^{-1}\triangleright\hat{t}_{A})\bar{\mathcal{F}}^{-1}_{(2)}$
(32) $\hat{\bar{t}}_{A}^{\mathcal{F}}=m[\mathcal{F}^{-1}(\triangleright\otimes
1)(\hat{\bar{t}}_{A}\otimes
1)]=(\mathcal{F}_{(1)}^{-1}\triangleright\hat{\bar{t}}_{A})\mathcal{F}^{-1}_{(2)}.$
(33)

The generators $\hat{t}_{A}^{\bar{\mathcal{F}}}$ describe the
$\mathcal{U}_{\bar{\mathcal{F}}}(\bar{i}su(2,2))$ Hopf algebra modules and
$\hat{\bar{t}}_{A}^{\mathcal{F}}$ span the module of Hopf algebra
$\mathcal{U}_{\mathcal{F}}(isu(2,2))$. Using the Hopf-algebraic action
consistent with relation (6)

$\hat{\bar{t}}_{A}\triangleright\hat{t}_{B}=\eta_{{A}B}\qquad\hat{t}_{A}\triangleright\hat{\bar{t}}_{B}=-\eta_{A{B}}$
(34)

one gets the following explicit formulae for conformal quantum twistors

$\mathcal{U}_{\bar{\mathcal{F}}}(\bar{i}su(2,2)):\qquad\hat{t}_{A}^{\bar{\mathcal{F}}}=\hat{t}_{A}+\frac{1}{\lambda}\bar{\Theta}_{A}{\\!}^{B}\hat{\bar{t}}_{B}$
(35)
$\mathcal{U}_{\mathcal{F}}(isu(2,2)):\qquad\hat{\bar{t}}_{A}^{\mathcal{F}}=\hat{\bar{t}}_{A}-\frac{1}{\lambda}\Theta_{A}{\\!}^{B}\hat{t}_{B}.$
(36)

Using the Formula (32) for $\bar{t}_{A}$ and (33) for $t_{A}$, one gets
additionally

$\hat{t}_{A}^{\mathcal{F}}=\hat{t}_{A}\qquad\hat{\bar{t}}_{A}^{\bar{\mathcal{F}}}=\hat{\bar{t}}_{A}.$
(37)

We obtain the following two algebras describing twist-deformed quantum
twistorial phase space coordinates
$(\hat{t}_{A}^{\mathcal{F}},\hat{\bar{t}}_{A}^{\mathcal{F}})$ and
$(\hat{t}_{A}^{\bar{\mathcal{F}}},\hat{\bar{t}}_{A}^{\bar{\mathcal{F}}})$

$\displaystyle[{\hat{\bar{t}}}_{A}^{\;\mathcal{{F}}},{\hat{\bar{t}}}_{B}^{\mathcal{\,{F}}}]=\frac{2}{\lambda}\Theta_{AB}\qquad[\hat{{t}}_{A}^{\mathcal{{F}}},\hat{{t}}_{B}^{\mathcal{{F}}}]=0\qquad[\hat{\bar{t}}_{A}^{\mathcal{{F}}},\hat{t}_{B}^{\mathcal{{F}}}]=\eta_{AB}$
(38)
$\displaystyle[\hat{t}_{A}^{\bar{\mathcal{F}}},\hat{t}_{B}^{{\bar{\mathcal{F}}}}]=\frac{2}{\lambda}\bar{\Theta}_{AB}\qquad[\hat{\bar{t}}_{A}^{{\bar{\mathcal{F}}}},\hat{\bar{t}}_{B}^{\bar{\mathcal{F}}}]=0\qquad[\hat{\bar{t}}_{A}^{{\bar{\mathcal{F}}}},\hat{t}_{B}^{{\bar{\mathcal{F}}}}]=\eta_{AB}.$
(39)

It should be stressed that the twistorial quantum phase space coordinates
$(\hat{t}_{A}^{\bar{\mathcal{F}}},\hat{\bar{t}}_{A}^{\mathcal{F}})$ satisfy
the following Hermitian-conjugated algebra

$\hat{t}_{A}^{\mathcal{F}}=(\hat{\bar{t}}_{A}^{\,\bar{\mathcal{F}}})^{\dagger}\qquad\hat{\bar{t}}_{A}^{\mathcal{F}}=(\hat{{t}}_{A}^{\bar{\mathcal{F}}})^{\dagger}.$
(40)

The shifts described by Formulae (35) and (36) are the examples of the
Bogolyubov transformation (see [41]) of twistorial oscillators satisfying the
relations (8) and (9).

###  2.3 The Twisted Conformal Covariance of Quantum Twistors and Born Duality
Map

The relations (39) and (38) are quantum-covariant under the action of the
twisted inhomogeneous conformal Hopf algebras
$\mathbb{H}_{\mathcal{F}}=\mathcal{U}_{\mathcal{F}}(isu(2,2))$ and
$\mathbb{H}_{\bar{\mathcal{F}}}=\mathcal{U}_{\bar{\mathcal{F}}}(\bar{i}su(2,2))$,
respectively. In order to demonstrate such a property, one should calculate
the twist-deformed coproducts using the familiar similarity maps

$\displaystyle\Delta_{\mathcal{F}}(\hat{g})=\mathcal{F}^{-1}\circ\Delta_{0}(\hat{g})\circ\mathcal{F}\qquad\hat{g}=(\hat{t},\hat{S}_{A\bar{B}})\in
isu(2,2)$ (41)
$\displaystyle\Delta_{{\mathcal{\bar{{F}}}}}(\hat{{g}})=\bar{\mathcal{F}}^{-1}\circ\Delta_{0}(\hat{{g}})\circ\bar{\mathcal{F}}\qquad\hat{g}=(\hat{\bar{t}},\hat{S}_{A\bar{B}})\in\bar{i}su(2,2)$
(42)

where $\Delta_{0}(\hat{g})=\hat{g}\otimes 1+1\otimes\hat{g}$ and the
generators $\hat{S}_{A\bar{B}}\in su(2,2)$ can be represented in terms of
twistor coordinates $(\hat{t}_{A},\hat{\bar{t}}_{A})$ (see (21)). From
Formulae (29), (30), (41) and (42), one obtains that

$\Delta_{\mathcal{F}}(\hat{t}_{A})=\Delta_{0}(\hat{t}_{A})\qquad\Delta_{\mathcal{\bar{F}}}(\hat{\bar{t}}_{A})=\Delta_{0}(\hat{\bar{t}}_{A})$
(43)

and

$\Delta_{\mathcal{F}}(\hat{S}_{A\bar{B}})=\Delta_{0}(\hat{S}_{A\bar{B}})+\lambda^{-1}(\Theta_{B}^{D}\hat{t}_{A}\otimes\hat{t}_{D}+\Theta^{C}_{B}\hat{t}_{C}\otimes\hat{t}_{A})$
(44)
$\Delta_{\bar{\mathcal{F}}}(\hat{S}_{A\bar{B}})=\Delta_{0}(\hat{S}_{A\bar{B}})+\lambda^{-1}(\bar{\Theta}_{B}^{D}\hat{\bar{t}}_{A}\otimes\hat{\bar{t}}_{D}+\bar{\Theta}^{C}_{B}\hat{\bar{t}}_{C}\otimes\hat{\bar{t}}_{A})$
(45)

