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\begin{document}
\centerline{\Large\bf Twisting $\kappa$-deformed phase
space\footnote{Supported by KBN grant No. 5 P03B 106
21}}\bigskip\bigskip\bigskip

\centerline{Piotr Czerhoniak} \bigskip\centerline{\it Institute of
Physics, University of Zielona G\'ora,}\centerline{\it ul.
Podg\'{o}rna 50, 65-246 Zielona G\'{o}ra, Poland}\bigskip\bigskip


\begin{abstract}
We briefly discuss the twisting procedure applied to the
$\kappa$-defor\-med space-time. It appears that one can consider
only two kinds of such twistings: in space and time directions.
For both types of twisitngs we introduce related phase spaces and
consider briefly their properties. We discuss in detail the
changes of duality relations under the action of twist.
\end{abstract}\bigskip


\section{Introduction}\bigskip\noindent Recently, some suggestions
appeared that the classical Lorentz invariance should be treated
as an approximate symmetry in ultra-high energy processes (shift
of GZK kinematic threshold) and the relativistic space-time
symmetries on Planck scale should be modified \cite{COM1}. There
are also some theoretical predictions coming from the string
theory and quantum gravity models that space-time at very short
distances of order Planck length should be quantum i.e.
noncommutative \cite{DFR}. One can modify the standard Lorentz or
Poincar\'{e} relativistic symmetry in two different ways:
obtaining a commutative space-time (as in the standard
relativistic theory) or noncommutative space-time with space and
time non-commuting variables.\medskip\\The first type of modified
relativistic symmetries follows from the concept of the double
special relativistic (DSR) theory introduced by Amelino-Camelia
\cite{GAC1}, where two observer-independent parameters (scales) --
$c , \lambda_p$ -- velocity of light and Planck length play the
fundamental role. In this framework, two basic models of such DSR
theory are considered: DSR1 theory proposed by Amelino-Camelia and
DSR2 theory considered by Magueijo and Smolin \cite{MS}. In both
proposals the energy-momentum vector space is extended to the
four-momentum algebra considered as enveloping algebra of this
linear energy-momentum space. In this algebra the nonlinear
transformation of the basis vectors (of energy-momentum space) is
realized according to DSR1 or DSR2 theory models. In this way we
get four non-linearly transformed vectors belonging to the
energy-momentum algebra which are assumed to represent physical
energy and momentum. Using this transformed four-momentum basis we
obtain for instance the standard dispersion relation in a modified
form which is, however, quite equivalent (under an inverse
nonlinear transformation) to the standard one. One can also
consider the action of the relativistic symmetry on physical
energy and momentum as a nonlinear realization of the Lorentz
group and express the addition laws of energy and momentum by the
coproduct (non-linearly transformed trivial coproduct) \cite{LN}
and extending in this way the energy-momentum algebra to a
bialgebra structure. Space-time related to this bialgebra
structure is defined as a dual bialgebra where  multiplication of
the space-time variables follows from the coproduct. The momentum
coproduct is symmetric, therefore space-time is commuting. In
consistency with conclusions in \cite{LN}, recently it has been
observed \cite{AT} that DSR theories with symmetric coproduct law
describes  just the standard special relativity framework in
nonlinear disguise.\medskip\\The second type of modified
relativistic symmetries bases on the concept of the Hopf algebra
(quantum group) \cite{ABE}, \cite{DR1} and quantum deformations of
the classical Lorentz or Poincar\'{e} symmetry \cite{M}. In this
approach there is a distinguished deformation, the so called
$\kappa$-deformation of the Poincar\'{e} symmetry \cite{LNR},
where $O(3)$-rotational symmetry is not deformed. The Hopf algebra
structure of this deformation will be discussed in detail in
Section 2. However, we would like to stress that the four-momentum
algebra for DSR1 theory and the $\kappa$-deformed theory (in
bicrossproduct basis) are the same, and give us the same physical
predictions. The main difference between both theories lies in
their coalgebra sectors, because in the $\kappa$-deformed
Poincar\'{e} algebra the coproduct of the four-momentum is
non-symmetric. This fact has important consequences for a
possibility of physical interpretation of the $\kappa$-deformed
addition law of energy-momentum. On the other hand, this
non-symmetricity of coproduct allows us to obtain
non-commutativity of space-time in the Hopf algebra
framework.\medskip\\Further, we shall consider $\kappa$-deformed
space-time defined as a dual Hopf algebra to the $\kappa$-deformed
four-momentum algebra considered as a subalgebra of
$\kappa$-Poincar\'{e} Hopf algebra i.e. the space-time is
understood in Majid and Ruegg sense \cite{MR}. Such a space-time
has a Hopf algebra structure given by coproduct of trivial form.
In this paper we shall consider a twisting operation acting on the
$\kappa$-deformed space-time algebra. The notion of twisting of
Hopf algebra was introduced by Drinfeld \cite{DR} and then applied
to enveloping algebras of simple Lie algebras \cite{RE} and the
applications to physical symmetry algebras one can find in
\cite{APL} and recently in $\kappa$-deformations of Weyl and
conformal symmetries \cite{LLM}.\\For a given pair of space-time
and four-momentum algebras one can define a phase-space algebra
using the notion of cross product algebra \cite{M}. It is well
known that although both paired algebras have a Hopf algebra
structure, the phase space algebra is no longer a Hopf algebra.
The case of $\kappa$-deformed phase space is considered in
\cite{LN1}, \cite{ALN}.\medskip\\In Section 2 we recall the Hopf
algebra structure of $\kappa$-Poincar\'{e} algebra in a very
useful bicrossproduct basis \cite{MR} in which the Lorentz algebra
sector has a classical form. We also discuss the $\kappa$-deformed
space-time underlying the role of duality relations.\medskip\\In
Section 3.1 we show that one can define only two kinds of twisting
operations on $\kappa$-deformed space-time: twisting in space
directions (SD) and in time direction (TD). For both cases we
derive their Hopf algebra structure. In Section 3.2 we extend our
dual pair to a phase space algebra and we find phase space
commutation relations. It appears that the SD-twisted phase space
can not be considered as a linear vector space because of
nonlinear phase space commutation relations.\medskip\\ Section 4
is devoted to the derivation of duality relations between the
twisted space-time algebra and the momentum algebra for simplicity
in two dimensional space-time. It is more a computational part of
the paper where we find that the space-time twisting effectively
changes only duality relations. In this way the basis of twisted
space-time algebra becomes non-orthogonal (one can say "twisted")
with respect to the momentum basis.

\section{$\kappa$-Poincar\'{e} algebra in the bicrossproduct
basis and $\kappa$-deformed space-time}\bigskip\noindent Let us
recall the structure of $\kappa$-Poincar\'{e} algebra given in the
bicrossproduct basis \cite{MR}\medskip\\- non-deformed ({\it
classical}) Lorentz algebra
($g_{\mu\nu}=(-1,1,1,1)$)\begin{equation} [M_{\mu\nu},
M_{\rho\tau}] = i(g_{\mu\rho}M_{\nu\tau}+
g_{\nu\tau}M_{\mu\rho}-g_{\mu\tau}M_{\nu\rho}
-g_{\nu\rho}M_{\mu\tau})\, .\label{r21}\end{equation}-  deformed
covariance relations ($M_i= \frac12\epsilon_{ijk} M_{jk}, \
N_i=M_{i0}$)
\begin{equation}
\begin{array}{lll}
[ M_i , p_j ] & = i\epsilon_{ijk}p_k\, ,\qquad & [ M_i , p_0 ]  =
0\, ,\medskip\cr [ N_i , p_0 ] & = i p_i\, , \qquad & [ p_\mu ,
p_\nu ] = 0\, ,\medskip\cr[ N_i , p_j ]& =
i\delta_{ij}\left[\kappa c\sinh(\frac {p_0}{\kappa c})
e^{-\frac{p_0}{\kappa c}}  + \frac1{2\kappa c} (\vec p)^2\right] -
\frac {i}{\kappa c} p_i p_j\,
.&\end{array}\label{r22}\end{equation}\bigskip\\where \quad
$\kappa$ - massive deformation parameter and\quad c - light
velocity.\\One can extend this algebra to a Hopf algebra defining
the coalgebra sector by {\it coproduct} $\Delta(X)$
\begin{equation}
\begin{array}{ll}
\Delta(M_i) = & M_i\otimes 1 + 1\otimes M_i\medskip\, ,\cr
\Delta(N_i) = & N_i\otimes 1 + e^{-\frac{p_0} {\kappa c}}\otimes
N_i + {\frac{1}{\kappa c}}\epsilon_{ijk}p_j\otimes M_k\, ,
\medskip\cr \Delta(p_0) = & p_0\otimes 1 + 1\otimes p_0\, ,\medskip\cr
\Delta(p_i) = & p_i\otimes 1 + e^{-{\frac{p_0}{\kappa c}}}\otimes
p_i\, .\end{array}\label{r23}\end{equation}with {\it antipode}
$S(X)$ and {\it counit} $\epsilon(X)$
\begin{equation}\begin{array}{lll}S(M_i) & = -M_i\, ,\qquad & S(N_i)
 = -e^{\frac{p_0}{\kappa c}}N_i+{\frac{1}{\kappa
c}}\epsilon_{ijk}e^{\frac{p_0}{\kappa c}}p_j M_k\, ,\medskip\cr
S(p_0)& = -p_0\, , & S(p_i) = -p_i e^{\frac{p_0}{\kappa c}}\, .
