# On Hopf algebroid structure of $\kappa$-deformed Heisenberg algebra

Jerzy Lukierski  Institute for Theoretical Physics, University of Wroclaw
Zoran Škoda  Faculty of Science, University of Hradec Králové  Mariusz
Woronowicz  Institute for Theoretical Physics, University of Wroclaw

###### Abstract

The $(4+4)$-dimensional $\kappa$-deformed quantum phase space as well as its
$(10+10)$-dimensional covariant extension by the Lorentz sector can be
described as Heisenberg doubles: the $(10+10)$-dimensional quantum phase space
is the double of $D=4$ $\kappa$-deformed Poincaré Hopf algebra $\mathbb{H}$
and the standard $(4+4)$-dimensional space is its subalgebra generated by
$\kappa$-Minkowski coordinates $\widehat{x}_{\mu}$ and corresponding commuting
momenta $\widehat{p}_{\mu}$. Every Heisenberg double appears as the total
algebra of a Hopf algebroid over a base algebra which is in our case the
coordinate sector. We exhibit the details of this structure, namely the
corresponding right bialgebroid and the antipode map. We rely on algebraic
methods of calculation in Majid-Ruegg bicrossproduct basis. The target map is
derived from a formula by J-H. Lu. The coproduct takes values in the bimodule
tensor product over a base, what is expressed as the presence of coproduct
gauge freedom.

###### pacs:

02.20.Sv, 02.20.Uw, 03.65.Fd

##  I Introduction

It is often convenient to consider covariant phase spaces, that is to include
symmetries along with the space (or space-time) generators. Noncommutative
$\kappa$-deformed Minkowski space has been complemented with appropriate
momenta into a noncommutative phase space in a number of works. It has been
recently argued that this noncommutative phase space has a structure of a
(topological) Hopf algebroid 1 ; 2  and the same for general Lie algebra type
phase spaces halg ; the covariant structure is however not included in those
works. It is known that both the standard and general $\kappa$-phase spaces
have a structure of a Heisenberg double luknow , a Hopf algebraic
generalization of a Heisenberg algebra prominently used in quantum group
theory Majid ; heisd . Furthermore, J-H. Lu Lu  proved that a finite
dimensional Heisenberg double has a structure of a Hopf algebroid. Her
description can be generalized to some infinite dimensional situations,
including to the $\kappa$-phase spaces; the advantage of covariant phase space
description as well as of Heisenberg double packaging are among the main
motivations for this work.

While Hopf algebras are quantum analogues of groups drinfeld ; wor ; frt ,
Hopf algebroids are quantum analogues of groupoids. Indeed, algebra of
functions on a group is a commutative Hopf algebra, while functions on a
groupoid form a commutative Hopf algebroid. To drive our expectations in
physics let us recall some example of a groupoid. Groupoid is a many object
version of a group: it consists of objects (units) and arrows between them.
Each arrow has a source and target object, we can compose two arrows, $g$ and
$f$ into $g\circ f$ if the target of $f$ is the same as the source of $g$;
each arrow has its composition inverse and the composition is associative. The
main example is a transformation (action) groupoid which for a group $G$
acting on a manifold $M$ encodes both space $M$ and the action in a single
groupoid. The objects are points of $M$ and arrows are pairs $(g,m)\in G\times
M$. The source of $(m,g)$ is $m$ and the target of $(m,g)$ is $g\cdot m$. This
way we know what happens to point $m$ when we act by $g$: the information is
in the target. Thus both the space and its symmetries are encoded in the
action groupoid. Notice again that the arrows form the product $G\times M$ of
a symmetry object and the space.

If we replace a group $G$ by Hopf algebra $\mathbb{H}$ of functions on a group
and manifold $\mathcal{M}$ by an algebra $B$ of coordinate functions on
$\mathcal{M}$, we expect that $G\times M$ will be replaced by some sort of a
tensor product $\mathbb{H}\otimes B$ and the action of $G$ on $M$ dualizes to
coaction of $\mathbb{H}$ on $B$. In a typical construction of that type, like
in this article, the tensor product is in fact a semidirect product (in Hopf
algebra literature called the smash product algebra). Such examples are
noncommutative analogues of transformation groupoids. In general a Hopf
algebroid over base $B$ is a $B$-bialgebroid with an antipode map (see Section
IV) where $B$-bialgebroid entails the following data.

  * •

The role of the quantum arrow space of a bialgebroid or Hopf algebroid
$\mathcal{H}$ is taken by the total algebra $H$ and the base $B$ is the base
algebra.

  * •

One supplies the (dual versions of) source map $\alpha$ and target map $\beta$
which are now a homomorphism $\alpha:B\to H$ and an antihomomorphism
$\beta:B\to H$ of algebras such that their images commute in $H$:

$[\alpha(b),\beta(c)]=0,\,\,\,\,b,c\in B.$

This way the formula $b.h.c:=h\beta(b)\alpha(c)$ equips $H$ with $B$-bimodule
structure (this choice leads to right $B$-bialgebroid as opposed to left
bialgebroids where $b.h.c=\alpha(b)\beta(c)h$).

  * •

There is a coproduct $\Delta:H\to H\otimes H$ which is however coassociative
only when projected to the equivalence classes in the tensor product of
$B$-bimodules $H\otimes_{B}H=(H\otimes H)/\mathcal{I}_{B}$ brzmil ; Lu ;
bohmHbk ; Xu  where $\mathcal{I}_{B}$ is the left ideal in $H\otimes H$
generated by all elements of the form

$\alpha(b)\otimes 1-1\otimes\alpha(b),\,\,\,\,\,\,b\in B.$

Regarding that $\mathcal{I}_{B}$ is not a 2-sided ideal brzmil ; Lu ; bohmHbk
; Xu  (it is 2-sided in some special cases, for instance for the undeformed
Heisenberg algebra), the quotient is not an algebra. Thus the Hopf algebraic
requirement that the effective coproduct $\Delta:H\to H\otimes_{B}H$ is
multiplicative (that is $\Delta(bc)=\Delta(b)\Delta(c)$) does not make sense
in general. A subtle solution to make sense of the multiplicativity of
$\Delta$ is the following: one has to require that the image of $\Delta$ is
within some subbimodule of $H\otimes_{B}H$ where the factorwise product is
still well defined; such a subspace exists, it is the Takeuchi product
$H\times_{B}H\subset H\otimes_{B}H$, see e.g. takeuchi ; brzmil ; bohm ;
bohmHbk ; halg .

Hopf algebras come not only from groups (as function algebras) but also from
Lie algebras (as universal enveloping algebras). This means that they can also
encode infinitesimal symmetries. Similarly, Hopf algebroids can also encode
the infinitesimal actions. This explains that the Heisenberg algebra which
entails coordinates but also their infinitesimal translation generators is a
candidate for a quantum action groupoid. We formalize this in the language of
Hopf algebroids and in the case of $\kappa$-deformations.