Due to the modification using (44) and (45) of the primitive coproducts of
$S_{AB}$, one can show that

$\mathbb{H}_{\mathcal{F}}:\qquad\qquad\hat{g}\triangleright([\hat{\bar{t}}_{A}^{\mathcal{F}},\hat{\bar{t}}_{B}^{\mathcal{F}}]-\frac{2}{\lambda}{\Theta}_{AB})=0$
(46)
$\mathbb{H}_{\bar{\mathcal{F}}}:\qquad\qquad\hat{\bar{g}}\triangleright([\hat{t}_{A}^{\bar{\mathcal{F}}},\hat{t}_{B}^{\bar{\mathcal{F}}}]-\frac{2}{\lambda}{\bar{\Theta}}_{AB})=0$
(47)

where we use the standard Hopf-algebraic formula defining the action
$\triangleright$ of generators $\hat{h}\in\mathbb{H}$ on the products
$\hat{a}\cdot\hat{b}(\hat{a},\hat{b}\in\mathbb{A})$

$\hat{h}\triangleright(\hat{a}\cdot\hat{b})=(\hat{h}_{(1)}\triangleright\hat{a})(\hat{h}_{(2)}\triangleright\hat{b}).$
(48)

where $\mathbb{A}$ is the $\mathbb{H}$-module algebra. We recall that, in the
case of noncommutative $\theta_{\mu\nu}$-deformed quantum space-times and
quantum four-momenta (see (19) and (20)), one gets analogously in coproducts
$\Delta_{F}(M_{\mu\nu})$ and $\Delta_{\bar{{F}}}(M_{\mu\nu})$ the additional
terms which are linear in $\theta_{\mu\nu}$ and bilinear in four-momenta (for
twist (16)) or bilinear in space-time coordinates (for twist (17)); these
terms are needed for the twisted quantum Poincaré invariance of the algebraic
relations (19) and (20) (see also [33]).

For coordinate and momenta twistors, one can consider the quantum covariance
under two different inhomogeneous twisted conformal Hopf algebras
$\mathcal{U}_{\mathcal{F}}(isu(2,2))$ and
$\mathcal{U}_{\bar{\mathcal{F}}}(\bar{i}su(2,2))$, but they can be mapped into
each other if we supplement the twistorial Born map (31) with the following
exchange relation ($\Theta_{AB}$ in the general case are complex, but it
should be observed that $\theta_{\mu\nu}$ in both Formulae (19) and (20) is
real and not changing under the map (49)).

$\Theta_{AB}\leftrightarrow\bar{\Theta}_{AB}$ (49)

The superposition of maps (31) and (49) leads to the following Born
substitution rule of the twist factors (29) and (30)

$\mathcal{F}\leftrightarrow\bar{\mathcal{F}}$ (50)

The relations (49) and (50) describe the Born partial duality (semi-duality)
of inhomogeneous conformal Hopf-algebras $\mathcal{U}_{\mathcal{F}}(isu(2,2))$
and $\mathcal{U}_{\bar{\mathcal{F}}}(\bar{i}su(2,2))$ (see, e.g., [42, 43,
44]) with interchanged subalgebras of twistorial momenta and coordinates.

##  3 Twistorial DSR Algebra as Deformed Smashed Product of $su(2,2)$ and
Twistorial Quantum Phase Space

###  3.1 Twistorial DSR (TDSR) Algebra

During 2000–2001, the notion of Double Special Relativity (DSR) was proposed,
with the postulate that the geometry of special relativity in the presence of
QG corrections is modified by quantum corrections, with Planck mass or Planck
length playing the role of mass-like deformation parameter. In fact, in Snyder
model [45], introducing first in the literature NC quantum space-time
coordinates $\hat{x}_{\mu}$ by means of the relation

$[\hat{x}_{\mu},\hat{x}_{\nu}]=gM_{\mu\nu}\qquad[g]=L^{2}$ (51)

one usually assumes that $g=\beta l^{2}_{p}$ (where $\beta$ is a dimensionless
constant) and one can consider Snyder model as the first historical example of
DSR model. In the Hopf-algebraic framework of quantum groups, the general DSR
algebra can be described as quantum algebra
$\mathcal{U}_{\lambda}(\mathbb{A}_{DSR})$, where $\mathbb{A}_{DSR}$ is given
by the Formula (2).

If we wish to introduce the twistorial counterpart of DSR theory, described by
the corresponding class of twistorial DSR algebras, one should replace the
algebra $\mathbb{A}_{DSR}$ (see (2)) by the algebra $\mathbb{A}_{TDSR}$

$\mathbb{A}_{TDSR}=\bar{i}su(2,2)\ltimes T^{4}\simeq
isu(2,2)\ltimes\bar{T}^{4}\simeq su(2,2)\ltimes H^{4,4}_{\hbar}$ (52)

where $H^{4,4}_{\hbar}=\bar{T}^{4}\rtimes_{\hslash}T^{4}$ denotes the quantum
twistorial phase space, described by the twistorial oscillators algebra (see
(8) and (9)). Subsequently, one can introduce the twistorial DSR (TDSR)
algebra as described by the following quantum deformations:

$\mathcal{U}_{\lambda}(\mathbb{A}_{TDSR})\equiv\mathcal{U}_{\lambda}(su(2,2)\ltimes_{\hslash}H^{4,4}_{\hslash}$
(53)

In Formula (53), we use the particular case of semi-direct product, called
smash product (see, e.g., [46, 47]) of the Hopf algebra $\mathbb{H}$ and its
module algebra $\mathbb{A}$

$\mathcal{H}=\mathbb{H}\ltimes\mathbb{A}.$ (54)

If $h,h^{\prime}\in\mathbb{H}$ and $a,b\in\mathbb{A}$, the multiplication rule
in $\mathcal{H}$ is described by the following formula $(h\otimes
a\in\mathcal{H};\Delta(h)\equiv h_{(1)}\otimes h_{(2)})$

$(h\otimes a)\cdot(h^{\prime}\otimes b)=h\cdot
h^{\prime}_{(1)}\otimes(a\triangleleft h^{\prime}_{(2)})b$ (55)

which uses as input the coalgebraic sector in $\mathbb{H}$.