\end{array}\label{r24}\end{equation} \begin{equation} \epsilon(X) \ = \
0\, ,\quad \epsilon(1) \ = \ 1\, ,\qquad for \qquad X= M_i, N_i,
p_\mu\, .\label{r25}\end{equation}Therefore, in the
$\kappa$-Poincar\'{e} Hopf algebra one can distinguish the
following three subalgebras\\- classical Lorentz algebra $\cal{L}
= \{M_{\mu \nu}\}$ given by (\ref{r21}),\\- non-deformed
$O(3)$-rotation algebra ${\cal{M}} = \{M_i , \Delta , S ,
\epsilon\}$ as a Hopf subalgebra with trivial coproduct
$\Delta(M_i)$,\\- abelian four-momentum algebra ${\cal{P}}_\kappa
= \{p_\mu , \Delta , S , \epsilon \}$ as a Hopf subalgebra with
non-symmetric coproduct $\Delta(p_\mu)$ given by the
relations\begin{equation}\begin{array}{lll}[ p_\mu , p_\nu ] & = \
0\, ,\qquad & \epsilon( p_\mu ) \ = \ 0\, ,\medskip\cr \Delta(p_0)
& = \ p_0\otimes 1 + 1\otimes p_0\, , & \Delta( p_i ) \ = \
p_i\otimes 1 + e^{-\frac{p_0}{\kappa c}}\otimes p_i\, ,\medskip\cr
S( p_0 ) & = \ -p_0\, , & S(p_i) \ = \ -p_i e^{\frac{p_0}{\kappa
c}}\, .\end{array}\label{r26}\end{equation}The Hopf four-momentum
algebra ${\cal{P}}_\kappa$ can be considered as a deformation of
universal enveloping algebra of classical translation algebra,
generated (in Drinfel'd sense \cite{DR1}) by polynomial functions
of linear momentum generators $p_\mu$ (translations).\\To define
the $\kappa$-deformed space-time we adopt Majid and Ruegg point of
view \cite{MR}, namely, using this algebra ${\cal{P}}_\kappa$ we
define $\kappa$-deformed space-time algebra ${\cal{X}}_\kappa$ in
a natural way, as a dual algebra to four-momentum translation
algebra ${\cal{P}}_\kappa$. One can assume the standard duality
relations between linear bases of both algebras, respecting Hopf
algebra structure.\\Let ${\cal{X}}_\kappa$ be generated by
space-time variables ${\cal{X}}_\kappa = \{x_0 , \vec{x} \}$ dual
to four-momentum $p_\mu$ satisfying the following duality
relations
\begin{eqnarray}&< p_\mu , x_\nu > = i\,g_{\mu\nu}\,
,\qquad\qquad & g_{\mu \nu} = (1 , -1 , -1 , -1 )\,
.\label{r27a}\medskip\\&< p_\mu , 1 > = < 1 , x_\mu > = 0\,
,\quad& < 1, 1 > = 1\, .\label{r27b}\medskip\\&< p\, q , x > = < p
\otimes q , \Delta(x) >\, , & < p , x\, y > = < \Delta(p) ,
x\otimes y >\, .\label{r27c}\end{eqnarray}where $p_\mu, p , q\in
{\cal{P}}_\kappa\, ,\quad x_\nu , x , y\in {\cal{X}}_\kappa$,\\or
using Sweedler coproduct notation (see for instance \cite{M})
$\Delta(a) = a_{(1)}\otimes a_{(2)}$ relations (\ref{r27c}) can be
rewritten as\begin{eqnarray}< p\, q , x > &=& < p \otimes q ,
\Delta(x) > \ = \ < p , x_{(1)} > < q , x_{(2)}
>\, .\label{r28a}\medskip\\< p , x\, y > &=& < \Delta(p) ,
 x\otimes y > \ = \ < p_{(1)} , x > < p_{(2)} , y >\, .
\label{r28b}\end{eqnarray}Relations (\ref{r26}-\ref{r27c}) do not
describe the Hopf algebra structure of space-time algebra
uniquely, in particular they describe a wide class of coproducts
$\Delta(x_\nu)$. The form of space-time coproduct depends on the
choice of higher oder duality relations and vice versa. If we
assume orthogonality of space-time and four-momentum monomial
basis in the form ($i , j , k , l , m , n = 1, 2,
3$)\begin{equation} < {p_i}^m\,{p_0}^n , {x_j}^k\,{x_0}^l
> \ = \ m!\,n!\,\delta_{m k} \delta_{n l} \delta_{i j} < p_0 ,x_0
>^n < p_i , x_i >^m\, .\label{r29}\end{equation}then we get the
following form of non-commuting space-time Hopf algebra \cite{MR}
\begin{eqnarray} [ x_0 , x_i ]& = \ < e^{-\frac{p_0}{\kappa c}} ,
x_0 > x_i \ = \ -\frac{i}{\kappa c} x_i\, ,\qquad\qquad\quad & [
x_i , x_j ] \ = \ 0\, .\label{r210a}\bigskip\\ \Delta_0(x_\mu)& =
\ x_\mu\otimes 1 + 1\otimes x_\mu\, ,\quad S(x_\mu) \ = \ -
x_\mu\, ,\quad & \epsilon(x_\mu) \ = \ 0\,
.\label{r210b}\end{eqnarray} with trivial, symmetric coproduct
$\Delta_0(x_\mu)$.\\In this place we would like to stress the
difference between the notion of space-time algebra and the
concept of deformed (quantum or $\kappa$-deformed) space-time --
the linear span of space-time variables $x_\mu$. For the commuting
(classical) space-time both notions are equivalent -- commuting
space-time algebra is simply the algebra of commuting (polynomial)
functions on space-time. Classical space-time can be regarded as a
dual to the four-momentum (translation) space because of trivial
momentum coproduct. However, in the deformed (quantum) case the
momentum coproduct (\ref{r26}) contains the exponential factor
which belongs to the momentum algebra. It is the reason why we use
the notion of space-time algebra for non-commuting
space-time.\medskip\\Because the space-time variables $x_\mu$ do
not commute among themselves, one can also choose a space-time
monomial basis with opposite ordering and satisfying the duality
relations \cite{LN1}\begin{eqnarray}< {p_i}^m\,{p_0}^n ,
{x_0}^l\,{x_j}^k
> &=& \frac{l!\,\delta_{k m} \delta_{i j}}{(l-n)!}< p_i , x_i >^m
< p_0 , x_0 >^n < e^{-\frac{m p_0}{\kappa c}} , x_0^{l-n}
>\, ,\bigskip\cr < {p_i}^m\,{p_0}^n , {x_0}^l\,{x_j}^k > &=&
0\qquad\qquad for\qquad n > l\,
.\label{r211}\end{eqnarray}therefore, for this ordering, we obtain
non-orthogonal duality relations for $0\leq n \leq l$. Both
relations (\ref{r29}) and (\ref{r211}) one can rewrite in a more
convenient form using the exponential generating function
\begin{eqnarray}< {p_i}^m\,e^{\xi p_0} ,\, {x_j}^k\, x_0^l
> &=& < e^{\xi p_0} ,\, x_0^l > < {p_i}^m , {x_j}^k >\, .\label{r212a}
\medskip\\ < {p_i}^m\,e^{\xi p_0} , \,x_0^l\, {x_j}^k > &=&
 < e^{\xi p_0 - \frac{m p_0}{\kappa}} ,\, x_0^l > < {p_i}^m ,\,
{x_j}^k > =\nonumber\medskip\\&=& < e^{\xi p_0} , (x_0 -
i\frac{k}{\kappa c} )^l > < {p_i}^m ,\, {x_j}^k >\,
.\label{r212b}\end{eqnarray}Comparing these formulae one can easy
obtain the general form of the space-time commutation
relations\begin{equation}[ {x_i}^k , {x_0}^l ] \ = \ {x_i}^k
\left\{ {x_0}^l - \left( x_0 - i\frac{k}{\kappa
c}\right)^l\right\}\, .\label{r213}\end{equation}in the
right-time-ordered basis (\ref{r29}) (with all powers of the time
variable $x_0$ on the right).\\We would like to notice, that the
duality relations (\ref{r29}) have the same form as in the case of
the classical Poincar\'{e} algebra with trivial Hopf structure and
its commuting dual space-time algebra. This duality relation can
be rewritten in an equivalent form (see
\cite{MR})\begin{equation}< f(p_i , p_0) , :\phi( x_j , x_0 ): > \
= \ f\left(-i\frac{\partial}{\partial x_i} , i
\frac{\partial}{\partial x_0}\right) \phi ( 0 , 0 )\,
.\label{r214}\end{equation}for polynomial functions $ f , \phi $
\, and $:\phi:$ denotes right-time-ordered polynomial.\\Further,
we shall consider the coproducts of space-time variables related
to $\Delta_0(x_\mu)$ by twisting procedure, and we shall discuss
twisted duality relations.