In Hopf-algebraic $\kappa$-deformation scheme the general covariant
$\kappa$-deformed phase space is provided by the Heisenberg double
$\mathcal{H}=\mathbb{H}\rtimes\widetilde{\mathbb{H}}$ (see e.g. Majid ; heisd
), where $\mathbb{H}=U_{\kappa}(\mathfrak{g})$ describe $\kappa$-deformed
Poincaré–Hopf algebra 13a ; majidruegg ; kosmas  and $\widetilde{\mathbb{H}}$
is the Hopf algebra describing dual $\kappa$-deformed quantum Poincaré group
zakrz . Heisenberg double is a special case of smash (or crossed) product
algebra $\mathbb{H}\rtimes V$, where $V$ is an $\mathbb{H}$-module algebra
Majid . In this paper we employ the general property (see e.g. Lu , Sect. 6)
that Heisenberg double algebra is equipped with the Hopf algebroid structure.
In recent literature (see e.g. 2 ) the bialgebroid structures of deformed
standard quantum phase spaces $(\widehat{x}_{\mu},\widehat{p}_{\mu})$ with
$\kappa$-Minkowski space–time sector

$[\widehat{x}_{0},\widehat{x}_{i}]=-\frac{i}{\kappa}\widehat{x}_{i},\qquad[\widehat{x}_{i},\widehat{x}_{j}]=0,$
(1)

and commuting fourmomenta $\widehat{p}_{\mu}$ were studied by embedding into
canonical quantum phase space algebra (we put $\hslash=1$)

$[x_{\mu},x_{\nu}]=[p_{\mu},p_{\nu}]=0,\qquad[x_{\mu},p_{\nu}]=i\eta_{\mu\nu}.$
(2)

Relation (1) permits the following general class of realizations of quantum
phase spaces

$\widehat{x}_{\mu}=f_{\,\mu}^{\nu}(p)x_{\nu},\qquad\widehat{p}_{\mu}=p_{\mu},$
(3)

where $f_{\,\mu}^{\nu}(p)$ are chosen in consistency with relations (1), (2)
(Jacobi identities) and provide large variety of quantum phase spaces with
space–time algebra described by relations (1). This approach lacks structural
indication how to obtain the covariant action of $\kappa$-deformed
Poincaré–Hopf algebra, which is a part of full definition of quantum
$\kappa$-deformed Minkowski space.

Here the $\kappa$-deformed quantum phase space is constructed as the
Heisenberg double of $D=4$ $\kappa$-deformed Poincaré–Hopf algebra; this
method is first presented in luknow . Such construction contains built-in
$\kappa$-covariance of $\kappa$-deformed quantum phase space first observed
for $\kappa$-Minkowski space–time sector in majidruegg . In Majid–Ruegg basis
majidruegg  we obtain that both $\kappa$-Poincaré–Hopf algebra $\mathbb{H}$
and $\kappa$-Poincaré group $\widetilde{\mathbb{H}}$ are described by two dual
bicrossproduct structures luknow ; Maj1988 ; lukx ; 13c , namely

$\mathbb{H}=U(so(1,3))\triangleright_{\\!\kappa}\\!\\!\\!\\!\blacktriangleleft\mathcal{T}^{4}\qquad\mathop{\longleftrightarrow}\limits^{\mathrm{duality}}\qquad\widetilde{\mathbb{H}}=\widetilde{\mathcal{T}}_{\kappa}^{4}\triangleright_{\\!\kappa}\\!\\!\\!\\!\blacktriangleleft\mathcal{L}^{6},$
(4)

where $\mathcal{L}^{6}$ describe the functions of Abelian Lorentz parameters
$\lambda_{\mu\nu}^{\quad}$ which are dual to $U(so(3,1))$ and
$\mathcal{T}^{4}$ is the fourmomenta sector dual to the algebra
$\widetilde{\mathcal{T}}_{\kappa}^{4}$ describing noncommutative functions of
$\kappa$-deformed Minkowski coordinates (see (1)). The $\kappa$-Poincaré
covariance of fourmomentum sector $\mathcal{T}^{4}$ can be derived from the
bicrossproduct structure of $\mathbb{H}$ (see (4)). The $\kappa$-deformed
Poincaré algebra $\mathbb{H}$ acts on standard $\kappa$-deformed quantum phase
space $\mathcal{T}^{4}\rtimes\widetilde{\mathcal{T}}_{\kappa}^{4}$ in a
covariant way. Further, the covariant action of $\mathbb{H}$ on
$\mathcal{L}^{6}$ follows from the duality of $\mathcal{L}^{6}$ and
$U(so(3,1))$ algebras as well as the semidirect product of the coalgebra
sectors in $\mathbb{H}$ and $\widetilde{\mathbb{H}}$.

In Section 6 of Lu , J-H. Lu has given explicit formulas for a Hopf algebroid
structure on a Heisenberg double of any finite dimensional Hopf algebra, which
is here replaced by $\kappa$-Poincaré Hopf algebra $\mathbb{H}$ (which is
$\infty$-dimensional as a Hopf algebra, but the recipes still work). To have
the formulas explicit, in this article we calculate the details of the Hopf
algebroid structure for the Heisenberg double
$\mathcal{H}^{(4,4)}\equiv\mathcal{H}_{(p,x)}=\mathcal{T}^{4}\rtimes\tilde{\mathcal{T}^{4}}$,
where the dual Hopf algebras $\mathcal{T}^{4},\tilde{\mathcal{T}}^{4}$
describe respectively the momentum and coordinate sectors

$\mathcal{T}^{4}\left\\{\begin{array}[]{l}[\widehat{p}_{\mu},\widehat{p}_{\nu}]=0,\\\
\Delta(\widehat{p}_{i})=\widehat{p}_{i}\otimes
e^{-{\frac{\widehat{p}_{0}}{\kappa}}}+1\otimes\widehat{p}_{i},\\\
\Delta(\widehat{p}_{0})=\widehat{p}_{0}\otimes
1+1\otimes\widehat{p}_{0},\end{array}\right.$ (5)
$\tilde{\mathcal{T}}^{4}\left\\{\begin{array}[]{l}[\widehat{x}_{0},\widehat{x}_{i}]=-\frac{i}{\kappa}\widehat{x}_{i},\qquad[\widehat{x}_{i},\widehat{x}_{j}]=0,\\\
\Delta(\widehat{x}_{\mu})=\widehat{x}_{\mu}\otimes
1+1\otimes\widehat{x}_{\mu}.\end{array}\right.$ (6)

Most earlier important examples of Hopf algebroids over noncommutative base
are also related to physics. It has been argued out in MackSchomerus  that low
dimensional QFT-s allow for weak Hopf algebra symmetries where weak roughly
means that the coproduct and unit are not compatible, $\Delta(1)\neq 1\otimes
1$; usually one dispenses with weak units passing to appropriate quotient Hopf
algebras, but at the cost of having nonphysical zero-norm states. The data of
a weak Hopf algebra has an equivalent description as a (special kind of a)
Hopf algebroid. Similarly, the dynamical quantum Yang-Baxter equation (dQYB)
which describes dynamical quantum group related with Lie algebra
$\mathfrak{g}$ are described GN ; EtingofNikshych  by a Hopf algebroid with a
commutative base algebra $B$ dual to the Cartan subalgebra of $\mathfrak{g}$
EV . It has been shown in Xu  how to reduce dQYB to the ordinary Yang-Baxter
equation in a Hopf algebroid framework. As a vector space, this algebroid is
built from the algebraic sector of the quantum group by tensoring with the
space encoding the dynamical parameter. This conceptually attractive approach
is useful for further generalizations and application of twists doninmudrov .

All algebras in the paper are over the field $\mathbb{C}$ of complex numbers
and the opposite algebra to $B$ is denoted by $B^{\mathrm{op}}$. We freely use
the Sweedler notation for the comultiplication $\Delta(b)=\sum b_{(1)}\otimes
b_{(2)}$ with or without summation sign. The antipode for a Hopf algebra is
denoted by $S$, but the antipode for a Hopf algebroid by $\tau$. We mention
that in this paper we follow research partly explained in our last publication
lukx ; however the derivation of the target map in Section III.4 is new.

##  II $\kappa$-Poincaré–Hopf algebra and its dual group

We present here the $\kappa$-Poincaré–Hopf algebra $\mathbb{H}$ as the basic
object and the $\kappa$-Poincaré group as its dual Hopf algebra. Their duality
will play the major role in the next section on the Heisenberg double.

