One Parameter Family of Jordanian Twists

One Parameter Family of Jordanian Twists

D. Meljanac, S. Meljanac, Z. Škoda and R. Štrajn

Daniel MELJANAC ${}^{{\dagger}^{1}}$, Stjepan MELJANAC ${}^{{\dagger}^{2}}$,
Zoran ŠKODA ${}^{{\dagger}^{3}}$ and Rina ŠTRAJN ${}^{{}^{{\dagger}^{4}}}$

${}^{{\dagger}^{1}}$ Division of Materials Physics, Institute Rudjer Bošković,
${}^{{\dagger}^{1}}$ Bijenička cesta 54, P.O. Box 180, HR-10002 Zagreb,
Croatia<EMAIL_ADDRESS>
${}^{{\dagger}^{2}}$ Theoretical Physics Division, Institute Rudjer Bošković,
${}^{{\dagger}^{2}}$ Bijenička cesta 54, P.O. Box 180, HR-10002 Zagreb,
Croatia<EMAIL_ADDRESS>
${}^{{\dagger}^{3}}$ Department of Teachers’ Education, University of Zadar,
${}^{{\dagger}^{3}}$ Franje Tudjmana 24, 23000 Zadar, Croatia<EMAIL_ADDRESS>
${}^{{\dagger}^{4}}$ Department of Electrical Engineering and Computing,
University of Dubrovnik,
${}^{{\dagger}^{4}}$ Ćira Carića 4, 20000 Dubrovnik, Croatia
<EMAIL_ADDRESS>
Received April 16, 2019, in final form October 19, 2019; Published online
October 25, 2019

We propose an explicit generalization of the Jordanian twist proposed in
$r$-symmetrized form by Giaquinto and Zhang. It is proved that this
generalization satisfies the 2-cocycle condition. We present explicit formulas
for the corresponding star product and twisted coproduct. Finally, we show
that our generalization coincides with the twist obtained from the simple
Jordanian twist by twisting by a 1-cochain.

noncommutative geometry; Jordanian twist 53D55; 16T05

##  1 Introduction

Drinfeld twists of Hopf algebras [7] provide a systematic way of producing new
examples in noncommutative geometry. Given a Hopf algebra $\mathcal{H}$ with a
coproduct $\Delta_{0}$, a counit $\epsilon_{0}$, and an element
$\mathcal{F}\in\mathcal{H}\otimes\mathcal{H}$ satisfying a 2-cocycle condition
and a normalization (counitality) condition [3, 8, 9, 27], often called a
Drinfeld twist, one defines a new coproduct,
$\Delta^{\mathcal{F}}(-)=\mathcal{F}^{-1}\Delta_{0}(-)\mathcal{F}$, which is
coassociative due to the 2-cocycle condition. Moreover, $\mathcal{H}$ as an
algebra, together with the new coproduct $\Delta^{\mathcal{F}}$ becomes a new
twisted Hopf algebra $\mathcal{H}^{\mathcal{F}}$. Along with a Hopf algebra,
many associated constructions like its representations, comodules, module
algebras and so on, are twisted as well, using standard formulas involving the
twist $\mathcal{F}$. The systematic nature of the twisting procedure makes it
suitable for finding new physical models with the Hopf algebra covariance
built in.

In 1989, a new construction of a deformation 2-cocycle is proposed by Coll,
Gerstenhaber and Giaquinto in [4]. Their construction starts with a
$k$-algebra $A$ with multiplication $\mu_{A}$ and two derivations
$\phi,\psi\colon A\to A$ satisfying $[\phi,\psi]=\lambda\psi$ for some
$\lambda\in k$. The action of the 2-dimensional Lie algebra $L$ generated by
$\phi$ and $\psi$ on $A$ by derivations extends to a unique action
$\triangleright$ of the Hopf algebra $U(L)$ on $A$ making it a $U(L)$-module
algebra. They prove [4, 12] that $\mu_{A}\circ(\phi\otimes\psi)$ is a
Hochschild 2-cocycle which may be integrated to yield a formal deformation of
$A$. The deformed multiplication on $A[[t]]$ is given by
$\mu_{A}\circ(1\otimes 1+t\phi\otimes
1)^{1\otimes\psi}\\!\circ(\triangleright\otimes\triangleright)$ for
$\lambda=1$. This formula involves the element