\section{Twisted $\kappa$-deformed space-time and phase
space}\subsection{Twisting procedure}It is well known, that
$\kappa$-deformed space-time (\ref{r210a}) can be extended to a
Hopf algebra up to similarity transformation (twisting) in the
coalgebra sector. The choice of coproduct in the form
(\ref{r210b}) is the simplest one, however one can also consider a
general class of twisted coproducts \cite{RE}, \cite{APL}
\begin{equation}\Delta^F( x_\mu ) \ = \ F \Delta_0( x_\mu )
F^{-1}\, .\label{r31}\end{equation}where $(a\otimes b)(c\otimes d)
= ac\otimes b d $ and an invertible twisting function $F \in
{\cal{X}}_\kappa\otimes {\cal{X}}_\kappa$ satisfies the additional
Hopf structure requirements which follow from the properties of
twisted coproduct $\Delta^F( x_\mu)$\medskip\\{\it algebra
homomorphism}
\begin{equation} [\Delta^F( x_0 ) , \Delta^F( x_i ) ] \ = \
-\frac{i}{\kappa c} \Delta^F( x_i )\, ,\qquad  [ \Delta^F( x_i ) ,
\Delta^F( x_j ) ] \ = \ 0\, .\label{r32}\end{equation}{\it
coassociativity}\begin{equation}(\Delta^F\otimes 1) \Delta^F
(x_\mu) \ = \ (1\otimes \Delta^F ) \Delta^F (x_\mu)\,
.\label{r33}\end{equation}{\it consistency
relations}\begin{eqnarray}(\epsilon\otimes 1)\circ \Delta^F( x_\mu
) &=& (1\otimes \epsilon)\circ \Delta^F( x_\mu ) \ = \ x_\mu\,
.\label{r34}\medskip\\( S^F\otimes 1 )\circ \Delta^F( x_\mu ) &=&
(1\otimes S^F)\circ \Delta( x_\mu ) \ = \ 0\,
.\label{r35}\end{eqnarray}where we denote $(a\otimes b)\circ
(c\otimes d) = a c b d $. We would like to notice, that twisted
space-time is defined by twisted coproduct $\Delta^F$  and
antipode $S^F$.\bigskip\\The coassociativity condition (\ref{r33})
can be rewritten in a more familiar form as a condition imposed on
the twisting function $F$. It is easy to derive the following
relations\begin{eqnarray} (1\otimes \Delta^F ) \Delta^F (x ) &=&
(1\otimes F) ( 1\otimes \Delta_0 ) \Delta^F ( x )\cr &=& (1\otimes
F) ( 1\otimes \Delta_0 ) F \Delta_0( x ) F^{-1}\, .\bigskip\\
(\Delta^F \otimes 1 ) \Delta^F (x ) &=& ( F\otimes 1) (\Delta_0
\otimes 1 ) \Delta^F ( x )\cr &=& ( F\otimes 1) ( \Delta_0\otimes
1 ) F \Delta_0( x ) F^{-1}\, .\label{r351}\end{eqnarray}Comparing
both formulae we derive frequently used condition imposed on the
function $F$\begin{equation}(1\otimes F) ( 1\otimes \Delta_0 ) F \
= \ ( F\otimes 1) ( \Delta_0\otimes 1 ) F\,
.\label{r352}\end{equation}Without loss of generality we assume
the following exponential form \cite{RE} of the twisting
function\begin{equation} F \ = \ \exp\left(\sum\,\phi_n\otimes
\phi^n\right)\, .\label{r353}\end{equation}where $\phi_n , \phi^n
\in {\cal{X}}_\kappa$. Then the coassociativity condition
(\ref{r352}) one can express by the formula\begin{equation}
e^{1\otimes \phi_n\otimes \phi^n} e^{\phi_n\otimes
\Delta_0(\phi^n)} \ = \ e^{\phi_n\otimes \phi^n\otimes 1}
e^{\Delta_0(\phi_n)\otimes \phi^n}\,
.\label{r354}\end{equation}which is satisfied if the following
relations hold\begin{equation}[ \phi_n\otimes \phi^n\,, \Delta_0(
\phi^m ) ] \ = \ 0\, ,\quad \Delta_0 ( \phi^n ) \ = \ \phi^n
\otimes 1 + 1\otimes \phi^n\,
.\label{r355}\end{equation}therefore, the elements $\phi_n ,
\phi^n$ are commuting ones with trivial coproduct.\bigskip\\The
$\kappa$-deformed space-time algebra contains just two abelian
subalgebras, first generated by the time variable $x_0$ and the
other one by the space variables $x_i$ (\ref{r210a}), both with
trivial coproduct (\ref{r210b}). Using this fact one can construct
two twisting functions satisfying the relations
(\ref{r31}-\ref{r34})\\{\it twisting of space directions}
(SD)\begin{equation}F_0( a ) \ = \ e^{a\,x_0\otimes x_0}\,
.\label{r36}\end{equation}{\it twisting of time direction}
(TD)\begin{equation}F( b ) \ = \ e^{b_{i j}\,x_i\otimes x_j}\,
.\label{r37}\end{equation}where $a,\, b_{ij}\in \C$ in general
case. If we consider the space-time Hopf algebra with involution
$*$ satisfying $(a\otimes b)^*=a^*\otimes b^*$ then, the
assumption of hermiticity of space-time generators $x^*_\mu =
x_\mu$ implies the unitarity of twisting functions i.e. $a,\,
b_{ij}\in i\,\R$ are pure imaginary complex numbers.

\noindent For the twisting function $F_0(a)$, using formulae
(\ref{r31}) and (\ref{r35}) we obtain SD-twisted space-time Hopf
algebra ${\cal{X}}_\kappa(\alpha)$ hermitian basis)
\begin{eqnarray} [ x_0 , x_i ] &=& -\frac{i}{\kappa c} x_i\, ,
\qquad\qquad  [ x_i , x_j ] \ = \ 0\,
,\nonumber\bigskip\\\Delta^{F_0(a)}(x_0) &=& \Delta_{\alpha}( x_0
) \ = \ x_0\otimes 1 + 1\otimes x_0\, ,\nonumber\bigskip\\
\Delta^{F_0(a)}(x_i) &=& \Delta_{\alpha}( x_i ) \ = \ x_i\otimes
e^{\alpha x_0} + e^{\alpha x_0}\otimes x_i\,
,\label{r38}\bigskip\\ S^{F_0(a)}( x_i) &=& S_\alpha( x_i) \ = \ -
x_i e^{\alpha(\frac{i}{\kappa c} - 2 x_0)}\, ,\nonumber\bigskip\\
S^{F_0(a)}( x_0) &=& S_\alpha( x_0 ) \ = \ - x_0\,
,\qquad\qquad\quad \epsilon( x_\mu ) \ = \ 0\,
.\nonumber\end{eqnarray}where\begin{equation}\alpha\equiv a
<e^{-\frac{p_0}{\kappa c}} , x_0 > \ = \ -\frac{i\,a}{\kappa c}\in
\R\, .\label{r39}\end{equation}and similarly, choosing the
twisting function as $F(b)$ we get the TD-twisted space-time
Hopf-algebra ${\cal{X}}_\kappa(\beta)$ hermitian basis)
\begin{eqnarray} [ x_0 , x_i ] &=& -\frac{i}{\kappa c} x_i\, ,
\qquad\qquad  [ x_i , x_j ] \ = \ 0\, ,\nonumber\bigskip\\
\Delta^{F(b)}(x_0) &=& \Delta_{\beta}( x_0 ) \ = \ x_0\otimes 1 +
1\otimes x_0 + \beta_{i j}\,x_i\otimes x_j\, ,\nonumber\bigskip\\
\Delta^{F(b)}(x_i) &=& \Delta_{\beta}( x_i ) \ = \ x_i\otimes 1 +
1\otimes x_i\, ,\label{r310}\bigskip\\ S^{F(b)}( x_i) &=& S_\beta(
x_i) \ = \ - x_i \, ,\nonumber\bigskip\\ S^{F(b)}( x_0) &=&
S_\beta( x_0) \ = \ - x_0 + \beta_{i j}\,x_i x_j\,
,\qquad\qquad\quad \epsilon( x_\mu ) \ = \ 0\, .\nonumber
\end{eqnarray} where\begin{equation}\beta_{i j} \ = \ \frac{2
i}{\kappa c}\,b_{i j}\in \R\, .\label{r311}\end{equation}One can
put an additional condition on the time variable $x_0$, to be the
rotational scalar, therefore one can assume\begin{equation}[
\Delta( M_k ) , \Delta_\beta( x_0 ) ] \ = \ 0\, .