One can propose two ways of constructing TDSR algebra (53), in analogy with
the two ways of describing the relativistic quantum NC phase space in Snyder
model (see [48, 49, 50]):

  1. (1)

by proposing the quantum twistorial map as given by Formulae (35) and (36)
(further in this section we link such a map with a cochain twist
quantization);

  2. (2)

by calculating for the quantum Hopf algebras $\mathcal{U}_{\lambda}(isu(2,2))$
and
$\mathcal{U}_{\lambda}(\bar{i}su(2,2))$ the Heisenberg double construction,
which provides the generalized twistorial quantum phase space spanned by the
quantum symmetry generators and the dual conformal quantum matrix group
coordinates (for $\kappa$-deformed Poincaré– Heisenberg double see [51, 11];
for $\theta_{\mu\nu}$-deformed Poincaré–Heisenberg double, see [52, 53]).

###  3.2 De Sitter Twistors and Length/Mass Dimensionalities

In order to introduce into the twistor algebra in (8) and (9) the
dimensionfull parameters, one can use the twistorial realization of conformal
algebra generators
$(P_{\mu},M_{(\alpha\beta)},M_{(\dot{\alpha}\dot{\beta})},\\\ K_{\mu},D)$,
where

$[P_{\mu}]=L^{-1}\qquad[M_{(\alpha\beta)}]=[M_{(\dot{\alpha}\dot{\beta})}]=[D]=L^{0},\qquad[K_{\mu}]=L$
(56)

describe the length dimensionalities $[L]=[M]^{-1}$ ($[M]$ describes the mass
dimensionality). Recalling twistorial realization of conformal algebra

$\displaystyle p_{\alpha\dot{\beta}}=\pi_{\alpha}\bar{\pi}_{\dot{\beta}}\qquad
M_{(\alpha\beta)}=(M_{(\dot{\alpha}\dot{\beta})})^{\dagger}=\pi_{(\alpha}\omega_{\beta)}$
(57) $\displaystyle D=\pi_{\alpha}\omega^{\alpha}+H.C.\qquad
K_{\alpha\dot{\beta}}=\omega_{\alpha}\bar{\omega}_{\dot{\beta}}$ (58)

we can easily deduce that the length dimensions of Weyl spinors
$(\pi_{\alpha},\omega_{\alpha})$,$(\bar{\pi}_{\dot{\alpha}},\bar{\omega}_{\dot{\alpha}})$
(see (4)) are the following

$[\pi_{\alpha}]=[\bar{\pi}_{\dot{\alpha}}]=L^{-\frac{1}{2}}\qquad[\omega_{\alpha}]=[\bar{\omega}_{\dot{\alpha}}]=L^{\frac{1}{2}}$
(59)

In particular, one can introduce the rescaled dimensionless
$([u]=[\bar{u}]=L^{0})$ twistor components as follows

$u_{A}=\left(\begin{array}[]{c}\lambda^{\frac{1}{2}}\pi_{\alpha}\\\
\lambda^{-\frac{1}{2}}\bar{\omega}^{\dot{\alpha}}\end{array}\right)\qquad\bar{u}_{A}=\left(\begin{array}[]{c}\lambda^{\frac{1}{2}}\bar{\pi}_{\dot{\alpha}}\\\
\lambda^{-\frac{1}{2}}\omega^{\alpha}\end{array}\right)$ (60)

where $\lambda$ is the fundamental length parameter. Generalizing the
particular choice in (11) and (14) for palatial twistors to the antisymmetric
tensorial matrix $\Theta_{AB}$

$\Theta_{AB}=\left(\begin{array}[]{cc}\theta_{\alpha\gamma}&\theta_{\alpha\dot{\delta}}\\\
\theta_{\dot{\beta}\gamma}&\theta_{\dot{\beta}\dot{\delta}}\end{array}\right)$
(61)

we postulate the following dimensionalities

$[\theta_{\alpha\gamma}]=L^{2}\qquad[\theta_{\alpha\dot{\delta}}]=[\theta_{\dot{\beta}\gamma}]=L\qquad[\theta_{\dot{\beta}\dot{\delta}}]=L^{0}.$
(62)

consistent with the assignment $\Theta_{AB}=I_{AB}$.

Using (59) and (29)–(30), one obtains (recall the $[\lambda]=L$) that the
expression

$f\equiv-2i\lambda\ln F=\Theta_{AB}{\hat{t}}^{A}\wedge{\hat{t}}^{B}=$
$\theta_{\alpha\gamma}\pi^{\alpha}\wedge\pi^{\gamma}+\theta_{\alpha\dot{\delta}}\pi^{\alpha}\wedge\bar{\omega}^{\dot{\delta}}+\theta_{\dot{\beta}\gamma}\bar{\omega}^{\dot{\beta}}\wedge\pi^{\gamma}+\theta_{\dot{\beta}\dot{\delta}}\bar{\omega}^{\dot{\beta}}\wedge\bar{\omega}^{\dot{\delta}}=\lambda\Theta^{(0)}_{AB}u^{A}\wedge
u^{B}$ (63)

where $\theta_{\alpha\gamma}=\lambda^{2}\theta^{(0)}_{\alpha\gamma}$,
$\theta_{\alpha\dot{\delta}}=\lambda\theta^{(0)}_{\alpha\dot{\delta}}$,
$\theta_{\beta\dot{\delta}}=\theta^{(0)}_{\beta\dot{\delta}}$ and
$\theta^{90)}_{AB}$ is dimensionless. It follows that $[f]=L$; similarly, one
gets $[\bar{f}]=L$.

Due to these numerical values of dimensionalities in front of the exponent
(63) in Formulae (29) and (30) the numerical factor $\lambda^{-1}$ appears. It
should be observed that, contrary to the case of space-time twists (16) and
(17), in both twistorial twists (29) and (30), the scaling normalization
factor is the same.

###  3.3 Twist Deformation of Twistors by Drinfeld Twist

One can construct the Drinfeld twist (see, e.g., [38, 55, 54]) by
multiplication of two-cocycle twists (29) and (30) in various ways, related by
BCH-type formulas. Such twist can be used for twist quantization of the
Heisenberg-conformal algebra (52), which becomes a quasi-bialgebroid described
by the smash product of the conformal $su(2,2)$ and canonical twistorial
Heisenberg algebra as the $su(2,2)$ module.

We consider the following cochain twist (see, e.g., [34];
$\lambda\sim\frac{1}{m_{p}}$ is real).

$\mathbb{F}=\exp\left[-\frac{1}{2\lambda}(\Theta^{AB}{\hat{t}}_{A}\wedge{\hat{t}}_{B}+\bar{\Theta}^{AB}{\hat{\bar{t}}}_{A}\wedge{\hat{\bar{t}}}_{B})\right]$
(64)

In Formula (64), the exponential factor is dimensionless. Because $\mathbb{F}$
does not satisfy the two-cocycle condition, the resulting twisted coproducts
are not coassociative and the twist quantization will generate the quasi-
bialgebroid structure.