$\displaystyle(1\otimes 1+t\phi\otimes
1)^{1\otimes\psi}=\sum_{n=0}^{\infty}t^{n}\phi^{n}\otimes\binom{\psi}{n}$
$\displaystyle\hphantom{(1\otimes 1+t\phi\otimes
1)^{1\otimes\psi}}{}=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}\phi^{n}\otimes\psi(\psi-1)\cdots(\psi-n+1)\in(U(L)\otimes
U(L))[[t]],$

which is a (Drinfeld) 2-cocycle twist for the Hopf algebra $U(L)[[t]]$. They
provide several examples. Their construction is reanalyzed in detail in [12]
and the 2-cocycle twist has been rediscovered in [35]. Algebras obtained by
variants of their method are now often called Jordanian deformations. Most
studied examples are a Jordanian deformation of the universal enveloping
algebra $U(\mathfrak{sl}(2))$ (and its dual) and more general Jordanian
quantum groups, leading to the corresponding Jordanian classical $r$-matrices
and quantum $R$-matrices (some of which were known before, e.g., [26, Example
1, due D. Gurevich] and [5, Section 2.2]). Regarding that
$U(\mathfrak{sl}(2))$ can be embedded into Yangian $Y(\mathfrak{sl}(2))$, it
is not surprising that more elaborate versions of Jordanian twists are used to
obtain new deformations of Yangians [20, 23, 37], with applications to
integrable models, chain models in particular [21]. A comprehensive study of a
related class of classical $r$-matrices can be found in [19].

Here we present another approach to new Jordanian deformations. Closer to the
setup of our paper, consider the universal enveloping algebra of the
2-dimensional solvable Lie algebra with generators $H$ and $E$ with $[H,E]=E$.
Define

$\displaystyle H^{\langle m\rangle}=H(H+1)\cdots(H+m-1),\qquad H^{\langle
0\rangle}=1.$

Giaquinto and Zhang in [10, Theorem 2.20]111The twist $\mathcal{F}$ in [10] is
renamed here as $\mathcal{F}^{-1}$. proposed the Jordanian twist [35] in
$r$-symmetrized form

$\displaystyle\mathcal{F}_{{\rm
GZ}}^{-1}=\sum_{m=0}^{\infty}\frac{t^{m}}{m!}\sum_{r=0}^{m}(-1)^{r}\binom{m}{r}E^{m-r}H^{\langle
r\rangle}\otimes E^{r}H^{\langle m-r\rangle}.$

This twist can also be written as

$\displaystyle\mathcal{F}_{{\rm
GZ}}^{-1}=\sum_{m=0}^{\infty}t^{m}\sum_{r=0}^{\infty}(-E)^{m-r}\binom{-H}{r}\otimes
E^{r}\binom{-H}{m-r}=\sum_{k,l=0}^{\infty}t^{k+l}(-E)^{k}\binom{-H}{l}\otimes
E^{l}\binom{-H}{k}.$

We shall use a different notation in this paper, namely

$\displaystyle E=P,\qquad H=-D,\qquad[D,P]=-P.$

This suggests an interpretation of $D$ as the relativistic dilation operator
and $P$ as the momentum in some applications. We introduce a family of twists
$\mathcal{F}_{{\rm GZ},u}^{-1}$, parametrized by parameter $u$, via an
explicit series (2.1). This family interpolates between the Jordanian twists
$\mathcal{F}^{-1}_{0}$ and $\mathcal{F}^{-1}_{1}$, where

$\displaystyle\mathcal{F}_{0}=\exp\left(-\ln\left(1-\frac{1}{\kappa}P\right)\otimes
D\right)\qquad\mbox{and}\qquad\mathcal{F}_{1}=\exp\left(-D\otimes\ln\left(1+\frac{1}{\kappa}P\right)\right).$
(1.1)

Our main interest in Jordanian twists is due to their appearance [2, 39] in
the study of $\kappa$-deformed Minkowski space (where the intepretation of $D$
and $P$ as the dilation and momentum operators also makes sense), where
$\kappa$ is viewed as being linked to the scale of quantum gravity [24, 25,
38].