###  II.1 $\kappa$-Poincaré–Hopf algebra $\mathbb{H}$

In the following we use the conventions for the indices
$\mu,\nu,\lambda,\sigma=0,1,2,3$; $i,j=1,2,3$ and metric
$g_{\mu\nu}=(-1,1,1,1)$. We denote the $\kappa$-Poincaré algebra generators by
$(\widehat{p}_{\mu},\widehat{m}_{\mu\nu})$ and set $\hslash=1$. Then the
$\kappa$-Poincaré–Hopf algebra $\mathbb{H}$ in bicrossproduct basis majidruegg
has the following form

– algebra sector:

$\begin{array}[]{l}[\widehat{m}_{\mu\nu},\widehat{m}_{\lambda\sigma}]=i\left(g_{\mu\sigma}\widehat{m}_{\nu\lambda}+g_{\nu\lambda}\widehat{m}_{\mu\sigma}-g_{\mu\lambda}\widehat{m}_{\nu\sigma}-g_{\nu\sigma}\widehat{m}_{\mu\lambda}\right)\\\
[\widehat{m}_{ij},\widehat{p}_{\mu}]=-i\left(g_{i\mu}\widehat{p}_{j}-g_{j\mu}\widehat{p}_{i}\right)\\\
[\widehat{m}_{i0},\widehat{p}_{0}]=i\widehat{p}_{i},\qquad[\widehat{p}_{\mu},\widehat{p}_{\nu}]=0\\\
[\widehat{m}_{i0},\widehat{p}_{j}]=i\delta_{ij}\left(\kappa\sinh({\frac{\widehat{p}_{0}}{\kappa}})e^{-{\frac{\widehat{p}_{0}}{\kappa}}}+{\frac{1}{2\kappa}}\overrightarrow{\widehat{p}}^{2}\right)-{\frac{i}{\kappa}}\widehat{p}_{i}\widehat{p}_{j}\end{array}$
(7)

– coalgebra sector:

$\begin{array}[]{l}\Delta(\widehat{m}_{ij})=\widehat{m}_{ij}\otimes
I+I\otimes\widehat{m}_{ij}\\\ \Delta(\widehat{m}_{k0})=\widehat{m}_{k0}\otimes
e^{-{\frac{\widehat{p}_{0}}{\kappa}}}+I\otimes\widehat{m}_{k0}+{\frac{1}{\kappa}}\widehat{m}_{kl}\otimes\widehat{p}_{l}\\\
\Delta(\widehat{p}_{0})=\widehat{p}_{0}\otimes I+I\otimes\widehat{p}_{0}\\\
\Delta(\widehat{p}_{k})=\widehat{p}_{k}\otimes
e^{-{\frac{\widehat{p}_{0}}{\kappa}}}+I\otimes\widehat{p}_{k}\end{array}$ (8)

– counit:

$\epsilon(\widehat{p}_{\mu})=0,\,\,\,\,\,\,\,\,\epsilon(\widehat{m}_{\mu\nu})=0.$
(9)

– Hopf algebra antipode:

$\begin{array}[]{cc}S(\widehat{m}_{ij})=-\widehat{m}_{ij},&S(\widehat{m}_{i0})=-\widehat{m}_{i0}+\frac{3i}{2\kappa}\widehat{p}_{i}\\\
S(\widehat{p}_{i})=-e^{{\frac{\widehat{p}_{0}}{\kappa}}}\widehat{p}_{i},&S(\widehat{p}_{0})=-\widehat{p}_{0}.\end{array}$
(10)

###  II.2 The concept of a Hopf pairing

We say that Hopf algebra $\mathbb{H}=(A,m,\Delta,S,\epsilon)$ is in duality
with Hopf algebra
$\tilde{\mathbb{H}}=(A^{*},m^{*},\Delta^{*},S^{*},\epsilon^{*})$ if there is a
vector space pairing $\langle,\rangle:A^{*}\otimes A\to\mathbb{C}$ Majid  such
that

$\begin{array}[]{l}\langle c^{*},ab\rangle=\langle\Delta(c^{*}),a\otimes
b\rangle,\\\ \langle c^{*}\otimes d^{*},\Delta(a)\rangle=\langle
c^{*}d^{*},a\rangle,\end{array}$ (11)

$\langle a^{*},1_{A}\rangle=\epsilon^{*}(a^{*})$, $\langle
1_{A^{*}},a\rangle=\epsilon(a)$ and $\langle S(b^{*}),a\rangle=\langle
b^{*},S(a)\rangle$. Any two Hopf algebras in duality act one on another.
Namely,

$a^{*}\triangleright a:=a_{(1)}\langle
a^{*},a_{(2)}\rangle,\,\,\,\,\,\,a^{*}\triangleleft a:=\langle
a^{*}_{(1)},a\rangle a_{(2)}^{*},$

Using (11) one can directly verify that these actions are Hopf (in other words
Majid , $A$ is a left $\tilde{\mathbb{H}}$-module algebra and $A^{*}$ is a
right $\mathbb{H}$-module algebra), that is

$\begin{array}[]{lcl}a^{*}\triangleright(ab)&=&(a_{(1)}^{*}\triangleright
a)(a_{(2)}^{*}\triangleright b)\\\ &=&a_{(1)}\langle
a^{*}_{(1)},a_{(2)}\rangle b_{(1)}\langle
a_{(2)}^{*},b_{(2)}\rangle\end{array}$

###  II.3 $\kappa$-Poincaré quantum group $\widetilde{\mathbb{H}}$

We introduce the $\kappa$-Poincaré quantum group as the dual vector space
$\tilde{\mathbb{H}}$ to $\mathbb{H}$ with generators
$\widehat{p}_{\nu},\widehat{m}_{\lambda\mu}$ via the following canonical
duality relations

$\begin{array}[]{c}<\widehat{x}^{\mu},\widehat{p}_{\nu}>=i\delta_{\nu}^{\mu}\\\
<\widehat{\lambda}{{}^{\mu}}_{\nu},\widehat{m}_{\lambda\rho}>=i(\delta_{\lambda}^{\mu}g_{\nu\rho}-\delta_{\rho}^{\mu}g_{\nu\lambda}-\delta_{\lambda}^{\nu}g_{\mu\rho}+\delta_{\rho}^{\nu}g_{\mu\lambda})\\\
\langle\widehat{x}^{\mu},\widehat{m}_{\lambda\rho}\rangle=\langle\widehat{\lambda}{{}^{\mu}}_{\nu},\widehat{p}_{\sigma}\rangle=0,\quad\langle\widehat{\lambda}{{}^{\mu}}_{\nu},I\rangle=\delta^{\mu}_{\nu}\end{array}$
(12)

The pairing for vector spaces has to be given for vector space basis but in
the case of Hopf algebra duality it is enough to specify the pairing just on
the algebra generators. Indeed, for products of generators we can use (11).

From (12) we obtain the commutation relations defining $\kappa$-Poincaré group
zakrz ; 13c  in the following form

– algebra sector:

$\begin{array}[]{l}[\widehat{x}^{\mu},\widehat{x}^{\nu}]={\frac{i}{\kappa}}(\delta_{0}^{\mu}\widehat{x}^{\nu}-\delta_{0}^{\nu}\widehat{x}^{\mu}),\,\,\,\,\,\,[\widehat{\lambda}_{\nu}^{\mu},\widehat{\lambda}_{\beta}^{\alpha}]=0\\\
[\widehat{\lambda}_{\nu}^{\mu},\widehat{x}^{\lambda}]=-{\frac{i}{\kappa}}\left((\widehat{\lambda}_{0}^{\mu}-\delta_{0}^{\mu})\widehat{\lambda}_{\nu}^{\lambda}+(\widehat{\lambda}_{\nu}^{0}-\delta_{\nu}^{0})g^{\mu\lambda}\right)\end{array}$
(13)

– coalgebra sector:

$\begin{array}[]{l}\Delta(\widehat{x}^{\mu})=\widehat{\lambda}{{}^{\mu}}_{\rho}\otimes\widehat{x}^{\rho}+\widehat{x}^{\mu}\otimes
I\\\
\Delta(\widehat{\lambda}{{}^{\mu}}_{\nu})=\widehat{\lambda}{{}^{\mu}}_{\rho}\otimes\widehat{\lambda}{{}^{\rho}}_{\nu}\end{array}$
(14)