One gets the twist-deformed quantum twistor variables
$\hat{\xi}_{R}=(\hat{t}_{A},\hat{\bar{t}}_{A})(R=1\cdots 8)$ as describing the
twist quantization of Hopf algebra module

$\hat{\xi}_{R}^{\mathbb{F}}=m(\mathbb{F}^{-1}\circ(\vartriangleright\otimes
1)\circ(\hat{\xi}_{R}\otimes 1)).$ (65)

If in (65), we insert (34) and (64), then calculate the contribution generated
by the linear $\lambda$-term in $f=\ln\mathbb{F}$, we get

$\displaystyle\hat{t}^{\mathbb{F}}_{A}=\hat{t}_{A}+\lambda^{-1}\bar{\Theta}_{A}^{B}\hat{\bar{t}}_{B}+o(\lambda^{-2})$
(66)
$\displaystyle\hat{\bar{t}}^{\mathbb{F}}_{A}=\hat{\bar{t}}_{A}-\lambda^{-1}{\Theta}_{A}^{B}\hat{t}_{B}+o(\lambda^{-2})$
(67)

i.e., the linear term in the $\lambda^{-1}$ power expression gives the
quantization maps analogous to (35) and (36). Taking only the linear term into
consideration, we obtain the following modification of twistorial CCR (see (8)
and (9))

$[\hat{\bar{t}}^{\mathbb{F}}_{A},\hat{t}^{\mathbb{F}}_{B}]=(\eta_{AB}-4\lambda^{-2}\Theta_{A}^{C}\bar{\Theta}_{BC})$
(68)
$[\hat{t}^{\mathbb{F}}_{A},\hat{t}^{\mathbb{F}}_{B}]=2\lambda^{-1}\bar{\Theta}_{AB}\qquad[\hat{\bar{t}}^{\mathbb{F}}_{A},\hat{\bar{t}}^{\mathbb{F}}_{B}]=2\lambda^{-1}{\Theta}_{AB}.$
(69)

The relations (66)–(69) without higher order terms in $\lambda^{-1}$ can be
treated as describing the quantization map for $\Theta_{AB}$-deformed
twistorial Heisenberg algebra.

The canonical conformal covariance relations for coordinate and momentum
twistors $(A,B=1\dots 4;\hat{S}_{A{B}}\in su(2,2))$

$\displaystyle[\hat{S}_{A{B}},\hat{t}_{C}]=\eta_{{B}C}\hat{t}_{A}$ (70)
$\displaystyle[\hat{S}_{A{B}},\hat{\bar{t}}_{\bar{C}}]=-\eta_{A{C}}\hat{\bar{t}}_{\bar{B}}$
(71)

after twisting by $\mathbb{F}$ do not remain valid. We arrive at higher order
terms in Formulae (66) and (67) and in (70) and (71) additional terms which
contain, besides the $su(2,2)$ generators $S_{A{B}}$ (see (22)), the bilinear
products $\hat{t}_{A}\hat{t}_{B}$, $\hat{\bar{t}}_{A}\hat{\bar{t}}_{B}$, which
together form the set of generators of $Sp(8;R)$ algebra, which is realized
linearly on the eight-dimensional real twistor space
$\rho_{R}=(\hat{t}_{A}+\hat{\bar{t}}_{A},i(\hat{t}_{A}-\hat{\bar{t}}_{A})$.
The algebra $Sp(8;R)$ is known as providing the generalization of $D=4$
conformal symmetries in the presence of tensorial central charges [56], and it
leads to numerous applications, e.g., in the Vasiliev higher spin algebras
[57, 58]. One can conclude therefore that the twist (64), which simultaneously
deforms both the twistorial coordinates and momenta, could be better adjusted
to the twist quantization of inhomogeneous generalized conformal algebra
$isp(8;R)=R^{8}\rtimes Sp(8;R)$.

We recall that, for the algebra (52) and its $\mathbb{F}$-twisted quantum
version, the coalgebraic sector can be defined only in the framework of quasi-
bialgebroids (quasi-Hopf algebroids) [8, 9, 10, 11, 12, 13, 59].

###  3.4 Heisenberg Doubles and Generalized Twistorial Quantum Phase Space

It can be shown that, in the $D=4$ space-time framework, both
$(4+4)$-dimensional quantum phase space as well as the $(10+10)$-dimensional
one which also contains the Lorentz sector can be described as the Heisenberg
doubles, providing various generalizations and extensions of Heisenberg
algebra.

The Heisenberg double is a special example of the smash product (54), when
$\mathbb{A}$ is identified with the dual Hopf algebra $\mathbb{H}^{\star}$. In
such a case, the nondegenerate bilinear Hopf pairing
$<\cdot,\cdot>:\mathbb{H}\otimes\mathbb{A}\rightarrow\mathbb{C}$ between two
Hopf algebra $\mathbb{H}$ and $\mathbb{H}^{\star}$ is used, with the following
action $\mathbb{H}\triangleright\mathbb{H}^{\star}$

$h\triangleright a=a_{(1)}<h,a_{(2)}>.$ (72)

Subsequently, one can write

$h\triangleright(ab)=a_{(1)}<h_{(1)},a_{(2)}>b_{(1)}<h_{(2)},b_{(2)}>=(h_{(1)}\triangleright
a)(h_{(2)}\triangleright b)$ (73)

in accordance with the action (48) on the Hopf algebra module. One can derive
in $\mathcal{H}$ the cross relations between the algebraic sectors of
$\mathbb{H}$ and $\mathbb{H}^{\star}$ $(h\equiv h\otimes 1,a\equiv 1\otimes
a)$

$h\cdot a=a_{(1)}<h_{(1)},a_{(2)}>h_{(2)}$ (74)

which completes the multiplication table in
$\mathbb{H}\otimes\mathbb{H}^{\star}$.

In applications of the Heisenberg double $\mathbb{H}\ltimes\mathbb{H}^{\star}$
to physical models, the Hopf-algebra $\mathbb{H}$ usually describes the
generalized quantum momenta, while the dual Hopf algebra $\mathbb{H}^{\star}$
provides the sector of generalized quantum positions. In the four-dimensional
space-time approach, one obtains the generalized $(10+10)$-dimensional quantum
phase space expressed as the Heisenberg double $\mathcal{H}^{(\mathcal{P})}$
($\mathbb{H}=\mathcal{U}(\hat{g})$ denotes the enveloping Hopf–Lie algebra and
$\mathbb{H}^{\star}=\mathcal{C}(\mathbb{G})$ is the Hopf algebra of functions
on the Lie group manifold $\mathbb{G}$):

$\mathcal{H}^{(\mathcal{P})}=\mathcal{U}(i\hat{o}(3,1))\ltimes\mathcal{C}(IO(3,1))$
(75)

where $i\hat{o}(3,1)$ describes Poincaré algebra and $IO(3,1)$ the dual
Poincaré group. In the quantum case, e.g., in applications to QG, both Hopf
algebras in (75) can be quantum-deformed in a way preserving the Hopf-
algebraic duality property, e.g., by twisting or $\kappa$-deformation [11, 51,
52].