Any Drinfeld twist $\mathcal{F}$ can be modified by any 1-cochain
$\omega\in\mathcal{H}$, producing a new twist
$\big{(}\omega^{-1}\otimes\omega^{-1}\big{)}\mathcal{F}\Delta(\omega)$, see
[27]. In an earlier paper [1], this procedure has been used to obtain a
certain twist $\mathcal{F}_{R,u}^{-1}$ for every $u$. In that context, it has
been written in the form of a product of three exponential factors, see also
reference [31]. Regarding that it is obtained from a 2-cocycle by modification
by a 1-cochain implies that it is itself a 2-cocycle.

Twists $\mathcal{F}_{{\rm GZ},u}$ and $\mathcal{F}_{R,u}$ generate the same
Hopf algebra. It is proved in this paper that our generalized Giaquinto–Zhang
twist $\mathcal{F}_{{\rm GZ},u}^{-1}$ satisfies the same differential identity
as $\mathcal{F}_{R,u}^{-1}$, including the initial condition; consequently the
two twists coincide. The importance of this result is that while
$\mathcal{F}_{{\rm GZ},u}^{-1}$ is introduced via an explicit series expansion
more suited for other calculations, the very construction of
$\mathcal{F}_{R,u}^{-1}$ ensures that it is a 2-cocycle; we however also
exhibit an elaborate proof of the 2-cocycle condition, directly from the
definition of $\mathcal{F}_{{\rm GZ},u}^{-1}$.

The exposition is organized as follows. In Section 2, we define an
interpolation $\mathcal{F}_{{\rm GZ},u}^{-1}$ via an explicit expansion and
show that it has the claimed limits at $u=0$ and $u=1$. In Section 2.1, we
prove directly from the definition that $\mathcal{F}_{{\rm GZ},u}^{-1}$
satisfies the 2-cocycle condition. In Section 2.2, we compute the
corresponding star product and in Section 2.3 the twisted coproduct $\Delta
p_{\mu}$. In Section 2.4 we introduce noncommutative coordinates and their
realizations. Section 3 is dedicated to the family $\mathcal{F}_{R,u}$ of
Jordanian twists obtained from a simple Jordanian twist $\mathcal{F}_{0}$
(1.1) via twisting by a 1-cochain. We start the section by introducing
$\mathcal{F}_{R,u}$ as a product of three exponential factors. Then we compute
the corresponding deformed Hopf algebra in Section 3.1, introduce the
corresponding noncommutative coordinates and realizations in Section 3.2 and
compute the star products in Section 3.3. In Section 4, we present two
different proofs both showing that $\mathcal{F}_{{\rm GZ},u}$ equals
$\mathcal{F}_{R,u}$. The first proof in Section 4.1 is by showing that they
solve the same Cauchy problem (an ordinary differential equation with initial
condition). The second proof in Section 4.2 uses a comparison among the star
products. The final Section 5 is the conclusion. Appendix A is added
presenting a proof of an identity used in the proof in Section 2.1 of the
2-cocycle condition for $\mathcal{F}^{-1}_{{\rm GZ},u}$.

##  2 Generalization of the Giaquinto–Zhang twist

We define the generalized Jordanian twist $\mathcal{F}_{{\rm GZ},u}^{-1}$ via
an explicit expansion,

$\displaystyle\mathcal{F}_{{\rm
GZ},u}^{-1}=\sum_{k,l=0}^{\infty}\left(\frac{1}{\kappa}\right)^{k+l}\left((u-1)P\right)^{k}\binom{D}{l}\otimes(uP)^{l}\binom{D}{k}.$
(2.1)

The twist $\mathcal{F}_{{\rm GZ},u}^{-1}$ interpolates between
$\mathcal{F}_{0}^{-1}$ and $\mathcal{F}_{1}^{-1}$. For $u\rightarrow 0$ one
can easily see [2] that (2.1) reduces to

$\displaystyle\mathcal{F}_{0}^{-1}=\sum_{m=0}^{\infty}\left(\frac{-1}{\kappa}\right)^{m}P^{m}\otimes\binom{D}{m}={\rm
e}^{\ln(1-\frac{1}{\kappa}P)\otimes D}.$

For $u=1$

$\displaystyle\mathcal{F}_{1}^{-1}=\sum_{m=0}^{\infty}\left(\frac{1}{\kappa}\right)^{m}\binom{D}{m}\otimes
P^{m}={\rm e}^{D\otimes\ln(1+\frac{1}{\kappa}P)}.$