–  counit $\epsilon$ and antipode $S$:

$\begin{array}[]{lll}\epsilon(\widehat{\lambda}{{}^{\mu}}_{\nu})=\delta{{}^{\mu}}_{\nu}&&S(\widehat{\lambda}{{}^{\mu}}_{\nu})=g^{\mu\rho}\widehat{\lambda}_{\
\rho}^{\sigma}g_{\nu\sigma}=\widehat{\lambda}_{\nu}{{}^{\mu}}\\\
\epsilon(\widehat{x}^{\mu})=0&&S(\widehat{x}^{\mu})=-\widehat{\lambda}_{\nu}{{}^{\mu}}\widehat{x}^{\nu}\end{array}$
(15)

In the Heisenberg double algebra
$\mathcal{H}^{(10,10)}=\mathbb{H}\rtimes\widetilde{\mathbb{H}}$ the
commutation relations (7) and (13) are supplemented by the following relations
obtained from (12), (8) and (14)

The generalized covariant $\kappa$-deformed phase space is described by sets
of commutators (7), (13), (17) and (18). The coproducts (8) and (14) realize
the coalgebraic homomorphism of relations (7) and (13), but the relations
(17),(18) will be mapped into the coalgebra only in the bialgebroid framework.

The Hopf subalgebra $\tilde{\mathcal{T}}^{4}_{\kappa}=U_{\kappa}(g)$
($\kappa$-Minkowski space-time sector) is the subalgebra generated by the 4
generators $\widehat{x}_{\mu}$ only.

##  III Heisenberg double

Heisenberg double is a construction of an associative algebra $A\rtimes A^{*}$
(the notation suggests that it is a special case of a smash product algebra)
containing a Hopf algebra $\mathbb{H}$ and its dual Hopf algebra
$\tilde{\mathbb{H}}$ as the analogues of the coordinate and momentum sectors
within the standard Heisenberg algebra. It is the tensor product vector space
$A\otimes A^{*}$ with the nontrivial algebra structure given by the cross
relations.

###  III.1 Cross relations

The two tensor factors do not commute, but satisfy the cross relations

$(1\otimes a^{\ast})(a\otimes 1)=(a^{\ast}_{(1)}\triangleright a)\otimes
a_{(2)}^{\ast}=a_{(1)}<a_{(1)}^{\ast},a_{(2)}>\otimes\,a_{(2)}^{\ast},$ (16)

while $(a\otimes 1)(1\otimes a^{*})=a\otimes a^{*}$. Regarding that we know
that $A$ is the first and $A^{*}$ the second tensor factor in practice we may
concatenate and skip the $\otimes$ sign. Thus $a\otimes a^{*}=aa^{*}$ and (16)
reads $a^{\ast}a=a_{(1)}<a_{(2)},a_{(1)}^{\ast}>a_{(2)}^{\ast}$.

Using coproducts (8), (14) we calculate meljcorr  the cross relations (16)

$\begin{array}[]{cc}[\widehat{p}_{k},\widehat{x}_{l}]=-i\delta_{kl},&[\widehat{p}_{0},\widehat{x}_{0}]=i\\\
[\widehat{p}_{k},\widehat{x}_{0}]=-{\frac{i}{\kappa}}\widehat{p}_{k},&[\widehat{p}_{0},\widehat{x}_{l}]=0,\\\
[\widehat{p}_{\mu},\widehat{\lambda}_{\rho}^{\lambda}]=0,&\end{array}$ (17)
$\begin{array}[]{l}[\widehat{m}_{\lambda\rho},\widehat{\lambda}_{\nu}^{\mu}]=i\left(\delta_{\rho}^{\mu}\widehat{\lambda}_{\lambda\nu}-\delta_{\lambda}^{\mu}\widehat{\lambda}_{\rho\nu}+\delta_{\lambda}^{\nu}\widehat{\lambda}_{\rho\mu}-\delta_{\rho}^{\nu}\widehat{\lambda}_{\lambda\mu}\right),\\\
[\widehat{m}_{\lambda\rho},\widehat{x}^{\mu}]=i\left(\delta_{\rho}^{\mu}\widehat{x}_{\lambda}-\delta_{\lambda}^{\mu}\widehat{x}_{\rho}\right)+{\frac{i}{\kappa}}\left(\delta_{\rho}^{0}\widehat{m}{{}_{\lambda}}^{\mu}-\delta_{\lambda}^{0}\widehat{m}{{}_{\rho}}^{\mu}\right)\end{array}$
(18)

where
$\widehat{m}{{}_{\lambda}}^{\mu}=g^{\mu\rho}\widehat{m}{{}_{\lambda\rho}}$ and
$\widehat{m}{{}^{\mu}}_{\lambda}=g^{\mu\rho}\widehat{m}{{}_{\rho\lambda}}$.

The general covariant $\kappa$-deformed quantum phase space
$\mathcal{H}^{(10,10)}$ is generated by $\mathbb{H}$, $\tilde{\mathbb{H}}$
with the above cross relations. The standard $\kappa$-deformed phase space
$\mathcal{H}^{(4,4)}\subset\mathcal{H}^{(10,10)}$ is its subalgebra generated
by $\widehat{x}^{\mu}$ and $\widehat{p}_{\nu}$ only. The quotient of
$\mathcal{H}^{(10,10)}$ by the relations $\widehat{\lambda}^{\mu}_{\
\nu}=\delta^{\mu}_{\nu}$ is the covariant $\kappa$-deformed DSR algebra. The
subalgebra with generators $\widehat{x}^{\mu}$, $\widehat{p}_{\nu}$ and
$\widehat{m}_{\lambda\sigma}$ is dual to the $\kappa$-deformed DSR algebra.

###  III.2 Heisenberg double as a right bialgebroid

The Heisenberg double $\mathcal{H}$ of any finite-dimensional Hopf algebra
$\mathbb{H}$ is shown in the Section 6 of Lu  to have the structure of a Hopf
$A^{*}$-algebroid where the base $B:=A^{*}$ is the underlying algebra of
$\mathbb{H}$. Our Hopf algebroid is an instance of an $\infty$-dimensional
version of that algebroid. We shall neglect the mathematical questions of
completions (see e.g. halg ).

The recipes of J-H. Lu Lu  are the following. The source
$\alpha(a^{\ast})=1\otimes a^{\ast}\in H=A\rtimes A^{*}$ for $a^{*}\in A^{*}$,
(usually we identify $A^{\ast}\cong{\mathbb{C}}\otimes A^{\ast}\hookrightarrow
A\rtimes A^{\ast}$). For the target Lu suggests a formula involving the
canonical element. In Section 6 of Lu Lu  a formula for the target map $\beta$
of the left bialgebroid structure on a Heisenberg double of finite dimensional
Hopf algebra is written out; this formula involves the canonical element in
$A\otimes A^{*}$. Given a vector space basis $\\{f^{I}\\}_{I}$ of $A$ and the
dual basis $\\{h_{I}\\}_{I}$ of $A^{*}$ characterized by $\langle
f^{I},h_{J}\rangle=\delta^{I}_{J}$, the canonical element is
$\sum_{I}f^{I}\otimes h_{I}$. If $A$ is infinite dimensional then the
dimension of $A^{*}$ is of even bigger cardinality than that of $A$, hence it
requires some care to make sense of the canonical element. Still, the dual
vectors $f^{I}$ are well defined and, in cases like ours, the infinite sum
converges in a (formal) completion $A\widehat{\otimes}A^{*}$. M. Stojić has
pointed to us a right bialgebroid version of Lu’s formula for $\beta$, namely

$\beta(t)=\sum_{I,J}f^{J}S^{-1}f^{I}\otimes h_{I}hh_{J}\in A\rtimes A^{\ast}$
(19)

where $\sum_{I}f^{I}\otimes h_{I}$ is the canonical element.