In the twistor approach, one can choose as the generalized twistorial quantum
phase space the following Heisenberg double

$\mathcal{H}^{(\mathcal{T})}=\mathcal{U}_{\lambda}(isu(2,2))\ltimes\mathcal{C}_{\lambda}(ISU(2,2))$
(76)

where the Planck length plays the role of dimensionfull deformation parameter.
In particular, one can consider in (76) the twist deformations with twists
$\mathcal{F},\bar{\mathcal{F}}$ (see (29) and (30)). If we observe that the
twistors $\hat{t}_{A},\hat{\bar{t}}_{A}$ as well as the twists (29) and (30)
are related by the Born map (see (31)), we obtain the following table of four
Hopf algebras, describing possible twist-deformed inhomogeneous $D=4$ quantum
conformal symmetries and $D=4$ inhomogeneous quantum conformal groups

$\begin{array}[]{ccc}\mathcal{U}^{\mathcal{F}}_{\lambda}(isu(2,2))&\overset{\hbox{Hopf}}{\underset{\hbox{duality}}{\rightleftarrows}}&\mathcal{C}^{\mathcal{F}}_{\lambda}(\bar{I}SU(2,2))\\\
\hbox{Born}\updownarrow\hbox{duality}&&\hbox{Born}\updownarrow\hbox{duality}\\\
\mathcal{U}^{\mathcal{\bar{F}}}_{\lambda}(\bar{i}su(2,2))&\overset{\hbox{Hopf}}{\underset{\hbox{duality}}{\rightleftarrows}}&\mathcal{C}^{\mathcal{\bar{F}}}_{\lambda}(ISU(2,2)).\end{array}$
(77)

By Hopf duality, the twist quantization of the algebra $isu(2,2)$
$(\bar{i}su(2,2))$ is mapped into the cotwist quantization of the group
$ISU(2,2)$ ($\bar{I}SU(2,2)$) with the following properties of algebraic and
coalgebraic sectors

$\begin{array}[]{ccc}\begin{array}[]{c}\hbox{Twist quantization}\\\ \hbox{of
}\mathcal{U}(isu(2,2))\end{array}&\xleftarrow{\\!}\\!\xrightarrow[\hbox{duality}]{\hbox{Hopf\,\,}}&\begin{array}[]{c}\hbox{cotwist
quantization}\\\ \hbox{of }\mathbb{C}(ISU(2,2))\end{array}\\\
\left(\begin{array}[]{c}\hbox{multiplication in}\\\ \hbox{algebra not
changed,}\\\ \hbox{coproducts
modified}\end{array}\right)&\longleftarrow\\!\longrightarrow&\left(\begin{array}[]{c}\hbox{multiplication
in}\\\ \hbox{algebra modified,}\\\ \hbox{coproducts
unchanged}\end{array}\right).\end{array}$ (78)

In $D=4$, the most general $(23+23)$-dimensional twistorial DSR (TDSR) algebra
can be described by the deformed Heisenberg double (76) with (4 + 4) NC
degrees of freedom in $T^{4}\oplus\bar{T}^{4}$ and (15 + 15)-dimensional
conformal sector as the subalgebra described by the $su(2,2)$ Heisenberg
double.

If the Hopf algebras $\mathcal{U}(isu(2,2)),\mathcal{U}(\bar{i}su(2,2))$ are
twist-deformed, the cotwist-deformed algebras
$\mathcal{C}(\bar{I}SU(2,2)),\mathcal{C}(ISU(2,2))$ provide noncommutative
matrix entries of quantum $SU(2,2)$ group. The noncommutativity of matrix
group elements $g^{A}_{B}\in SU(2,2)$ is determined by RTT relations (see,
e.g., [26, 7])

$R^{AC}_{BD}g^{B}_{E}g^{D}_{F}=g^{D}_{C}g^{A}_{B}R^{BD}_{EF}$ (79)

where the $R$-matrix is expressed by the following cotwist formula

$R^{AC}_{BD}=(\mathcal{F}^{T}\mathcal{F}^{-1})^{AC}_{BD}.$ (80)

The cotwist $\mathcal{F}^{AC}_{BD}$ dual to the twist $\mathbb{F}$ is
determined by the following evaluation map:

$\mathcal{F}^{AC}_{BD}(g,g^{\prime})\equiv<\mathbb{F}|g^{A}_{B}\otimes
g^{\prime C}_{D}>.$ (81)

The noncommutative multiplication formula of cotwisted $SU(2,2)$ matrix
elements is given by the formula

$g\circ g^{\prime}=\mathcal{F}(g_{(1)},g_{(1)})\cdot g_{(2)}\cdot
g^{\prime}_{(2)}\cdot\mathcal{F}^{-1}(g_{(3)},g^{\prime}_{(3)}).$ (82)

Because the twistor coordinates and momenta $T^{4},\bar{T}^{4}$ as well as the
complex Minkowski space-time coordinates are expressed by the element of
$SU(2,2)$ group (see (25)–(27)), the NC multiplication rule (82) defines the
noncommutativity of cotwist-deformed twistor coordinates
$\hat{t}_{A},\hat{\bar{t}}_{A}$ as well as the complex quantum Minkowski
coordinates $\hat{z}^{\dot{\alpha}\beta}$. Further, using the cotwist-deformed
multiplication rule (82), one can also derive the cotwist deformation of
incidence relations (7).

##  4 Outlook

The aim of this paper is the presentation of some aspects of the NC framework
for quantum-deformed twistors. Our inspiration came from the paper by Penrose
[27] who under the name of palatial twistors introduced the “physical” class
of $\Theta_{AB}$-deformed dS (de Sitter) twistors, with the parameters
$\theta_{AB}$ determined geometrically by real de Sitter infinity twistor
$I_{AB}$ (see (11)).

In our scheme, we reduce the multiparameter deformations effectively to the
one-Parametric ones by using the geometric degree of freedom which describes
the variable Planck length or variable Planck mass. As an example, one can
provide the generalized $\kappa$-deformations depending on the constant four-
vector $a_{\mu}$, generating the following $a_{\mu}$-dependent quantum space-
times [60]:

$[\hat{x}_{\mu},\hat{x}_{\nu}]=i(a_{\mu}x_{\mu}-a_{\nu}x_{\mu})$ (83)

where $a_{\mu}=\lambda a^{(0)}_{\mu}$ ($[\lambda]=L$, $[a^{(0)}_{\mu}]=L^{0}$)
and the fourvector $a^{(0)}_{\mu}$ are chosen as normalized, namely
$(a^{(0)}_{\mu})^{2}=-1$ for standard time-like $\kappa$-deformation,
$a_{\mu}^{(0)2}=1$ for tachyonic space-like $\kappa$-deformation and
$a_{\mu}^{(0)2}=0$ for light cone $\kappa$-deformation.