For $u=\frac{1}{2}$ this reduces to the twist introduced in [10], where

$\displaystyle t=\frac{1}{2\kappa},\qquad E=P,\qquad\text{and}\qquad H=-D.$

###  2.1 2-cocycle condition

######  Theorem 2.1.

For arbitrary $u$, twists $\mathcal{F}_{{\rm GZ},u}^{-1}$ satisfy the
$2$-cocycle condition given by

$\displaystyle\big{(}(\Delta_{0}\otimes 1)\mathcal{F}^{-1}_{{\rm
GZ},u}\big{)}\big{(}\mathcal{F}^{-1}_{{\rm GZ},u}\otimes
1\big{)}=\big{(}(1\otimes\Delta_{0})\mathcal{F}^{-1}_{{\rm
GZ},u}\big{)}\big{(}1\otimes\mathcal{F}^{-1}_{{\rm GZ},u}\big{)}.$ (2.2)

######  Proof 2.2.

If we write

$\displaystyle
f_{n}:=\sum_{k+l=n}\left((u-1)P\right)^{k}\binom{D}{l}\otimes(uP)^{l}\binom{D}{k}$

then

$\displaystyle\mathcal{F}^{-1}_{{\rm
GZ},u}=\sum_{n=0}^{\infty}\left(\frac{1}{\kappa}\right)^{n}f_{n}$

with $f_{n}$ not depending on $\kappa$ and $f_{0}=1\otimes 1$. In terms of
$f_{i}$, the 2-cocycle condition becomes a sequence of equations for all $n$,

$\displaystyle\sum_{i=0}^{n}\left((\Delta_{0}\otimes
1)f_{i}\right)(f_{n-i}\otimes
1)=\sum_{i=0}^{n}\left((1\otimes\Delta_{0})f_{i}\right)(1\otimes f_{n-i}).$

In the first order in $1/\kappa$,

$\displaystyle f_{1}=(u-1)P\otimes D+uD\otimes P,$
$\displaystyle(\Delta_{0}\otimes 1)f_{1}+f_{1}\otimes
1=(1\otimes\Delta_{0})f_{1}+1\otimes f_{1},$

and in the second order,

$\displaystyle f_{2}=(u-1)^{2}P^{2}\otimes\binom{D}{2}+(u-1)uPD\otimes
PD+u^{2}\binom{D}{2}\otimes P^{2},$ $\displaystyle(\Delta_{0}\otimes
1)f_{2}+((\Delta_{0}\otimes 1)f_{1})(f_{1}\otimes 1)+f_{2}\otimes
1=(1\otimes\Delta_{0})f_{2}+((1\otimes\Delta_{0})f_{1})(1\otimes
f_{1})+1\otimes f_{2}.$

For general order $n$, it should hold that

$\displaystyle\sum_{\begin{subarray}{c}k_{1},k_{2},l_{1},l_{2}=0\\\
k_{1}+k_{2}=k,\,l_{1}+l_{2}=l,\,k+l=n\end{subarray}}^{\infty}\left[\Delta_{0}\left(P^{k_{1}}\binom{D}{l_{1}}\right)\otimes
P^{l_{1}}\binom{D}{k_{1}}\right]\left[P^{k_{2}}\binom{D}{l_{2}}\otimes
P^{l_{2}}\binom{D}{k_{2}}\otimes 1\right]$
$\displaystyle\qquad{}=\sum_{\begin{subarray}{c}k_{1},k_{2},l_{1},l_{2}=0\\\
k_{1}+k_{2}=k,\,l_{1}+l_{2}=l,\,k+l=n\end{subarray}}^{\infty}\left[P^{k_{1}}\binom{D}{l_{1}}\otimes\Delta_{0}\left(P^{l_{1}}\binom{D}{k_{1}}\right)\right]\left[1\otimes
P^{k_{2}}\binom{D}{l_{2}}\otimes P^{l_{2}}\binom{D}{k_{2}}\right].$