\label{r312}\end{equation}and using (\ref{r23}) we find the
condition $\beta_{i j} = \beta\,\delta_{i j}$, therefore
consequently also in this case twisting in time direction is
described by one parameter $b$. It is obvious, that in the limit
$\alpha,\,\beta\rightarrow 0$ we obtain a $\kappa$-deformed
space-time algebra given by (\ref{r210a}-\ref{r210b}).

\subsection{Phase space as cross product algebra}Let us notice,
that we have two pairs of dual Hopf algebras
${\cal{X}}_\kappa(\alpha)\otimes {\cal{P}}_\kappa$ and
${\cal{X}}_\kappa(\beta)\otimes {\cal{P}}_\kappa$ which in the
non-deformed limit $\kappa\to\infty$ turn out to be classical
space-time and momentum algebras with multiplication defined by
the commutator, therefore we should get the quantum-mechanical
phase space with standard Heisenberg commutation relations.\\In
order to construct such a deformed phase space algebra
$\Pi_\kappa$ isomorphic as a vector space to $\Pi_\kappa \sim
{\cal{X}}_\kappa\otimes {\cal{P}}_\kappa$ one has to extend the
commutation relations (\ref{r26}) and (\ref{r38}) or (\ref{r310})
by adding a cross commutators between ${\cal{X}}_\kappa$ and
${\cal{P}}_\kappa$.\\It appears that a consistent construction of
phase space $\Pi_\kappa$ can be done using the notion of a left
(right) cross product (smash product) algebra \cite{M}. For
simplicity, we shall consider only left cross product
algebra.\medskip\\One can define a {\it left action}
(representation) of the momentum algebra ${\cal{P}}_\kappa$ on the
space-time algebra ${\cal{X}}_\kappa$ as a linear map
\begin{equation}\triangleright :\, {\cal{P}}_\kappa\otimes
{\cal{X}}_\kappa\rightarrow {\cal{X}}_\kappa :\, p\otimes
x\rightarrow p\triangleright x\, .\label{r313}\end{equation}such
that\begin{equation} (p\tilde{p})\triangleright x \ = \
p\triangleright (\tilde{p} \triangleright x)\, ,\quad
1\triangleright x \ = \ x\, .\label{r314}\end{equation}We choose
the following left action\begin{equation} p\triangleright x \ = \
x_{(1)} < p\,, x_{(2)} >\, .\label{r315}\end{equation} therefore
${\cal{X}}_\kappa$ is a left ${\cal{P}}_\kappa$-module or even a
left ${\cal{P}}_\kappa$-module algebra, because ${\cal{X}}_\kappa$
and ${\cal{P}}_\kappa$ are also Hopf algebras and the left action
(\ref{r315}) satisfies\begin{equation} p\triangleright
(x\tilde{x}) \ = \ (p_{(1)}\triangleright x) (p_{(2)}
\triangleright \tilde{x})\, ,\quad p\triangleright 1=\epsilon(p)
1\, .\label{r316}\end{equation}This implies that we can regard the
twisted space-time as the left $\kappa$-deformed momentum
${\cal{P}}_\kappa$-module algebra for both choices of twisting
functions (\ref{r36}) and (\ref{r37}) with the following left
action (\ref{r315}) in the case of SD-twisted space-time
(\ref{r38})\begin{eqnarray} p_0 \triangleright x_0 & = \ i\,
,\qquad\qquad & p_i \triangleright x_0 \ = \ 0\, ,\cr p_0
\triangleright x_i & = \ i\,\alpha x_i\, ,\qquad\,\, & p_i
\triangleright x_j \ = \ -i\,\delta_{i j} e^{\alpha x_0}\, .
\label{r317}\end{eqnarray}and in the case of TD-twisted space-time
(\ref{r310}) we get\begin{eqnarray} p_0 \triangleright x_0 & = \
i\, ,\qquad & p_k \triangleright x_0 \ = \ -\,i \beta_{i k} x_i\,
,\cr p_0 \triangleright x_i & = \ 0\, ,\qquad\,\, & p_k
\triangleright x_i \ = \ -i\,\delta_{k i} \,
.\label{r318}\end{eqnarray}We recall the definition of a {\it left
cross product algebra} \cite{M}.\medskip\\Let ${\cal{P}}_\kappa$
be a Hopf algebra and ${\cal{X}}_\kappa$ a left
${\cal{P}}_\kappa$-module algebra. A left {\it cross product
algebra} $\Pi_\kappa = {\cal{X}}_\kappa
>\!\!\!\triangleleft{\cal{P}}_\kappa$ is a vector space
${\cal{X}}_\kappa\otimes
{\cal{P}}_\kappa$ with the product ({\it left cross
product})\begin{equation}(x\otimes p)(\tilde{x}\otimes \tilde{p})
\ = \ x(p_{(1)}\triangleright \tilde{x})\otimes p_{(2)}\tilde{p}\,
.\label{r319}\end{equation}and the unit element $1\otimes 1$,
where $x , \tilde{x} \in {\cal{X}}_\kappa$ and $p , \tilde{p} \in
{\cal{P}}_\kappa$. It appears that $\Pi_\kappa = {\cal{X}}_\kappa
>\!\!\!\triangleleft{\cal{P}}_\kappa$ is an associative algebra,
however it can not be extended to a Hopf algebra
(\cite{M}).\medskip\\The obvious isomorphism ${\cal{X}}_\kappa\sim
{\cal{X}}_\kappa\otimes 1$, ${\cal{P}}_\kappa\sim 1\otimes
{\cal{P}}_\kappa$ allows us to define the commutator for the whole
phase space $\Pi_\kappa$
\begin{equation}[x,p] \ = \ x\circ p-p\circ x\, ,
\qquad  x\circ p \ = \ x\otimes p\, , \qquad p\circ
x=(p_{(1)}\triangleright x)\otimes p_{(2)}\,
.\label{r320}\end{equation}Using this definition and formulae
(\ref{r38}) and (\ref{r317}) we obtain the commutation relations
for the linear basis of SD-{\it twisted phase space}
$\Pi_\kappa(\alpha) = {\cal{X}}_\kappa(\alpha)
>\!\!\!\triangleleft{\cal{P}}_\kappa$\begin{equation}
\begin{array}{lll} [ x_0 , x_i ] \ = & -\frac{i}{\kappa c} x_i\, ,
\qquad\qquad & [ x_i , x_j ] \ = \ 0\, ,\medskip\cr [ x_i , p_0 ]
\ = & -i\,\alpha x_i\, , & [ x_0 , p_i ] \ = \ \frac{i}{\kappa c}
p_i\, ,\medskip\cr [ x_0 , p_0 ] \ = & - i\, , & [ p_\mu , p_\nu ]
\ = \ 0\, ,\bigskip\cr [ x_i , p_j ] \ = & i\,\delta_{i j}
e^{\alpha x_0} + \left(1 - e^{-\frac{i \alpha}{\kappa c}}\right)
x_i\,p_j& \, .\end{array}\label{r321}\end{equation}Let us notice
that because of the exponential term in the last relation, the
SD-twisted phase space can be considered only as an algebra.\\In
the limit $\alpha \to 0$ we obtain the standard $\kappa$-deformed
phase space considered in \cite{LN1}, a deformed generalization of
Heisenberg algebra. It is interesting to notice that one can also
consider the limit $\kappa\to\infty , \alpha=const.$ (i.e. one can
assume the linear dependence of the twisting parameter $a$ on the
deformation parameter $\kappa$, see (\ref{r39})) of phase space
$\Pi_\kappa(\alpha) \to \Pi_\infty(\alpha)$ given by the
non-vanishing commutators\begin{equation}[ x_0 , p_0 ] \ = \ -i\,
,\quad [ x_i , p_j ] \ = \ i\,\delta_{i j} e^{\alpha x_0}\, ,\quad
[ x_i , p_0 ] \ = \ -i\,\alpha x_i\,
.\label{r322}\end{equation}with commuting space-time and
momentum.\medskip\\ Similarly, from formulae (\ref{r310}) and
(\ref{r318}) we obtain the commutation relations for the linear
basis of TD-{\it twisted phase space} $\Pi_\kappa(\beta) =
{\cal{X}}_\kappa(\beta)>\!\!\!\triangleleft
{\cal{P}}_\kappa$\begin{equation}
\begin{array}{lll} [ x_0 , x_i ] \ = & -\frac{i}{\kappa c} x_i\, ,
\qquad\qquad & [ x_i , x_j ] \ = \ 0\, ,\medskip\cr [ x_i , p_0 ]
\ = &  0\, , & [ x_0 , p_i ] \ = \  \frac{i}{\kappa c} p_i +
\beta_{j i} x_j\, ,\medskip\cr [ x_0 , p_0 ] \ = & - i\, , & [
p_\mu , p_\nu ] \ = \ 0\, ,\bigskip\cr [ x_i , p_j ] \ = &
i\,\delta_{i j}  \, .&\end{array}\label{r323}\end{equation}We see
that commutators are given by linear combinations of $p_i\,, x_i$
therefore one can regard these formulae as defining phase space
(not an algebra) in the classical sense. Also in this case we can
consider the limit $\kappa\to\infty, \beta=const.$ (see
(\ref{r311})) of phase space $\Pi_\kappa(\beta) \to
\Pi_\infty(\beta)$ given by the non-vanishing
commutators\begin{equation}[ x_0 , p_0 ] \ = \ -i\, ,\quad [ x_i ,
p_j ] \ = \ i\,\delta_{i j}\, ,\quad [ x_0 , p_i ] \ = \
i\,\beta_{j i} x_j\, .\label{r324}\end{equation}with commuting
space-time and momentum.\medskip\\It turns out that both phase
spaces $\Pi_\infty(\alpha)$ and $\Pi_\infty(\beta)$ can be
realized by the standard position and momentum operators
$\hat{x}_\mu ,\, \hat{p}_\nu$ satisfying the Heisenberg
commutation relations $[ \hat{x}_\mu , \hat{p}_\nu ] = -i\,g_{\mu
\nu}$\begin{equation} x_i \ = \ \hat{x}_i e^{\alpha \hat{x}_0}\,
,\quad x_0 \ = \ \hat{p}_0\, ,\quad p_\mu \ = \ \hat{p}_\mu\,
,\qquad for \qquad \Pi_\infty(\alpha)\, .