For the effective comultiplication $\Delta:H\to H\otimes_{B}H$ (where $B$ is
the base) J-H. Lu Lu  defines

$\Delta(a\otimes b^{\ast})=\sum(a_{(1)}\otimes 1)\otimes_{B}(a_{(2)}\otimes
b^{\ast}),\,\,\,\,\,\,\,\,\,\,\,\,\,a\in A,b\in A^{\ast},$ (20)

where $\Delta_{\mathbb{H}}(a)=\sum a_{(1)}\otimes a_{(2)}$ is the coproduct in
the Hopf algebra $\mathbb{H}$. Of course, in (19) we can simply write
$a_{(1)}\otimes a_{(2)}b^{\ast}$.

In the case of $\mathcal{H}^{(4,4)}$, this coproduct $\Delta$ is given on the
generators by the formulae

$\begin{array}[]{ll}\Delta(\widehat{x}^{\mu})=1\otimes\widehat{x}^{\mu}&\\\
\Delta(\widehat{p}_{k})=\widehat{p}_{k}\otimes
e^{\\-\frac{\widehat{p}_{0}}{\kappa}}+I\otimes\widehat{p}_{k},&\Delta(\widehat{p}_{0})=\widehat{p}_{0}\otimes
I+I\otimes\widehat{p}_{0}.\end{array}$ (21)

###  III.3 Counit and Fock actions

The property of the counit $\epsilon:H\to B$ which one wants to preserve from
the Hopf algebra case is its relation to the coproduct, that is

$(\epsilon\otimes_{B}\mathrm{id})\Delta=(\mathrm{id}\otimes_{B}\epsilon)\Delta=\mathrm{id}$
(22)

This requirement makes sense in view of the canonical identification of the
$B$-bimodules $B\otimes_{B}H\cong H\otimes_{B}B\cong H$. Taking into account
the $B$-bimodule structure on $H$ (used in the bimodule tensor product over
$B=A^{\ast}$), this amounts to
$h_{(1)}\alpha(\epsilon(h_{(2)}))=h=h_{(2)}\beta(\epsilon(h_{(1)}))$. However,
it is generally not possible to force the counit to be an algebra
homomorphism. Instead it satisfies the weaker properties brzmil ; bohmHbk

$\epsilon\left(\alpha(\epsilon(h))h^{\prime}\right)=\epsilon(hh^{\prime})=\epsilon\left(\beta(\epsilon(h))h^{\prime}\right)$
(23)

In the case of the Heisenberg double $A\rtimes A^{\ast}$ define

$\epsilon(a\otimes
b^{*})=\epsilon_{A}(a)b^{*},\,\,\,\,\,\,\,\,\,\,\,\,\,a\otimes b^{*}\in
A\rtimes A^{\ast}$ (24)

Then $(f,h)\mapsto f\blacktriangleleft h=\epsilon(\alpha(f)h)$ is a right
action, where $f\in\widetilde{\mathcal{T}_{\kappa}^{4}}$ and
$h,h^{\prime}\in\mathcal{H}^{(4,4)}=\mathcal{T}^{4}\rtimes\widetilde{\mathcal{T}_{\kappa}^{4}}$.
This action should be viewed as a deformed Fock space where the unit
$1=:\langle 0|$ is the right Fock vacuum. Indeed, $\langle
0|\widehat{p}_{\mu}=0$ and $\langle 0|\widehat{x}^{\nu}=\widehat{x}^{\nu}$ and
the usual normal ordering and commuting procedures for the evaluation of long
expressions in coordinates and momenta on the vacuum apply.

The counit $\epsilon$ satisfies defining equations (22) and on the generators
of $\mathcal{H}^{(4,4)}$ is given by

$\epsilon(\widehat{x}_{\mu})=\widehat{x}_{\mu},\qquad\epsilon(\widehat{p}_{\mu})=0,\qquad\epsilon(1)=1.$
(25)

In our case, the identity
$\epsilon(\alpha(\widehat{x}_{\mu}))=\epsilon(\beta(\widehat{x}_{\mu}))=\widehat{x}_{\mu}$
holds.

###  III.4 Calculating the target map

To apply the formula (19) for $\beta$ to the $(4+4)$-dimensional
$\kappa$-phase space we first compute the canonical element. This is easier in
the undeformed case; the results is then used to compute the deformed case.
Then the monomials $x^{I}$ in $x^{\mu}$ form a basis, and the monomials
$\widehat{p}_{I}$ in momentum operators $\widehat{p}_{\mu}$ are the dual
elements, up to proportionality constants, namely by (12) and (11) we obtain

$\langle x^{J},\widehat{p}_{I}\rangle_{0}=i^{|I|}I!$

where $I=(i_{0},i_{1},i_{2},i_{3})$ is a multiindex,
$|I|=-i_{0}+i_{1}+i_{2}+i_{3}$ and $I!=i_{0}!i_{1}!i_{2}!i_{3}!$. The
canonical element is therefore

$\sum_{i_{0},i_{1},i_{2},i_{3}=0}^{\infty}\frac{i^{|I|}}{I!}\widehat{p}_{I}\otimes
x^{I}=\exp(i\sum_{\mu=0}^{3}\widehat{p}_{\mu}\otimes x^{\mu}).$ (26)

There is now an isomorphism $\xi$ of coalgebras heisd  from the undeformed
coordinate algebra (polynomials in $x^{\mu}$) to the deformed coordinate
algebra (polynomials in $\widehat{x}^{\mu}$). The inverse $\xi^{-1}$ of this
isomorphism can be computed from the commutation relations as follows. In the
right bialgebroid case, $\widehat{x}^{\mu}$ is realized as the operator
$-i[\widehat{x}^{\mu},\widehat{p}_{\sigma}]x^{\sigma}$ where
$\widehat{p}_{\sigma}=-i\partial_{\sigma}$ and the operators act to the right.
Then, an arbitrary expression $h$ in $\widehat{x}^{\mu}$ acts from the right
as a differential operator to (the right-hand version of) the Fock vacuum, the
result is $\xi^{-1}(h)=\langle 0|h_{\mathrm{oper}}$. In our case,

$\widehat{x}_{k}=x_{k},\,\,\,\,\,\,\,\,\,\,\widehat{x}_{0}=x_{0}-\frac{i}{\kappa}\sum_{j=1}^{3}\partial_{j}x_{j}.$

Using this, one can calculate that

$\langle
0|\exp(-ip_{0}\widehat{x}_{0})\exp(ip_{j}\widehat{x}_{j})=\exp(-ip_{0}x_{0}+ip_{j}x_{j})$
(27)

where $p_{0},p_{j}$ are just numbers – not operators and the summation over
$i$, $\sum_{j=1}^{3}p_{i}x_{i}$ is understood.

In fact we should find some functions $F^{\mu}(k_{0},k_{1},k_{2},k_{3})$ so
that

$\langle 0|\exp(iF^{\mu}(k)\widehat{x}_{\mu})=\exp(-ik_{0}x_{0}+ik_{j}x_{j})$

but this involves more involved calculations of functions $F^{\mu}$. We
instead take a simpler looking product of exponentials, on the expense of need
for further care of the ordering in the remaining computation below.

The crucial observation heisd  is now that the deformed pairing
$\langle,\rangle$ can be expressed in terms of the undeformed pairing
$\langle,\rangle_{0}$ and the map $\xi$,

$\langle\xi(x^{J}),\widehat{p}_{I}\rangle=\langle
x^{J},\widehat{p}_{I}\rangle_{0}.$

Regarding that $\xi$ is an isomorphism of vector spaces, this means that it is
wise to take as basis $\xi(x^{J})$ and preserve the dual basis and we get the
canonical element
$\sum_{I}\frac{(-i)^{|I|}}{I!}\widehat{p}_{I}\otimes\xi(x^{I})=(1\otimes\xi)\exp(-i\widehat{p}_{\mu}\otimes
x^{\mu})=\exp(+i\widehat{p}_{0}\otimes\widehat{x}_{0})\exp(-i\widehat{p}_{i}\widehat{x}_{i})$.