The following are some directions in which one can continue the studies
presented in this paper:

  1. 1.

If we consider the twistor correspondence with complexified space-times, one
should introduce the pair of dual twistors $(t_{A},w_{A};t_{A}w^{A}=0)$ called
ambitwistors, not linked by complex conjugation (Hermitian conjugation in
quantized case), which provide the description of complex null geodesies in
complexified Minkowski space $\mathbb{M}_{4}^{\mathbb{C}}$ [61, 62, 63]. In
such a case, if $w^{A}=(\lambda^{\alpha},\mu_{\dot{\alpha}})$, one can
introduce the symplectic 2-form (see (13))

$\tilde{\Omega}_{2}=d(\tilde{I}^{AB}w_{A}dw_{B})=\tilde{I}^{AB}dw_{A}\wedge
dw_{B}=\frac{\lambda}{6}d\lambda_{\alpha}\wedge
d\lambda^{\alpha}+d\mu^{\dot{\alpha}}\wedge d\mu_{\dot{\alpha}}$ (84)

where $\lambda=\frac{1}{r^{2}}$ appears as the second cosmological constant.
In such a case, the curvatures $R$ and $r$ associated with twistors $t_{A}$
and $w_{A}$ can be different; in particular, if $R\gg r$ ,they may provide the
tool to describe two de Sitter geometries characterizing the cosmological
macroscopic distances and the ultrashort Planckian ones. It appears that the
duality map $t_{A}\leftrightarrow w_{A}$, which implies the interchange
relation $R\leftrightarrow r$, can be linked with Born duality relation (see,
e.g., [39, 65, 64, 40]). One can speculate that the presence of the pair of
dual radii (r,R) in ambitwistor framework can lead to the description of
quantum effects simultaneously at ultrashort (radius r) and at macroscoping
(radius R) cosmological distances.

  2. 2.

The $D=4$ twistorial construction presented here can be quite easily
generalized to $D=3$ and $D=6$ twistors, described by the $D=3$ and $D=6$
conformal groups $Sp(4;\mathbb{R})\simeq O(3,2)$ and
$U_{\alpha}(4;\mathbb{H})\simeq O(6,2)$. We add that the $D=4$ conformal group
$SU(2,2)$ can also be described as the antiunitary one
$U_{\alpha}(4;\mathbb{C})$ [66, 67]. In such a way, we deal with the
antiunitary family of groups $U_{\alpha}(4;\mathbb{F})$, where field F = R, C,
H. In addition, since the 1970s, supertwistors [68] have been studied, which
are a well recognized tool in the studies of superparticles, superstrings,
supersymmetric gauge theories and supergravity.

  3. 3.

Various quantum deformations of $SU(2,2)$ and of its complexification
$SL(4;\mathbb{C})$ have been used since the 1990s ([69, 70, 71, 72]; see also
[73]). One can recall that S. Zakrzewski, after classifying the $D=4$ Lorentz
matrices [74], proposed the algebraic technique to classify the classical
r-matrices of Poincaré algebras [75] After providing the classical SU(2,2)
r-matrices, it should be possible to obtain also the $r$-matrices for
inhomogeneous (pseudo) unitary algebras.

  4. 4.

Recently, the twistorial field-theoretic approach to formulate gauge theories
and gravity in twistor space has been promoted (see, e.g., [76, 77]), with the
dynamics described by twistorial actions. By using local twistor geometry, one
obtains in a natural way conformal gravity [78]; the twistorial model of
Einstein gravity with non-zero cosmological constant can also be obtained by
embedding into twistorial conformal gravity [76, 77]. The formulation of QG in
twistorial framework, by analogy with the approach presented in [5], may
require as well the noncommutative twistorial quantum geometry.

The paper was supported by Polish National Science Center, project
2017/27/B/ST2/01902.

## Acknowledgements

The author would like to thank Mariusz Woronowicz for the collaboration at
initial stage of this paper; Andrzej Borowiec and Mariusz Woronowicz for
valuable discussions; and Evgenij Ivanov and Maciek Dunajski, the organizers,
respectively, of Workshop SQ’19 (Erevan, August 2019) and the Conference
“Loops meet twistors” (Marseille-Luminy, September 2019), where some of the
results presented in this article have been presented.