\label{3r25}\end{equation}and assuming $\beta_{i j} =
\beta\,\delta_{i j}$\begin{equation} x_\mu \ = \ \hat{x}_\mu\,
,\quad p_0 \ = \ \hat{p}_0\, ,\quad p_i \ = \ \hat{p}_i - \beta
\hat{p}_0\, \hat{x}_i\, ,\qquad for \quad \Pi_\infty(\beta)\,
.\label{r326}\end{equation}We notice that both algebras
${\cal{X}}_\kappa$ and ${\cal{P}}_\kappa$ possess the Hopf
structure therefore one can also consider a left action of
space-time algebra on the momentum algebra $ \triangleright :
{\cal{P}}_\kappa\otimes {\cal{X}}_\kappa\rightarrow
{\cal{X}}_\kappa$ formally a changing the position and momentum
generators $ x\leftrightarrow p$. It corresponds in quantum
mechanical language to the exchange of momentum for positions
representation. In this case one can define a phase space as the
cross product algebra ${\cal{P}}_\kappa
>\!\!\!\triangleleft {\cal{X}}_\kappa$ (see \cite{LN1}) with slightly
different cross commutation relations. However, we do not consider
twisting of this kind of phase space.

\section{Duality for twisted D=2 space-time}\bigskip\noindent
Considering the formulae (\ref{r27c}) we notice that different
choices of coproducts $\Delta(x)$ or $\Delta(p)$ provide changes
in duality relations $< p^m\,q^n , x >$ and $< p , x^k\,y^l >$,
respectively. Therefore, we can expect the modified duality
relations $< p_i^m\,p_0^n , x_j >$ between four-momentum $p_\mu$
in the bicrossproduct basis (\ref{r24}) and SD-twisted or
TD-twisted space-time given by relations (\ref{r38}) or
(\ref{r310}). We find these twisted duality relations in the case
of two dimensional (D=2) space-time. This simplification is
convenient because of tedious calculations, however, one can
immediately generalize the obtained results to the
four-dimensional space-time.
\subsection{SD-twisted space-time}
In the case of the two dimensional SD-twisted space-time relations
(\ref{r38}) take the following form (we assume $c=1$ in order to
simplify the notation)\\- {\it space-time}
\begin{eqnarray}
[ x_0 , x ] &=& < f(p_0) , x_0 > \,x\, .\label{401}\\ \Delta (x_0)
&=& x_0\otimes 1 + 1\otimes x_0\, ,\qquad \Delta(x) \ = \ x\otimes
e^{\alpha x_0} + e^{\alpha x_0}\otimes x\,
.\label{402}\end{eqnarray}- {\it momentum space}
\begin{eqnarray}[ p_0 , p ] &=& 0\, .\label{403}\\
 \Delta(p_0) &=& p_0\otimes 1 + 1\otimes p_0\,
,\qquad \Delta(p) \ = \ p\otimes 1 + f(p_0)\otimes p\,
.\label{404}\end{eqnarray} where we denote $f(p_0) =
\exp\left(-{p_0/\kappa}\right)$.\\- {\it duality relations} ($\mu
, \nu = 0, 1$)
\begin{eqnarray}&< p_\mu , x_\nu > =
i\,g_{\mu\nu}\, ,\qquad\qquad & g_{\mu \nu} = (1 , -1)\,
.\label{405}\\&< p_\mu , 1> = < 1 , x_\mu
> = 0\, ,\quad& < 1, 1 > = 1\, .\label{406}\\&< p\, q , x > =
< p \otimes q , \Delta(x) >\, , & < p , x\, y > = < \Delta(p) ,
x\otimes y >\, .\label{407}\end{eqnarray} \noindent Taking into
account the coproduct relations (\ref{402}) and (\ref{404}) we can
immediately generalize the relations (\ref{406}) and obtain
\begin{equation}
< p^m_\mu , 1 > \  = \ < 1 , x^m_\mu > = \delta_{m 0}\,\qquad m =
0 , 1, 2 ,\dots\, .\label{408}\end{equation}In order to find other
duality relations we apply the useful relation which expresses the
coassociativity of coproduct (see \cite{M})\begin{eqnarray}<
p_\mu^m , x_\nu^k > &=& < \Delta^{(k-1)}(p_\mu^m) , x_\nu^{\otimes
k} > \ = \ < \left(\Delta^{(k-1)} (p_\mu)\right)^m ,
x_\nu^{\otimes k}
>\, =\nonumber\medskip\\ &=& < p_\mu^{\otimes m} , \Delta^{(m-1)} (x_\nu^k)
>
\ = \ < p_\mu^{\otimes m} , \left(\Delta^{(m-1)} (x_\nu)\right)^k
>\, .\label{409}\end{eqnarray}where\begin{equation}\Delta^{(n)} \ = \
(I^{\otimes (n-1)}\otimes \Delta ) \Delta^{(n-1)} \ = \
(\underbrace{1\otimes\cdots\otimes1}_{(n-1)-times}\otimes \Delta)
\Delta^{(n-1)}\,
.\label{410}\end{equation}\begin{equation}x_\nu^{\otimes k} \ = \
\underbrace{x_\nu\otimes x_\nu\otimes\cdots \otimes x_\nu}_{k -
times}\, ,\qquad p_\mu^{\otimes m} \ = \ \underbrace{p_\mu\otimes
p_\mu\otimes \cdots \otimes p_\mu}_{m - times}\,
.\label{411}\end{equation}In particular\begin{eqnarray}
\Delta^{(m-1)} (x) = \sum_{i=1}^{m} x_i^{(m-1)}\, ,\qquad
x_i^{(m-1)} = (e^{\alpha x_0})^{\otimes (i-1)}\otimes x\otimes
(e^{\alpha x_0})^{\otimes (m-i)}\,.&&\label{412}\\
\Delta^{(m-1)}(x_0) = \sum_{i=1}^{m} (x_0)_i^{(m-1)}\,
,\qquad\quad (x_0)_i^{(m-1)} = I^{\otimes (i-1)}\otimes x_0\otimes
I^{\otimes (m-i)}\, .&&\label{413}\\ \Delta^{(k-1)} (p) =
\sum_{i=1}^{k} p_i^{(k-1)}\, ,\quad\qquad\qquad p_i^{(k-1)} =
f^{\otimes (i-1)}\otimes p\otimes I^{\otimes (k-i)}\,
.\qquad&&\label{414}\\ \Delta^{(k-1)} (p_0) = \sum_{i=1}^{k}
(p_0)_i^{(k-1)}\, ,\qquad\qquad (p_0)_i^{(k-1)} = I^{\otimes
(i-1)}\otimes p_0\otimes I^{\otimes (k-i)}\, .