This can be rewritten as

$\displaystyle\sum_{k_{1}=0}^{k}\sum_{l_{1}=0}^{l}\left[\Delta_{0}\left(P^{k_{1}}\binom{D}{l_{1}}\right)\otimes
P^{l_{1}}\binom{D}{k_{1}}\right]\left[P^{k-k_{1}}\binom{D}{l-l_{1}}\otimes
P^{l-l_{1}}\binom{D}{k-k_{1}}\otimes 1\right]$
$\displaystyle\qquad{}=\sum_{k_{1}=0}^{k}\sum_{l_{1}=0}^{l}\left[P^{k_{1}}\binom{D}{l_{1}}\otimes\Delta_{0}\left(P^{l_{1}}\binom{D}{k_{1}}\right)\right]\left[1\otimes
P^{k-k_{1}}\binom{D}{l-l_{1}}\otimes P^{l-l_{1}}\binom{D}{k-k_{1}}\right],$
$\displaystyle\sum_{k_{1}=0}^{k}\sum_{l_{1}=0}^{l}\left[\Delta_{0}\big{(}P^{k_{1}}\big{)}\big{(}P^{k-k_{1}}\otimes
P^{l_{2}-l_{1}}\big{)}\binom{(D-k+k_{1})\otimes
1+1\otimes(D-l+l_{1})}{l_{1}}\otimes P^{l_{1}}\binom{D}{k_{1}}\right]$
$\displaystyle\qquad\quad{}\times\left[\binom{D}{l-l_{1}}\otimes\binom{D}{k-k_{1}}\otimes
1\right]$
$\displaystyle\qquad{}=\sum_{k_{1}=0}^{k}\sum_{l_{1}=0}^{l}\big{[}P^{k_{1}}\otimes\big{(}\Delta_{0}\big{(}P^{l_{1}}\big{)}\big{(}P^{k-k_{1}}\otimes
P^{l-l_{1}}\big{)}\big{)}\big{]}$
$\displaystyle\qquad\quad{}\times\left[\binom{D}{l_{1}}\otimes\binom{(D-k+k_{1})\otimes
1+1\otimes(D-l+l_{1})}{k_{1}}\left(\binom{D}{l-l_{1}}\otimes\binom{D}{k-k_{1}}\right)\right].$

Let us compare the terms of type $P^{A}\otimes P^{B}\otimes P^{C}$ with
$A+B+C=k+l=n$ on both sides. We see only the terms with $C=l_{1}$ on the left-
hand side and only the terms with $A=k_{1}$ on the right-hand side. We also
need to take into account
$\Delta_{0}\big{(}P^{k_{1}}\big{)}=\sum\limits_{a=0}^{k_{1}}\binom{k_{1}}{a}P^{k_{1}-a}\otimes
P^{a}$ on the left-hand side and
$\Delta_{0}\big{(}P^{l_{1}}\big{)}=\sum\limits_{b=0}^{l_{1}}\binom{l_{1}}{b}P^{b}\otimes
P^{l_{1}-b}$ on the right-hand side to obtain

$\displaystyle\sum_{\begin{subarray}{c}k_{1}=0\\\
k-a=A\end{subarray}}^{k}\sum_{a=0}^{k_{1}}\binom{k_{1}}{a}\binom{(D-k+k_{1})\otimes
1+1\otimes(D-l+C)}{C}\left(\binom{D}{l-C}\otimes\binom{D}{k-k_{1}}\right)\otimes\binom{D}{k_{1}}$
$\displaystyle=\sum_{\begin{subarray}{c}l_{1}=0\\\
l-b=C\end{subarray}}^{l}\sum_{b=0}^{l_{1}}\binom{l_{1}}{b}\binom{D}{l_{1}}\otimes\binom{(D-k+A)\otimes
1+1\otimes(D-l+l_{1})}{A}\left(\binom{D}{l-l_{1}}\otimes\binom{D}{k-A}\right),$
$\displaystyle\sum_{\begin{subarray}{c}k-a=A\\\
k_{1}=0\end{subarray}}^{k}\binom{k_{1}}{k-A}\binom{(D-k+k_{1})\otimes
1+1\otimes(D-l+C)}{C}\left(\binom{D}{l-C}\otimes\binom{D}{k-k_{1}}\right)\otimes\binom{D}{k_{1}}$
$\displaystyle=\sum_{\begin{subarray}{c}l-b=C\\\
l_{1}=0\end{subarray}}^{l}\binom{l_{1}}{l-C}\binom{D}{l_{1}}\otimes\binom{(D-k+A)\otimes
1+1\otimes(D-l+l_{1})}{A}\left(\binom{D}{l-l_{1}}\otimes\binom{D}{k-A}\right)$

for every $k,l\in\mathbf{N}_{0}$, and all $A\leq k$, $C\leq l$.