## References

  * [1] Oriti, D. (Ed.) Approaches to Quantum Gravity. In _Collection of Papers_ ; Cambridge University Press: Cambridge, UK, 2009.
  * [2] Ashtekar, A.; Lewandowski, J. Background-independent quantum gravity: A status report. _Class. Quant. Grav._ 2004, _21_ , R53. [CrossRef]
  * [3] Rovelli, C. Quantum Gravity; Cambridge University Press: Cambridge, UK, 2004.
  * [4] Ambjorn, J.; Jurkiewicz, J.; Loll, R. Causal Dyamical Triangulations and the Quest for Quantum Gravity. In _Foundations of Space-Time_ ; Ellis, G., Murugen, J., Weltman, W., Eds.; Cambridge University Press: Cambridge, UK, 2012.
  * [5] Beggs, E.; Majid, S. Quantum Riemannian Geometry; Serie in Mathematics; Springer: Berlin/Heidelberg, Germany, 2020; Volume 355.
  * [6] Drinfeld, V.G. Quantum Groups. In Proceedings of the XXth Math. Congress, Berkeley, CA, USA, 3–11 August 1986; Berkeley Press: Berkeley, CA, USA, 1986; Volume 1, p. 798.
  * [7] Majid, S. Foundations of Quantum Groups Theory; Cambridge University Press: Cambridge, UK, 2000.
  * [8] Lu, J.-H. Hopf Algebroids and Quantum Grupoids. _Int. J. Math._ 1996, _7_ , 47–70. [CrossRef]
  * [9] Brzezinski, T.; Militaru, G. Bialgebroids. $\times_{A}$-bialgebras and duality. _J. Algebra_ 2002, _251_ , 279–294. [CrossRef]
  * [10] Juric, T.; Kovacevic, D.; Meljanac, S. $\kappa$-deformed phase space, Hopf algebroid and twisting. _SIGMA_ 2014, _10_ , 106. [CrossRef]
  * [11] Lukierski, J.; Skoda, Z.; Woronowicz, M. $\kappa$-deformed covariant quantum phase spaces as Hopf algebroids. _Phys. Lett. B_ 2015, _750_ , 401–406. [CrossRef]
  * [12] Borowiec, A.; Pachol, A. Twisted bialgebroids versus bialgebroids from a Drinfeld twist. _J. Phys. A Math. Theor._ 2017, _50_ , 055205. [CrossRef]
  * [13] Lukierski, J.; Meljanac, S.; Woronowicz, M. Quantum twist-deformed $D=4$ phase spaces with spin sector and Hopf algebroid structures. _Phys. Lett. B_ 2019, _789_ , 82–87. [CrossRef]
  * [14] Amelino-Camelia, G. Relativity in space-times with short-distance structure governed by an observer-independent (Planckian) length scale. _Int. J. Mod. Phys. D_ 2002, _11_ , 35–59. [CrossRef]
  * [15] Kowalski-Glikman, J. Observer-independent quanta of mass and length. _Phys. Lett. A_ 2001, _286_ , 391–394. [CrossRef]
  * [16] Kowalski-Glikman, J. Introduction to Doubly Special Relativity. In _Planck Scale Effects in Astrophysics and Cosmology_ ; Lecture Notes in Physics; Springer: Berlin/Heidelberg, Germany, 2005; Volume 669, pp. 131–159.
  * [17] Borowiec, A.; Pachol, A. $\kappa$-Minkowski space-times and DSR algebras: Fresh look and old Problems. _SIGMA_ 2010, _6_ , 86. [CrossRef]
  * [18] Bronstein, M. Quantum theory of weak gravitational fields. _JETP_ 1936, _9_ , 140157.
  * [19] Kowalski-Glikman, J.; Lee, S. Triply Special Relativity. _Phys. Rev. D_ 2004, _70_ , 065020. [CrossRef]
  * [20] Lukierski, J.; Ruegg, H.; Nowicki, A.; Tolstoy, V.N. q-deformation of Poincaré algebra. _Phys. Lett. B_ 1991, _264_ , 331–338. [CrossRef]
  * [21] Ballesteros, A.; Herranz, F.; del Olmo, M.; Santander, M. A new null-plane quantum Poincaré algebra. _Phys. Lett. B_ 1995, _351_ , 137–145. [CrossRef]
  * [22] Majid, S.; Ruegg, H. Bicrossproduct structure of $\kappa$-Poincaré group and non-commutative geometry. _Phys. Lett. B_ 1994, _3–4_ , 348–354. [CrossRef]
  * [23] Penrose, R. Twistor quantization and curved spacetime. _Int. J. Theor. Phys._ 1968, _1_ , 61–99. [CrossRef]
  * [24] Kapustin, A.; Kuznetzov, A.; Orlov, D. Noncommutative instantons and twistor transforms. _Comm. Math. Phys._ 2001, _221_ , 385–432. [CrossRef]
  * [25] Hannabus, K.C. Noncommutative twistor space. _Lett. Math. Phys._ 2001, _58_ , 153–166. [CrossRef]
  * [26] Brain, S.; Majid, S. Quantization of twistor theory by cocycle twist. _Commun. Math. Phys._ 2008, _284_ , 713. [CrossRef]
  * [27] Penrose, R. Palatial twistor theory and the twistor googly problem. Philos. Trans. R. Soc. A 2015, _373_ , 20140237. [CrossRef] [PubMed]
  * [28] Lukierski, J.; Woronowicz, M. Noncommutative space-time from quantized twistors. In Proceedings of the Conference in Honour of 90th Birthday of Freeman Dyson, Singapore, 26–29 September 2013; Phua, K.K., Kwek, L.C., Chang, N.P., Chan A.H., Eds.; World Scientific: Singapore, 2014; p. 407.
  * [29] Mercalli, M.; Penrose, R. Gluing noncommutative twistor space. _Q. J. Math._ 2021, _72_ , 417–454. [CrossRef]
  * [30] Penrose, R.; MacCallum, M.A.H. Twistor theory: An Approach to the quantization of fields and space-time. _Phys. Rep._ 1972, _6_ , 241–315. [CrossRef]
  * [31] Penrose, R. Nonlinear Gravitons and Curved Twistor Theory. _Gen. Relativ. Gravit._ 1976, _7_ , 31–52. [CrossRef]
  * [32] Atiyah, M.; Dunajski, M.; Mason, L. Twistor theory at fifty: From contour integrals to twistor strings. Proc. R. Soc. Lond. A 2017, _473_ , 20170530. [CrossRef]
  * [33] Chaichian, M.; Kulish, P.P.; Nishijima, K.; Tureanu, A. On a Lorentz-invariant interpretation of noncommutative space-time and its implications on noncommutative QFT. _Phys. Lett. B_ 2004, _604_ , 98–102. [CrossRef]
  * [34] Aschieri, P.; Lizzi, F.; Vitale, P. Twisting all the way: From Classical Mechanics to Quantum Fields. _Phys. Rev. D_ 2008, _77_ , 025037. [CrossRef]
  * [35] Aschieri, P.; Borowiec, A.; Pachol, A. Observables and dispersion relations in $\kappa$-Minkowski spacetime. _JHEP_ 2017, _10_ , 152. [CrossRef]
  * [36] Doplicher, S.; Fredenhagen, K.; Roberts, J. Space-time quantization induced by classical gravity. _Phys. Lett. B_ 1994, _331_ , 39–44. [CrossRef]
  * [37] Doplicher, S.; Fredenhagen, K.; Roberts, J. The quantum structure of spacetime at the Planck scale and quantum fields. _Commun. Math. Phys._ 1995, _172_ , 187–220. [CrossRef]
  * [38] Drinfeld, V.G. Quasi-Hopf algebras. _Algebra i Anal._ 1989, _1_ , 114. (In Russian)
  * [39] Born, M. Reciprocity theory of elementary particles. _Rev. Mod. Phys._ 1949, _21_ , 463. [CrossRef]