&&\label{415}\end{eqnarray}Using the coproduct formulae
(\ref{402}) for space-time variables and duality relations
(\ref{405}) and relations (\ref{412}-\ref{413}) we
obtain\begin{eqnarray} < p_0^n , x_0 > =& \delta_{n 1} <p_0 , x_0
>\, ,\quad & < p^m , x_0 > \ = \ 0\, .\label{416}\\ <p_0^n , x > =&
0\, ,\qquad\qquad\qquad & < p^m , x > \ = \ \delta_{m 1} < p , x
>\, .\label{417}\end{eqnarray}For instance\begin{eqnarray} < p^m ,
x_0 > &=& < p^{\otimes m} , \Delta^{(m-1)} ( x_0) > = \cr &=&
\sum_{i=1}^{m}< p\otimes \cdots \otimes p , I^{\otimes
(i-1)}\otimes x_0 \otimes I^{\otimes (m-i)} > =\cr &=& m < p , 1
>^{m-1} < p , x_0> \ = \ 0\, .\nonumber\end{eqnarray} similarly
\begin{eqnarray} < p^m , x > &=& < p^{\otimes m} , \Delta^{(m-1)}
 (x) > = \cr &=& < p\otimes \cdots\otimes p , (e^{\alpha x_0})^{\otimes
(i-1)}\otimes x\otimes (e^{\alpha x_0})^{\otimes (m-i)} > = \cr
&=& m < p , e^{\alpha x_0} >^{m-1} < p , x > \ = \ \delta_{m 1} <
p , x >\, .\nonumber\end{eqnarray} Analogously, using the momentum
coproduct (\ref{404}) and formulae (\ref{414}-\ref{415}) we derive
\begin{eqnarray} < p_0 , x_0^l > = & \delta_{l 1} <p_0 , x_0
>\, ,\quad & < p , x_0^n > \ = \ 0\, .\label{418}\\ <p_0 , x^k > =
& 0\, ,\qquad\qquad\qquad  & < p , x^k > \ = \ \delta_{k 1} < p ,
x >\, .\label{419}\end{eqnarray} The relations
(\ref{416})-(\ref{417}) or (\ref{418})-(\ref{419}) can be
generalized to the following form\begin{eqnarray} < p_0^n , x_0^l
> &=& n!\,<p_0 , x_0>^n \delta_{n l}\, ,\quad < p^m , x_0^l > \ =
\ \delta_{m 0} \delta_{l 0}\, .\label{420}\\ < p^m , x^k > &=&
m!\,<p , x >^m \delta_{m k}\, ,\quad < p_0^n , x^k > \ = \
\delta_{n 0}\delta_{k 0}\, .\label{421}\end{eqnarray}From
(\ref{415}) and the trivial form of coproduct (\ref{404}) we
compute
\begin{eqnarray}< p_0^n , x_0^l > &=& <
\Delta^{(l-1)}(p_0^n) , x_0^{\otimes l} > \ = \ <
\left(\sum_{i=1}^{l}(p_0)_i^{(l-1)}\right)^n , x_0\otimes \cdots
\otimes x_0 > = \cr &=&  n! < p_0^{\otimes n} , x_0^{\otimes l} >
\ = \ n! < p_0 , x_0 >^l \delta_{n l}\, .\nonumber\end{eqnarray}
\begin{eqnarray}< p_0^n , x^k > &=& < \Delta^{(k-1)}(p_0^n) ,
x^{\otimes k} > \ = \ < (\Delta^{(k-1)}(p_0))^n , x^{\otimes k} >
= \cr &=& < \left(\sum_{i=1}^{k} (p_0)_i^{(k-1)}\right)^n , x
\otimes \cdots \otimes x > = \cr &=& < (p_0\otimes I^{\otimes
(k-1)} + \cdots + I^{\otimes (k-1)}\otimes p_0 )^n , x^{\otimes k}
> \ = \ \delta_{k 0}\delta_{n 0}\, .\nonumber\end{eqnarray}and
taking into account (\ref{414}) and the relation $<p_0^n , x > =
0$ (\ref{417}) we obtain
\begin{eqnarray}< p^m , x^k > &=& <
\Delta^{(k-1)}(p^m) , x^{\otimes k} > \ = \ <
\left(\Delta^{(k-1)}(p)\right)^m , x\otimes \cdots \otimes x > =
\cr &=& < \left(\sum_{i=1}^{k} f^{\otimes (i-1)}\otimes p\otimes
I^{\otimes (k-i)}\right)^m , x\otimes \cdots \otimes x> = \cr &=&
< \left(\sum_{i=1}^{k} I^{\otimes (i-1)}\otimes p\otimes
I^{\otimes (k-i)}\right)^m , x\otimes \cdots \otimes x> = \cr &=&
m!\,\delta_{k m} < p\otimes \cdots \otimes p , x\otimes \cdots
\otimes x > = m!\, <p , x>^m \delta_{k m}\,
.\nonumber\end{eqnarray}
\begin{eqnarray}< p^m , x_0^l > &=& <
\Delta^{(l-1)}(p^m) , x_0^{\otimes l} > \ = \ <
\left(\Delta^{(l-1)}(p)\right)^m , x_0\otimes \cdots \otimes x_0 >
= \cr &=& < \left(\sum_{i=1}^{l} f^{\otimes (i-1)}\otimes p\otimes
I^{\otimes (l-i)}\right)^m , x_0^{\otimes l} > \ = \ \delta_{l 0}
\delta_{m 0}\, .\nonumber\end{eqnarray}

\noindent We would like to stress that the duality relations
(\ref{420})-(\ref{421}) do not depend on SD-twisting
transformation and they have the same form in both,
$\kappa$-deformed and non-deformed cases.

\noindent Let us derive the relation which depends on the twisting
parameter.  Using (\ref{420})-(\ref{421}) and coproduct formula
(\ref{402}) for $\Delta(x)$ we find
\begin{eqnarray}&&< p_0^n p^m , x > \ = \ < p_0^n\otimes p^m ,
\Delta(x)> \ = \ < p_0^n\otimes p^m , x\otimes e^{\alpha x_0} +
e^{\alpha x_0}\otimes x >=\cr && \qquad \ = \ < p_0^n , x > < p^m
, e^{\alpha x_0} > + < p_0^n , e^{\alpha x_0} > < p^m , x
>=\cr && \qquad \ = \ \delta_{m 1} < p , x > < p_0^n , e^{\alpha
x_0} > = \delta_{m 1}\,\sum_{k=0}^{\infty}\frac{\alpha^k}{k!} < p
, x > < p_0^n , x_0^k >=\cr && \qquad \ = \ \alpha^n \delta_{m 1}
< p , x > < p_0 , x_0
>^n\, .\label{422}\end{eqnarray} \noindent or equivalently, for any
polynomial function $g(p_0)$ we get
\begin{equation}
< g(p_0) p^m , x > \ = \ \delta_{m 1} < p , x > g(\alpha <p_0 ,
x_0>).\label{423}\end{equation}It is convenient to use instead the
power function $p_0^n\,$, its generating function $g(p_0)=e^{\xi
p_0}$. Then we obtain \begin{equation} < p^m\,e^{\xi p_0} , x > \
= \ \delta_{m 1}\,< p , x > e^{\alpha \xi <p_0,x_0>}\,
.\label{424}\end{equation}or using (\ref{414}) we find a more
general formula\begin{equation} < p^m\,e^{\xi p_0} , x^k
> \ = \ k! (< p , x >)^k \delta_{m k }\,e^{k \alpha \xi <p_0 ,
x_0>}\, f^{\frac{1}{2} k(k-1)}(\alpha <p_0 , x_0>)\,
.\label{425}\end{equation}From (\ref{420}) we can easy calculate
the following duality relation
\begin{eqnarray}< p^m\,e^{\xi
p_0} , x_0^l > &=& \sum_{s=0}^{l} < p^m\otimes e^{\xi p_0} ,
x_0^{l-s}\otimes x_0^s > =\cr &=& \sum_{s=0}^{l} < p^m , x_0^{l-s}
> < e^{\xi p_0} , x_0^s > =\cr &=& \delta_{m 0} < e^{\xi p_0} ,
x_0^l >  \ = \ (\xi <p_0 , x_0 >)^l \delta_{m 0}\,
.\label{426}\end{eqnarray}and finally we find\begin{eqnarray}&&<
p^m\,e^{\xi p_0} , x^k x_0^l > \ = \ < \Delta(p^m\,e^{\xi p_0}) ,
x^k\otimes x_0^l >= \qquad\qquad\qquad\qquad\qquad\cr &&= \
\sum_{s=0}^{m} \left(\begin{array}{c}m\\s\end{array}\right)<
p^{m-s} f^s(p_0)\,e^{\xi p_0}\otimes p^s e^{\xi p_0} , x^k\otimes
x_0^l
>\, =\cr &&= \ \sum_{s=0}^{m} \left(\begin{array}{c}m \\
s\end{array}\right)
< p^{m-s} f^s(p_0) e^{\xi p_0} , x^k > < p^s e^{\xi p_0} , x_0^l
>\, = \cr &&= \ (\xi<p_0 ,x_0>)^l \sum_{s=0}^{m}
\left(\begin{array}{c}m\\s\end{array}\right)< p^{m-s}
f^s(p_0)\,e^{\xi p_0} , x^k > \delta_{s 0}\, = \cr &&= \ (\xi<p_0
, x_0>)^l < p^m\,e^{\xi p_0} , x^k
>\, = \nonumber\end{eqnarray}
\begin{equation} = k! (<p , x>)^k \delta_{m
k} (\xi<p_0 , x_0>)^l e^{k \alpha \xi <p_0,x_0>}
f^{\frac{1}{2}k(k-1)}(\alpha<p_0,x_0>)\,
.\label{427}\end{equation}or equivalently (expanding the
generating function in powers of $p_0^n$)\begin{eqnarray}< p^m
p_0^n , x^k x_0^l > &=& k! n! (<p , x>)^k (<p_0 , x_0>)^l
\delta_{m k}\times\cr &\times & \frac{(i k\alpha)^{n-l}}{(n-l)!}\,
f^{\frac{1}{2}k(k-1)}(\alpha<p_0,x_0>)\,
.\label{428}\end{eqnarray}In the limit $\alpha \to 0$ of the
twisting parameter we obtain the duality relations (\ref{r29}).