Here we used that the first tensor product commutes with the second, hence
behaving as formal commuting variable in the calculation above.

Note that the equality of the sums $\sum_{I}S^{-1}f^{I}\otimes
h_{I}=\sum_{I}f^{I}\otimes S^{-1}h_{I}$. Indeed, the antipode $S$ of is an
isomorphism as well and $\langle Sf,S^{-1}h\rangle=\langle
f,SS^{-1}h\rangle=\langle f,h\rangle$ because the pairing is Hopf (11).
Therefore $\tilde{f}^{I}=Sf^{I}$ also form a basis and
$\tilde{h}_{I}=S^{-1}h_{I}$ are the dual vectors; regarding that the canonical
element does not depend on the choice of basis, $\sum_{I}S^{-1}f^{I}\otimes
h_{I}=\sum_{I}S^{-1}Sf^{I}\otimes S^{-1}h_{I}=\sum_{I}f^{I}\otimes
S^{-1}h_{I}$. In our case, $h_{I}$ are the elements in the enveloping algebra
and the antipode contributes to the sign, thus for $\sum_{I}f^{I}\otimes
S^{-1}h_{I}$ we get the same group-like exponential (26) with the minus sign,
that is its multiplicative inverse. Noting that $A$ is in our case
commutative, the order of $f_{I}$ and $f_{J}$ can interchange in the Lu
formula (19) and we obtain

$\beta(h)=\zeta\left(e^{i\widehat{p}_{j}\otimes\widehat{x}_{j}}e^{-i\widehat{p}_{0}\otimes\widehat{x}_{0}}(1\otimes
h)e^{i\widehat{p}_{0}\otimes\widehat{x}_{0}}e^{-i\widehat{p}_{j}\otimes\widehat{x}_{j}}\right)$
(28)

where $\zeta$ is the isomorphism of vector spaces sending the tensor product
algebra $A\otimes A^{\ast}=\mathcal{T}\otimes\tilde{\mathcal{T}}^{4}$ to the
smash product algebra $\mathcal{T}\rtimes\tilde{\mathcal{T}}^{4}$. This
isomorphism is the identity on the vector space level but not a homomorphism
of algebras, hence the expressions in terms of products of algebra generators
may look different.

We want to apply this formula (28) to the generators $\widehat{x}_{0}$ and
$\widehat{x}_{i}$; to this end first we apply the conjugation with the inner
exponentials and then with the outer exponentials. Using ad-expansion
$e^{X}Ye^{-X}=e^{\mathrm{ad}\,X}(Y)$ and inductive calculations with the
exponential series, in the case of $h=\widehat{x}_{i}$ the inner conjugation
in (28) gives
$\exp(-i\widehat{p}_{0}\otimes\widehat{x}_{0})(1\otimes\widehat{x}_{i})\exp(i\widehat{p}_{0}\otimes\widehat{x}_{0})=\exp(-\frac{\widehat{p}_{0}}{\kappa})\otimes\widehat{x}_{i}$
and the outer conjugation leaves the result intact, hence after applying
$\zeta$ we obtain

$\beta(\widehat{x}_{i})=\exp(-\frac{\widehat{p}_{0}}{\kappa})\widehat{x}_{i}.$
(29)

For $h=\widehat{x}_{0}$ the inner conjugation is not affecting $(1\otimes h)$
and the outer conjugation gives

$\beta(\widehat{x}_{0})=\zeta\left(e^{i\widehat{p}_{k}\otimes\widehat{x}_{k}}(1\otimes\widehat{x}_{0})e^{-i\widehat{p}_{k}\otimes\widehat{x}_{k}}\right)=\zeta\left(1\otimes\widehat{x}_{0}-\frac{1}{\kappa}\widehat{p}_{k}\otimes\widehat{x}_{k}\right)=\widehat{x}_{0}-\frac{1}{\kappa}\widehat{p}_{k}\widehat{x}_{k}.$
(30)

###  III.5 The coproduct gauge freedom

If we view the coproduct as an algebra map $\Delta:H\to
H\otimes_{\mathbb{C}}H$ then it is coassociative only up to elements in
certain subspace of $H\otimes_{\mathbb{C}}H\otimes_{\mathbb{C}}H$ and $\Delta$
itself can also be redefined up to elements in $\mathcal{I}_{B}\subset
H\otimes_{\mathbb{C}}H$. This freedom of coproducts is parametrized by the
coproduct gauges. If we consider coproduct as means to build the realizations
of algebra $H$ on tensor spaces, which corresponds in the physical context to
the Fock space realizations and multiparticle states, various possibilities of
such tensoring depending on the physical choice of particular dynamical model
are possible (see also Section V). In such a framework the gauge-invariant
object is the algebra $H$, and its tensorial realizations are gauge-dependent,
different for various dynamical models with Fock-like extensions. The Hopf
algebroid structure in mathematical sense deals with strictly coassociative
coproduct, which we may call effective, where the freedom in the left ideal
$\mathcal{I}_{B}$ is quotiented out brzmil ; Xu ; Lu , hence $\Delta:H\to
H\otimes_{B}H=(H\otimes_{\mathbb{C}}H)/{\mathcal{I}}_{B}$.

Let us compute the ideal $\mathcal{I}_{B}$ using $\beta$ from (29) and (30):

$\begin{array}[]{l}\alpha(\widehat{x}_{i})\otimes
1-1\otimes\beta(\widehat{x}_{i})=\widehat{x}_{i}\otimes 1-1\otimes
e^{-\frac{\widehat{p}_{0}}{\kappa}}\widehat{x}_{i},\\\
\alpha(\widehat{x}_{i})\otimes
1-1\otimes\beta(\widehat{x}_{i})=\widehat{x}_{i}\otimes
1-1\otimes\widehat{x}_{0}+1\otimes\frac{1}{\kappa}\widehat{p}_{k}\widehat{x}_{k}.\end{array}$
(31)

The expressions on the right hand side of equations (31) generate
$\mathcal{I}_{B}$ as a left ideal.

We can arrive to the same ideal or its parts by considering coproduct freedom
using elementary arguments with tensors. Thus, if
$\Delta(\widehat{x}^{\mu})=1\otimes\widehat{x}^{\mu}$ then any other
homomorphism $\tilde{\Delta}$ which is of the form

$\widetilde{\Delta}(\widehat{x}_{\mu})=\Delta(\widehat{x}_{\mu})+\Lambda_{\mu}(\widehat{x},\widehat{p})=\widehat{x}_{\rho}\otimes\theta_{\mu}^{\rho}(\widehat{p}),$
(32)

where $\Delta(\widehat{x}^{\mu})=1\otimes\widehat{x}^{\mu}$ and
$\theta_{\mu}^{\rho}(\widehat{p})$ is the tensor to be determined satisfies

$\displaystyle[\Delta(\widehat{x}_{[\mu}),\Lambda_{\nu]}]+[\Lambda_{\mu},\Lambda_{\nu}]=C_{\mu\nu}^{(\kappa)\rho}\Lambda_{\rho},$
(33) $\displaystyle[\Delta(\widehat{p}_{\mu}),\Lambda_{\nu}]=0.$ (34)

where
$C_{\mu\nu}^{(\kappa)\rho}=\frac{1}{\kappa}(\delta_{\mu}^{0}\eta_{\nu}^{\rho}-\delta_{\nu}^{0}\eta_{\mu}^{\
\rho})$ are the structure constants,
$[\widehat{x}_{\mu},\widehat{x}_{\nu}]=C_{\mu\nu}^{(\kappa)\rho}\widehat{x}_{\rho}$.