  * [40] Freidel, L.; Kowalski-Glikman, J.; Leigh, R.G.; Minic, D. The Theory of metaparticles. _Phys. Rev. D_ 2019, _99_ , 066011. [CrossRef]
  * [41] Bogolyubov, N.N. On a new method in the theory of superconductivity. _Nuovo Cim._ 1958, _7_ , 794–805. [CrossRef]
  * [42] Majid, S.; Schroers, B.J. q-Deformation and Semidualisation in 3d Quantum Gravity. _J. Phys. A_ 2009, _42_ , 425402. [CrossRef]
  * [43] Osei, P.K.; Schroers, B.J. On the semiduals of local isometry groups in 3rd gravity. _J. Math. Phys._ 2012, _53_ , 073510. [CrossRef]
  * [44] Osei, P.K.; Schroers, B.J. Classical r-matrices via selfdualisation. _J. Math. Phys._ 2013, _54_ , 101702. [CrossRef]
  * [45] Snyder, H.S. Quantized space-time. _Phys. Rev._ 1947, _71_ , 38. [CrossRef]
  * [46] Klimyk, A.; Schmudgen, K. _Quantum Groups and Their Representations_ ; Springer: Berlin, Germany, 1997.
  * [47] Blattner, R.; Cohen, M.; Montgomery, S. Crossed products and inner actions of Hopf algebras. _Trans. Am. Math. Soc._ 1986, _298_ , 671–711. [CrossRef]
  * [48] Girelli, F.; Livine, E.R. Scalar field theory in Snyder space-time: Algernatives. _JHEP_ 2011, _1103_ , 1–31. [CrossRef]
  * [49] Meljanac, S.; Pachol, A. Heisenberg doubles for Snyder type models. _arXiv_ 2021, arXiv:2101.02512.
  * [50] Meljanac, S.; Mignemi, S. Unification of $\kappa$-Minkowski and extended Snyder spaces. _Phys. Lett. B_ 2021, _814_ , 136117 [CrossRef]
  * [51] Lukierski, J.; Nowicki, A. Heisenberg double description of $\kappa$-Poincaré algebra and $\kappa$ -deformed phase space. In Proceedings of the XXI International Colloquium on Group Theoretical Methods in Physics, Goslar, Germany, 15–20 July 1996; Dobrev, V.K., Doebner, H.D., Eds.; Heron Press: Sofia, Bulgaria, 1997; p. 186.
  * [52] Borowiec, A.; Pachol, A. Heisenberg Doubles of Quantized Poincaré algebras. _Theor. Math. Phys._ 2011, _169_ , 1620–1628. [CrossRef]
  * [53] Lukierski, J.; Woronowicz, M. Two $\theta_{\mu\nu}$-deformed covariant relativistic phase spaces as Poincaré -Hopf algebroids. _Phys. Rev. D_ 2020, _101_ , 126003. [CrossRef]
  * [54] Beggs, E.; Majid, S. Nonassociative Riemann geometry by twisting. _J. Phys. Conf. Ser._ 2010, _254_ , 012002; [CrossRef]
  * [55] Beggs, E.J.; Majid, S. Quantizaton by cochain twists and nonassociative differentials. _J. Math. Phys._ 2010, _51_ , 05322 [CrossRef]
  * [56] Bandos, I.; Lukierski, J.; Sorokin, D. Superparticle models with tensorial central charges. _Phys. Rev. D_ 2000, _61_ , 045002 [CrossRef]
  * [57] Vasiliev, M.A. Higher-Spin Theories and $Sp(2M)$ Invariant Space-Time. In Proceedings of the 3rd Sakharov Conference, Moscow, Russia, 24–29 June 2002.
  * [58] Vasiliev, M.A. Higher Spin Superalgebras in any Dimension and their Representations. _JHEP_ 2004, _0412_ , 046 [CrossRef]
  * [59] Bulacu, D.; Panaite, F.; Van Oystaeyen, F. Quasi-Hopf algebra actions and smash products. _Commun. Algebra_ 2000, _28_ , 631–651. [CrossRef]
  * [60] Kosiński, P.; Maślanka, P. The kappa-Weyl group and its algebra. In From Quantum Field Theory to Quantum Groups; Jancewicz, B., Sobczyk, J., Eds.; World Scientific Publishing Company: Singapore, 2006; p. 41.
  * [61] LeBrun, C. Spaces of complex null geodesics in complex Riemannian geometry. _Trans. Am. Math. Soc._ 1983, _278_ , 209–231. [CrossRef]
  * [62] Geyer, Y.; Lipstein, A.E.; Mason, L.J. Ambitwistor Strings in Four Dimensions. _Phys. Rev. Lett._ 2014, _113_ , 081602 [CrossRef] [PubMed]
  * [63] Baston, R.J.; Mason, L.J. Conformal Gravity, the Einstein Equations and Spaces of Complex Null Geodesics. _Class. Quant. Grav._ 1987, _4_ , 815. [CrossRef]
  * [64] Freidel, L.; Rudolph, F.J.; Svoboda, D. A Unique Connection for Born Geometry. _Commun. Math. Phys._ 2019, _372_ , 119–150. [CrossRef]
  * [65] Govaerts, J.; Jarvis, P.D.; Morgan, S.O.; Low, S.G. World-line quantisation of a reciprocally invariant system. _J. Phys. A_ 2007, _40_ , 12095 [CrossRef]
  * [66] Tits, J. _Tabellen zu den Einfachen Lie Gruppen und Ihren Darstellungen_ ; Lecture Notes in Mathematics; Springer: Berlin/Heidelberg, Germany, 1987; Volume 40.
  * [67] Lukierski, J.; Nowicki, A. Quaternionic Supergroups and $D=4$ Euclidean Extended Supersymmetries. _Ann. Phys._ 1986, _166_ , 164–188. [CrossRef]
  * [68] Ferber, A. Supertwistors and conformal supersymmetry. _Nucl. Phys. B_ 1978, _132_ , 55–64. [CrossRef]
  * [69] Dobrev, V. Quantum deformations of noncompact Lie (super)algebras, Gőttingen Univ. preprint, July 1991. _J. Phys. A_ 1993, _26_ , 1317. [CrossRef]
  * [70] Lukierski, J.; Nowicki, A. Quantum deformations of $D=4$ Poincaré and Weyl algebra from $q$-deformed $D=4$ conformal algebra. _Phys. Lett. B_ 1992, _279_ , 299–307. [CrossRef]
  * [71] Lukierski, J.; Nowicki, A.; Sobczyk, J. All real forms of $Uq(sl(4;C))$ and $D=4$ conformal quantum algebras. _J. Phys. A_ 1993, _26_ , 4047. [CrossRef]
  * [72] Frydryszak, A.; Lukierski, J.; Mozrzymas, M.; Minnaert, P. On quantum deformations of $D=4$ conformal algebra. In Proceedings of the VII-th Colloquium in Quantum Groups and Integrable Systems, Prague, Czech Republic, 18–20 June 1998. [CrossRef]
  * [73] Tolstoy, V.N. Twisted quantum deformations of Lorentz and Poincaré algebra. _Bulg. J. Phys._ 2008, _35_ , 441–459.
  * [74] Zakrzewski, S. Poisson structures on the Lorentz group. _Lett. Math. Phys._ 1994, _32_ , 11. [CrossRef]
  * [75] Zakrzewski, S. Poisson structures on the Poincaré groups. _Commun. Math. Phys._ 1997, _185_ , 285–311. [CrossRef]
  * [76] Adamo, T.; Mason, L. Conformal and Einstein Gravity from twistor action. _Class. Quant. Grav._ 2014, _31_ , 045014. [CrossRef]
  * [77] Adamo, T. Twistor Actions for Gauge Theory and Gravity. Ph.D. Thesis, University of Oxford, Oxford, UK, 2013
  * [78] Howe, P.S.; Lindström, U. Superconformal geometries and local twistors. _arXiv_ 2020, arXiv:2012.03282.