Therefore, we see that the twisting map destroys some orthogonal
duality relations i.e. roughly speaking twisting changes the
orthogonal basis of dual momentum and space-time algebras to a
non-orthogonal one.\\

\noindent Because twisted space-time is a non-commuting algebra,
one can also consider a polynomial basis with opposite ordering of
space and time variables i.e. left-time ordered polynomials. In
order to find the duality relations for this case, we would like
to stress that space-time commutation relations (\ref{r213}) are
related to the momentum coproduct in the bicrossproduct basis and
do not depend on the twisting map, therefore we can use this
relations to derive duality relations for the opposite ordering.
For D=2 space-time the formula (\ref{r213}) reads
\begin{equation} x_0^l x^k \ = \ x^k ( x_0 - k <f , x_0> )^l \ = \
\sum_{r=0}^{l}\left(\begin{array}{c}l\\r\end{array}\right) x^k
x_0^{l-r} (-k <f , x_0>)^r\, .\label{429}\end{equation}and for
$f(p_0)= \exp(-p_0/\kappa)$ we obtain the identity\begin{equation}
< f^k , x_0 >^n \ = \ (- k <f , x_0 >)^n\,
.\label{430}\end{equation}Therefore, from (\ref{427}) we
obtain\begin{eqnarray}&&< p^m e^{\xi p_0} , x_0^l x^k
>  = \sum_{r=0}^{l}\left(\begin{array}{c} l \\ r\end{array}\right)
(-k <f , x_0>)^r < p^m e^{\xi p_0} , x^k x_0^{l-r} > =\cr &&=
k!\,(<p,x>)^k\,\delta_{m k} e^{k \alpha \xi<p_0,x_0>}
f^{\frac{1}{2}k(k-1)}(\alpha<p_0,x_0>)\times\cr && \qquad\times
\sum_{r=0}^{l} \left(\begin{array}{c} l\\ r\end{array}\right) (-k
<f , x_0>)^r (\xi <p_0, x_0>)^{l-r}\, =\cr &&=
k!\,(<p,x>)^k\,\delta_{m k} e^{k \alpha \xi<p_0,x_0>}
f^{\frac{1}{2}k(k-1)}(\alpha<p_0,x_0>)\times\cr &&\qquad \times
\sum_{r=0}^{l} \left(\begin{array}{c} l
\\ r\end{array}\right) < f^k , x_0 >^r (\xi <p_0, x_0>)^{l-r}\,
=\cr && = k!\,(<p,x>)^k e^{k \alpha \xi<p_0,x_0>}
f^{\frac{1}{2}k(k-1)}(\alpha<p_0,x_0>)\,\delta_{m k}\times\cr &&
\qquad \times < f^k(p_0) , (x_0 + \xi<p_0,x_0>)^l
>\, .\label{431}\end{eqnarray}Taking into account (\ref{405}) and
the explicit form of the function $f(p_0)$ we obtain the SD-{\it
twisted duality relations}\begin{eqnarray}< p^m\,e^{\xi p_0} , x^k
x_0^l > & = & k! (-i)^k (i \xi)^l \delta_{m k} e^{i k \alpha
\left(\xi + \frac{1-k}{2\kappa}\right)}\, .\\< p^m e^{\xi p_0} ,
x_0^l x^k > &=&  k!\,(-i)^k \delta_{m k}  e^{i k \alpha \left(\xi
+ \frac{1-k}{2\kappa}\right)} < e^{-\frac{k p_0}{\kappa}} , (x_0 +
i\xi)^l >\, .\label{432}\end{eqnarray}or using the
equality\begin{equation}< (p_0 + ik\alpha)^n , x_0^l > \ = \
\frac{n!}{(n-l)!}\,(<p_0 , x_0>)^l (ik\alpha)^{n-l}\, .\label{433}
\end{equation}we find the equivalent form\begin{eqnarray}< p^m p_0^n
 , x^k x_0^l >  &=& n! <p^m, x^k > (< p_0 , x_0 >)^l\,
\frac{(i k \alpha)^{n-l}}{(n-l)!}\,\, e^{-\frac{i\alpha}{2\kappa}
k(k-1)}\,\cr &=& k!\,\delta_{k m} (-i)^k
e^{-\frac{i\alpha}{2\kappa} k(k-1)} <(p_0 + ik\alpha)^n , x_0^l
>\, .\label{434}\end{eqnarray} and the relation (\ref{431})
 can be written as a non-vanishing duality relation only for $n\leq l$
\begin{eqnarray}< p^m p_0^n , x_0^l x^k >  &=& n!l! <p^m, x^k >
(< p_0 , x_0 >)^le^{-\frac{i\alpha}{2\kappa} k(k-1)}
\times\nonumber\\&\times&\sum_{r=0}^{n} \frac{(i m \alpha)^r}{r!
(n-r)!(l-n+r)!}\left(-\frac{m}{\kappa}\right)^{l+r-n}\,
=\nonumber\end{eqnarray}\begin{equation}= \ k!\,\delta_{k m}
(-i)^k e^{-\frac{i\alpha}{2\kappa} k(k-1)} < e^{-\frac{m
p_0}{\kappa}} (p_0 + im\alpha)^n , x_0^l >\, .\label{435}
\end{equation}

\subsection{TD-twisted space-time}
Similarly, in the two-dimensional case the TD-twisted space-time
(\ref{r310}) is given by the relations\begin{eqnarray}[ x_0 , x ]
&=& < f(p_0) , x_0
> x\, ,\cr \Delta( x_0 ) &=& x_0\otimes 1 + 1\otimes x_0 + \beta\,x\otimes
x\,
,\qquad \Delta( x ) \ = \ x\otimes 1 + 1\otimes x\,
.\label{436}\end{eqnarray}and additional formulae
(\ref{403})-(\ref{407}) describing the momentum space and duality
relations. Also in this case the relations (see
(\ref{420})-(\ref{421}))
\begin{equation} < p_0^n , x_0^l
> = n!\,<p_0 , x_0>^n \delta_{n l}\, ,\quad <
p^m , x^k > = m!\,<p , x >^m \delta_{m k}\, .
\label{437}\end{equation} are valid and the remaining ones are
changed. It is easy to observe, that the form of coproduct
$\Delta(x_0)$ (\ref{436}) implies a vanishing duality relation for
any odd power of the momentum
\begin{eqnarray}<p^m , x_0 > &=& < p^2 ,
x_0 > \delta_{m 2} \ = \ \beta <p , x>^2 \delta_{m 2}\, ,\cr <
p^{2k+1} , x_0^l > &=& 0\quad \Leftrightarrow\quad < \sinh(\xi p)
, x_0^l > \ = \ 0\, .\label{438}\end{eqnarray}Let us derive the
duality relations for even power of the momentum. Using the
generating function we obtain
\begin{eqnarray}< e^{\xi p} , x_0^l > &=& < \cosh(\xi p) , x_0^l >
\ = \  < \Delta^{(l-1)} \cosh(\xi p) , x_0^{\otimes l} >= \cr &=&
< \cosh\left( \xi \Delta^{(l-1)}(p)\right) , x_0^{\otimes l} >=<
\cosh\left( \xi \sum_{i=1}^{l} p_i^{(l-1)}\right) , x_0^{\otimes
l} > =\cr &=& \bigcap_{i=1}^{l} < \cosh\left( \xi
p_i^{(l-1)}\right) , x_0^{\otimes l} >\,
.\label{439}\end{eqnarray}The non-vanishing duality relations are
implied by the square power of the momentum variable (\ref{438}),
therefore we can use the expansion\begin{equation}\cosh\left( \xi
p_i^{(l-1)}\right) \ \sim \ I^{\otimes l} + \frac{1}{2} \xi^2
\left( p_i^{(l-1)}\right)^2\, .\label{440}\end{equation}and we
get\begin{eqnarray}&&<\cosh(\xi p) , x_0^l > \ = \
\bigcap_{i=1}^{l} < \left(I^{\otimes l} + \frac{1}{2} \xi^2 \left(
p_i^{(l-1)}\right)^2\right) , x_0^{\otimes l} >\, =\cr&& = \
\delta_{l\,0} \ + \ \sum_{k=1}^{l} \frac{1}{2^k} \xi^{2k}
D_{k}^{l}\, .\label{441}\end{eqnarray} where\begin{equation}
D_k^{l} \equiv D_k^l(\kappa , \beta) \ = \ \sum_{i_1\neq
i_2\neq\cdots i_{k-1}}^{l-1} < (p_{i_1}^{(l-1)})^2
(p_{i_2}^{(l-1)})^2\cdots (p_{l}^{(l-1)})^2 , x_0^{\otimes l} >\,
. \label{442}\end{equation}satisfying\begin{equation} D_k^l(0 , 0)
\ = \ D_k^l( \kappa , 0 ) \ = \ 0\, .\label{442a}\end{equation}and
comparing the appropriate left and right terms in (\ref{441}) we
find\begin{equation}< p^{m} , x_0^l
> \ = \ \delta_{m 0}\,\delta_{l 0} + \delta_{m\, 2s} \sum_{k=1}^{l}
 \frac{(2k)!}{2^k} D_k^{l}(\kappa , \beta) \delta_{s k}\, .