The relations (33)–(34) are required if the transformation
$\Delta(\widehat{x}_{\mu})\longrightarrow\widetilde{\Delta}(\widehat{x}_{\mu})$
is todescribe the coproduct gauge. Postulating that
$(\widetilde{\Delta}(\widehat{x}^{\mu}),\Delta(\widehat{p}_{\nu}))$ satisfies
the quantum phase space algebra relations which
$\widehat{x}^{\mu},\widehat{p}_{\nu}$ do one algebraically derives the
conditions fixing the tensor $\theta_{\mu}^{\rho}(\widehat{p})$, namely

$\widetilde{\Delta}(\widehat{x}_{i})=\widehat{x}_{i}\otimes
e^{\frac{\widehat{p}_{0}}{\kappa}},\quad\widetilde{\Delta}(\widehat{x}_{0})=\widehat{x}_{0}\otimes
1+\frac{1}{\kappa}\widehat{x}_{i}\otimes
e^{\frac{\widehat{p}_{0}}{\kappa}}\widehat{p}_{i}.$ (35)

As follows from (32) one gets

$\begin{array}[]{l}\Lambda_{i}=\widehat{x}_{i}\otimes
e^{\frac{\widehat{p}_{0}}{\kappa}}-1\otimes\widehat{x}_{i},\\\
\Lambda_{0}=\widehat{x}_{0}\otimes
1-1\otimes\widehat{x}_{0}+\frac{1}{\kappa}\widehat{x}_{i}\otimes
e^{\frac{\widehat{p}_{0}}{\kappa}}\widehat{p}_{i}.\end{array}$ (36)

Thus we obtained that $\Lambda$-s are in the ideal $\mathcal{I}_{B}$, namely
they are of the form

$\begin{array}[]{l}\Lambda_{\mu}(\widehat{x},\widehat{p})=(1\otimes\theta^{\rho}_{\mu})(\alpha(\widehat{x}_{\rho})\otimes
1-1\otimes\beta(\widehat{x}_{\rho}))\in\mathcal{I}_{B}\end{array}$ (37)

Similar tensors $R_{\mu}=\widehat{x}_{\mu}\otimes
1-\tilde{\theta}_{\mu}^{\rho}(\widehat{p})\otimes\widehat{x}_{\rho}$, where
$\tilde{\theta}_{\mu}^{\nu}$ is the matrix inverse to
$\theta_{\lambda}^{\rho}$ (see (32)), have been introduced in 1  for the
canonical twisted Heisenberg algebra and considered in 2  for the
$\kappa$-deformed quantum phase space generated by the $\kappa$-deformed
Poincaré–Hopf algebra with classical Poincaré algebra sector. Note that our
$\widehat{x}^{\mu}$ corresponds to $\hat{y}^{\mu}$ from 1  rather than their
$\hat{x}^{\mu}$ which is a more convenient generator for the description of
the left bialgebroid structure.

Analogous analysis as for $\Lambda_{\mu}$ can be made for higher order tensors
lukx .

##  IV Antipode

Hopf algebroid is a bialgebroid with an antipode which is a linear map
$\tau:H\to H$. In bialgebroids, the source $\alpha$ and target $\beta$
together play the role of the unit and it is natural to require
$m(\tau\otimes_{B}\mathrm{id})\Delta=\alpha\epsilon$ and
$m(\mathrm{id}\otimes_{B}\mathrm{id})\Delta=\beta\epsilon$ as an extension of
the axioms for the antipode in Hopf algebra theory. The first of the two
equations is however problematic:
$m(\tau\otimes_{B}\mathrm{id})(\mathcal{I}_{B})\neq 0$ in general, hence the
right hand side with effective $\Delta$ being defined up to gauge in
$\mathcal{I}_{B}$ is not well defined; only a subclass of gauges could satisfy
this equation. J-H. Lu Lu  avoids this by restricing to a subclass of gauges;
in fact she introduces a linear map $\gamma:H\otimes_{B}H\to H\otimes H$ which
is a section of the projection $H\otimes H\to H\otimes_{B}H$, so that
$\Delta_{\gamma}=\gamma\circ\Delta$ is a choice of gauge. Her axioms (here in
the right bialgebroid version) are thus nonsymmetric:

$\displaystyle\tau\beta=\alpha,$ (38) $\displaystyle
m(\tau\otimes_{\mathbb{C}}\mathrm{id})(\gamma\circ\Delta)=\alpha\epsilon,$
(39) $\displaystyle m(\mathrm{id}\otimes_{B}\tau)\Delta=\beta\epsilon\tau.$
(40)

Now the left hand side of (40) does not depend on coproduct gauge for
$\Delta$. An approach by G. Böhm bohmHbk  is symmetric: she supplies both a
right $B$-bialgebroid $\mathcal{H}_{R}$ and a left
$B^{\mathrm{op}}$-bialgebroid $\mathcal{H}_{L}$ structure on the same total
algebra $\mathcal{H}=\mathcal{H}_{R}=\mathcal{H}_{L}$ and the antipode
$\tau:\mathcal{H}\to\mathcal{H}$ is a linear antiautomorphism satisfying a
number of axioms involving both left and right bialgebroid structures; the
coproducts of $\mathcal{H}_{L}$ and $\mathcal{H}_{R}$ are also compatible as
well as the counit, source and target data for the two bialgebroids. In 4  a
subalgebra $\mathcal{B}$ in $H\otimes_{\mathbb{C}}H$ is singled out within
which all gauge choices satisfy (39), i.e. $\gamma\circ\Delta$ can be replaced
by any $\tilde{\Delta}$ landing within $\mathcal{B}$. The latter approach has
been axiomatized in twosha .

To compute the antipode, we can either again follow the Heisenberg double
formulas for $\tau$ in Section 6 of Lu  which, like for $\beta$, also involve
the canonical elements. But this recipe boils down to the following direct
reasoning. For the momentum sector $\mathcal{T}^{4}$, $\tau$ agrees with the
Hopf-algebraic antipode:
$\tau(\widehat{p}_{0})=S(\widehat{p}_{0})=-\widehat{p}_{0}$ and
$\tau(\widehat{p}_{i})=S(\widehat{p}_{i})=-e^{{\frac{\widehat{p}_{0}}{\kappa}}}\widehat{p}_{i}$
(see (10)). For the coordinate sector, formulas for $\tau$ are forced by the
equations
$\tau(\beta(\widehat{x}_{\mu}))=\alpha(\widehat{x}_{\mu})=\widehat{x}_{\mu}$
which, using that $\tau$ is an antihomomorphism of algebras and inserting its
values on $\widehat{p}_{i}$, yield

$\displaystyle\tau(\widehat{x}_{i})=e^{-\frac{\widehat{p}_{0}}{\kappa}}\widehat{x}_{i}=\tau^{-1}(\widehat{x}_{i})=\beta(\widehat{x}_{i}),$
(41)
$\displaystyle\tau(\widehat{x}_{0})=\widehat{x}_{0}-\frac{1}{\kappa}\widehat{x}_{i}\widehat{p}_{i}=\beta(\widehat{x}_{0})-\frac{3i}{\kappa},$
(42)
$\tau^{2}(\widehat{p}_{\mu})=\widehat{p}_{\mu},\,\,\,\,\,\,\,\,\,\,\,\,\,\tau^{2}(\widehat{x}_{i})=\widehat{x}_{i},\,\,\,\,\,\,\,\,\,\,\,\,\,\tau^{2}(\widehat{x}_{0})=\widehat{x}_{0}-\frac{3i}{\kappa}.$
(43)

##  V Discussion and Outlook

The general aim of this paper is to promote the application of Hopf algebroid
structure to the description of quantum–deformed phase spaces. We consider
here the algebraic model of phase spaces constructed as Heisenberg doubles or
equivalently smash products of Drinfeld quantum enveloping algebra
$\mathbb{H}$ acting upon dual quantum group $\widetilde{\mathbb{H}}$. In
bialgebroid framework an important difference with bialgebra is the
generalization of unity to the noncommutative base algebra. If we consider the
quantum-deformed phase spaces as Heisenberg doubles the base algebra is
provided by the whole algebraic sector of quantum group
$\widetilde{\mathbb{H}}$. We add that our basic calculational result consists
in detailed presentation of $\kappa$-deformed quantum phase space as
physically interesting example of Hopf bialgebroid (see also lukx ).