\label{443}\end{equation} Now, we can derive a more general
formula\begin{eqnarray}< e^{\xi p_0}\,p^m , x_0^l > &=& < e^{\xi
p_0}\otimes p^m , \Delta^l(x_0) > \ = \ < e^{\xi p_0}\otimes p^m ,
\Delta_0^l(x_0) > =\cr &=& < e^{\xi p_0}\otimes p^m , (x_0\otimes
1 + 1\otimes x_0)^l > =\cr &=&
\sum_{s=0}^{l}\left(\begin{array}{c} l \\ s\end{array}\right)<
e^{\xi p_0}\otimes p^m , x_0^{l-s}\otimes x_0^s > =\cr &=&
\sum_{s=0}^{l}\left(\begin{array}{c} l\\s\end{array}\right)<
e^{\xi p_0}, x_0^{l-s} > < p^m , x_0^s
>\, .\end{eqnarray}or expanding we get non-vanishing duality relations
for $n \leq l$\begin{equation}< p_0^n p^m , x_0^l> \ = \
\frac{l!}{(l-n)!} <p_0 , x_0 >^n < p^m , x_0^{l-n} >\,
.\end{equation} Finally, using this relation and (\ref{437}) and
coproduct formula (\ref{404}) we derive the general duality
relations for the TD-{\it twisted space-time}
\begin{eqnarray}< p^m p_0^n , x^k x_0^l
> &=& < \Delta^m(p) \Delta^n(p_0) , x^k\otimes x_0^l > \cr &=&
\frac{m!\,n! <p , x>^k <p_0 , x_0>^n}{(m - k)! (l - n)!} < p^{m -
k} , x_0^{l - n} >\, .\end{eqnarray}or equivalently
\begin{eqnarray}&&< p^m p_0^n , x^k x_0^l > \ = \ <
\Delta^m(p) \Delta^n(p_0) , x^k\otimes x_0^l >\, =\cr&&= \ m!\,n!
<p , x>^k <p_0 , x_0>^n\,\delta_{m\,k}\,\delta_{n\,l} \ +
\nonumber\bigskip\\ &&+ \ \frac{m!\,n! <p , x>^k <p_0 , x_0>^n}{(m
- k)! (l - n)!}\, \delta_{m-k\,2s} \sum_{r=1}^{l-n}
\frac{(2r)!}{2^r} D_r^{l-n}(\kappa , \beta) \delta_{s r}\,
.\end{eqnarray}In the limit $\beta\to 0$ using (\ref{442a}) we
obtain the duality relation (\ref{r29}). Therefore we see that
TD-twisting appears as an additional term to the conventional
duality relations (\ref{r29}). One can also easily find the
duality relations for the opposite space-time ordering using
formula (\ref{429}) but in a complicated form.

\section{Final remarks}\bigskip\noindent  We consider the case
of SD-twisting in the two-dimensional case. First we notice that
the duality relations (\ref{429}) allow us to define a linear
(because of the bilinear form $< \ , \ >$) transformation
$\Phi_\alpha$ in the momentum algebra ${\cal{P}}_\kappa$
corresponding to the twist operation in the space-time
${\cal{X}}_\kappa(\alpha)$ as follows\begin{equation}< p^m p_0^n ,
x^k(\alpha) x_0^l(\alpha)> \ = \ < \Phi_\alpha(p^m p_0^n) , x^k
x_0^l > \ = \ < f_{m n}(\alpha) , x^k x_0^l >\,
.\label{51}\end{equation}where ($x(\alpha)\,, x_0(\alpha)$) are
space-time variables (\ref{401})-(\ref{402}) generating the
twist\-ed algebra ${\cal{X}}_\kappa(\alpha)$ and $x\,,x_0 \in
{\cal{X}}_\kappa$ (see (\ref{r210a})-(\ref{r210b})) or in explicit
form\begin{equation} \Phi_\alpha(p^m\,p_0^n) \ = \ f_{m\,
n}(\alpha) \ = \ ( p_0 + i m \alpha )^n p^m\,e^{-\frac{i\alpha}{2
\kappa} m(m-1)}\, .\label{52}\end{equation}In
particular\begin{eqnarray} \Phi_\alpha( p^m ) &=& f_{m\,0}(\alpha)
\ = \ p^m\,e^{-\frac{i\alpha}{2 \kappa} m(m-1)}\,
.\label{53}\medskip\\ \Phi_\alpha ( p_0^n ) &=& f_{0\,n}(\alpha) \
= \ p_0^n\, .\label{54}\medskip\\ \Phi_\alpha( 1 ) &=& 1\,
.\label{55}\end{eqnarray}therefore, the action of $\Phi_\alpha$ on
the momentum space (generated linearly  by $p_0\,, p$) is trivial
because $p_0 =f_{0 1}(\alpha)\,, p=f_{1 0}(\alpha)$  and their
coproduct is given by (\ref{r26}). The action of $\Phi_\alpha$ on
the momentum algebra basis $p^m p_0^n$ allows us to extend this
transformation onto an arbitrary polynomial function of momentum
$\phi = \phi_{m\,n} p^m p_0^n$ in a natural way, by the
replacement $p^m p_0^n\rightarrow f_{m\,n}(\alpha)$. Thus, one can
consider the dual pair (${\cal{X}}_\kappa(\alpha),
{\cal{P}}_\kappa$) of the twisted space-time algebra and the
$\kappa$-deformed momentum algebra or equivalently the pair of a
algebras (${\cal{X}}_\kappa , {\cal{P}}_\kappa(\alpha)$) of
$\kappa$-deformed space-time and $\Phi_\alpha$-transformed
momentum algebra with the same duality relations (\ref{51}) in
both cases. The essential difference in both dual constructions
lies in their coalgebra sectors.\medskip\\ Our construction of the
SD-twisted phase space algebra $\Pi_\kappa(\alpha)$ (\ref{r321})
leans on the notion of the cross algebra where the multiplication
(\ref{r319}) depends on both coproducts (\ref{402}) and
(\ref{404}). Therefore, for the second pair of algebras
(${\cal{X}}_\kappa , {\cal{P}}_\kappa(\alpha)$) we obtain the
different phase space algebra although both pairs are equivalent
as far as the duality relations are concerned. In derivation of
phase space commutation relations we use only the first order
duality relations (for instance in the definition of the left
action (\ref{r317})) therefore, in the case of the pair
(${\cal{X}}_\kappa , {\cal{P}}_\kappa(\alpha)$) we obtain a
two-dimensional version of commutation relations (\ref{r321}) for
$\alpha=0$ i.e. the standard $\kappa$-deformed phase space. The
same conclusions one can obtain by considering TD-twisting
space-time.\medskip\\Therefore, one can find the linear
transformation of the momentum algebra which corresponds the twist
operation in the space-time algebra, this construction however
does not provide the twisted phase space.\medskip\\In Section 3 we
described two possible twistings (in space and time directions) of
the space-time algebra and derived corresponding phase
spaces.\medskip\\The duality relations obtained in Section 4 for
two-dimensional space-time and momentum algebras one can
immediately extend to the four-dimensional case. They describe
explicitly the effect of the twisting operation in the space-time
algebra.\bigskip\bigskip\\{\bf Acknowledgements:} The author
wishes to thank Anatol Nowicki for inspiration, advice and
support.

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\end{document}