In the paper we introduced the notion of coproduct gauges describing a freedom
of coproducts in the Hopf algebroid framework. For quantum phase spaces the
coproduct gauges describe model – dependent ways of composing the global
two–particle phase space from the phase space coordinates of two its
constituents.

In a simple case of D=3 nonrelativistic phase space the momenta coproducts
describe global 2–particle momentum as follows

$\Delta(p_{i})=p_{i}\otimes 1+1\otimes p_{i}\quad\longleftrightarrow\quad
p^{(1+2)}_{i}=p^{(1)}_{i}+p^{(2)}_{i}$ (44)

and the formula for 2–particle center–of–mass coordinate is given by
well–known formula

$x^{(1+2)}_{i}=\frac{m_{1}}{m_{1}+m_{2}}\,x^{(1)}_{i}+\frac{m_{2}}{m_{1}+m_{2}}\,x^{(2)}_{i}$
(45)

which can be represented as well by the following one–parameter class of
coproduct gauges $\Lambda_{i}$ (see also lukx , formula (34) in the limit
$\kappa\to\infty$)

$\begin{array}[]{l}\Delta(x_{i})=x_{i}\otimes
1+\widehat{\Lambda}_{i}\\\\[10.0pt]
\widehat{\Lambda}_{i}=(\alpha-1)(x_{i}\otimes 1-1\otimes
x_{i})\qquad\quad\alpha=\frac{m_{1}}{m_{1}+m_{2}}\,.\end{array}$ (46)

The coproduct gauges describe different ways of composing center-of-mass
coordinate $x_{i}^{(1+2)}$ for different composite systems, characterized by
the masses ($m_{1},m_{2}$) of its constituents. From this point of view
therefore the coproduct gauge freedom is not unphysical as in the standard
gauge theories, but characterizes different choices of dynamical systems
leading to the same phase space algebra (Poisson algebra in classical
mechanics, (deformed) Heisenberg algebra in quantum theory).

One can also consider the formulae describing $D=4$ relativistic center of
mass coordinates $\widehat{x}_{\mu}^{(1+2)}$ (see e.g. 40 ; 41 ) which can be
described as the bialgebroid coproduct, with coproduct gauges depending as
well on negative powers of energy and three–momenta components. In such a way
by considering the $(10+10)$ dimensional phase space
${\mathcal{H}}^{(10,10)}$, containing Lorentz algebra and the dual Lorentz
group parameters, one can describe in principle various relativistic center-
of-mass coordinates for two particles with arbitrary spin.

Finally we present an outlook. We shall point out some ways of extending the
presented result, employing the notion of bialgebroid structure:

i) The bialgebroid formulae for various phase space algebra bases.

In this paper (see also lukx ) we consider the $\kappa$-deformed phase space
generated by Majid–Ruegg basis majidruegg  of corresponding $\kappa$-deformed
Poincaré–Hopf algebra. In such a basis the standard $\kappa$-deformed phase
space is described by centrally extended 8–dimensional Lie algebra. Such Lie
algebraic structure is preserved only under the linear change
($\widehat{p}_{\mu}\to\widehat{p}{\,}^{\prime}_{\mu}=\alpha_{\mu}^{\
\nu}\widehat{p}_{\nu}$) of momenta basis. The bicrossproduct structure of
Poincaré–Hopf algebra which implies quantum covariance properties of phase
space remains however valid if we change the fourmomenta basis in arbitrary
nonlinear way

$\widehat{p}_{\mu}\to\widehat{p}{\,}^{\prime}_{\mu}=F_{\mu}(\widehat{p})\qquad\qquad
F_{\mu}(0)=0\,.$ (47)

In particular one can choose $F_{\mu}(\widehat{p})$ in a way leading to
classical Poincaré algebra in algebraic sector of $\kappa$-deformed Poincaré
algebra 33 –35 . In such a way we get more complicated nonabelian formulae for
the composition law of fourmomenta, which unfortunately are not endowed with
physical interpretation.
ii) Hopf algebroid structure of generalized phase space
${\mathcal{H}}^{(10,10)}$.

In order to describe the global phase coordinates for two particles with
relativistic spin one should include into the phase space algebra the Lorentz
group parameters $\widehat{\lambda}^{\mu}_{\,\nu}$ and dual Lorentz algebra
generators $\widehat{m}_{\mu\nu}$ (for undeformed case these ideas were
considered earlier 17 –22 ). The noncanonical pair of dual generators
($\widehat{\lambda}_{\mu}^{\ \nu},\widehat{m}^{\rho}_{\ \lambda}$) satisfying
the relation (17) can be also replaced by a canonical pair
($\widehat{\omega}^{\mu}_{\ \nu},\widehat{\pi}^{\rho}_{\ \lambda}$) where
$\widehat{\lambda}^{\mu}_{\ \nu}=\delta^{\mu}_{\
\nu}+\widehat{\omega}^{\mu}_{\ \nu}$ and ($\eta_{\mu\nu}=diag(-1,1,1,1)$)

$\left[\widehat{\lambda}_{\mu}^{\ \nu},\widehat{\omega}^{\rho}_{\
\lambda}\right]=i\left(\delta_{\mu}^{\ \rho}\,\delta^{\nu}_{\
\lambda}-\delta_{\mu}^{\ \lambda}\,\delta^{\nu}_{\ \rho}\right)$ (48)

consistent with $\widehat{\omega}^{\rho}_{\
\lambda}=-\widehat{\omega}^{\lambda}_{\ \rho}$ and the formula
$\widehat{m}^{\rho}_{\ \lambda}=\widehat{\omega}^{\rho}_{\
\nu}\,\widehat{\pi}^{\nu}_{\ \lambda}-\widehat{\omega}^{\lambda}_{\
\nu}\,\widehat{\pi}^{\nu}_{\ \rho}$. The calculations of the target map and
coproduct gauge freedom for ${\mathcal{H}}^{(10,10)}$ are now under
consideration.
iii) Hopf superalgebroid structure of deformed supersymmetric quantum –
deformed phase space.

Following the analogy respecting the “sign rules” between Hopf algebras and
Hopf superalgebras one can define the $Z_{2}$-graded algebraic structure of
Hopf superbialgebroids (see e.g. karx ). In order to extend supersymmetrically
the $\kappa$-deformed phase space presented in this paper one can use already
known results about the supersymmetric extension of $\kappa$-deformed Poincaré
and its dual quantum Poincaré supergroup 13c ; Wessx . For such a case one
should firstly employ the supersymmetric extension of Heisenberg double, in
particular for $\kappa$-deformed Poincaré–Hopf algebra.
iv) Twisted bialgebroids and the deformation of quantum phase spaces.

One can extend the twist deformation technique of Hopf algebras to Hopf
algebroids Xu . In such a way one can obtain from Hopf algebroids with
commutative base algebra (e.g. standard Heisenberg algebra) the Hopf algebroid
with noncommutative base, which can be effectively calculated in terms of
twist factor by so–called $\star$-product formalism. In particular one can
apply this technique to dynamical quantum deformations, described by parameter
– dependent classical $r$-matrices satisfying dynamical YB equation (for the
link of dynamical quantum deformations with Hopf algebroids see e.g. Xu ;
doninmudrov ).

###### Acknowledgements.

J. L. and M. W. have been supported by Polish National Science Center project
2014/13/B/ST2/04043.

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