In terms of the new variables

$\displaystyle x=D\otimes 1\otimes 1,\qquad y=1\otimes D\otimes 1,\qquad
z=1\otimes 1\otimes D,$

and taking into account that $k+l=n$, we reduce the 2-cocycle condition to the
identity

$\displaystyle\binom{x}{l-C}\sum_{k_{1}=k-A}^{k}\binom{k_{1}}{k-A}\binom{x+y-k-l+k_{1}+C}{C}\binom{y}{k-k_{1}}\binom{z}{k_{1}}$
$\displaystyle\qquad{}=\binom{z}{k-A}\sum_{l_{1}=l-C}^{l}\binom{l_{1}}{l-C}\binom{y+z-k-l+l_{1}+A}{A}\binom{x}{l_{1}}\binom{y}{l-l_{1}}$
(2.3)

for all $A\leq k$ and $C\leq l$. This is restated as (A.1), and then proved,
in Appendix A.

For $C=0$ the identity (2.3) reduces to

$\displaystyle\sum_{k_{1}=k-A}^{k}\binom{k_{1}}{k-A}\binom{y}{k-k_{1}}\binom{z}{k_{1}}=\binom{z}{k-A}\binom{y+z-k+A}{A}.$

###  2.2 Star product

We now introduce an action $\triangleright$ of $P$ and $D$ on the space of
formal power series in variables $x_{\mu}$, where $\mu=0,1,\ldots,n$, by
formulas

$\displaystyle(P\triangleright f)(x)=-{\rm i}v_{\mu}\frac{\partial
f(x)}{\partial x_{\mu}},\qquad(D\triangleright f)(x)=x_{\mu}\frac{\partial
f(x)}{\partial x_{\mu}},$

where the constants $v_{\mu}$ are such that $v^{2}\in\\{-1,0,1\\}$ and the
Einstein summation rule is understood. We also denote $x=(x_{\mu})$ and
$\partial_{\mu}=\frac{\partial}{\partial x_{\mu}}$.

A star product $*$ is then defined as

$\displaystyle f*g=m\big{(}\mathcal{F}_{{\rm
GZ},u}^{-1}(\triangleright\otimes\triangleright)(f\otimes g)\big{)}$

for all formal power series $f$, $g$ in $x_{\mu}$ [30]. In particular, for
$f={\rm e}^{{\rm i}kx}$ and $g={\rm e}^{{\rm i}qx}$,

$\displaystyle{\rm e}^{{\rm i}kx}*{\rm e}^{{\rm
i}qx}=m\big{(}\mathcal{F}_{{\rm
GZ},u}^{-1}(\triangleright\otimes\triangleright)\big{(}{\rm e}^{{\rm
i}kx}\otimes{\rm e}^{{\rm i}qx}\big{)}\big{)}=:{\rm
e}^{\mathcal{A}(u;k,q,x)},$ (2.4)

where

$\displaystyle kx=k_{\alpha}x_{\alpha}\qquad\mathrm{and}\qquad
qx=q_{\alpha}x_{\alpha}$

are elements of the Minkowski space-time algebra, the function $\mathcal{A}$
is implicitly defined by (2.4) and $m$ denotes the multiplication map on usual
functions. Using the actions of $P$ and $D$ on ${\rm e}^{{\rm i}kx}$, it
follows that

$\displaystyle P\triangleright{\rm e}^{{\rm i}kx}=(v\cdot k){\rm e}^{{\rm
i}kx},\qquad P\triangleright{\rm e}^{{\rm i}qx}=(v\cdot q){\rm e}^{{\rm
i}qx},$

where $(v\cdot k)=v_{\alpha}k_{\alpha}$, $(v\cdot q)=v_{\alpha}q_{\alpha}$.
For $j<l$,

$\displaystyle\left.\left(P^{j}\binom{D}{l}\triangleright{\rm e}^{{\rm
i}kx}\right)\right|_{x=0}=0,$
$\displaystyle\left.\left(P^{n}\binom{D}{n}\triangleright{\rm e}^{{\rm
i}kx}\right)\right|_{x=0}=(v\cdot k)^{n},$
$\displaystyle\left.\left(P^{n+1}\binom{D}{n}\triangleright{\rm e}^{{\rm
i}kx}\right)\right|_{x=0}=(n+1)(v\cdot k)^{n+1}.$

The following identities hold

$\displaystyle\binom{D}{n}\triangleright{\rm e}^{{\rm i}kx}=\frac{({\rm
i}kx)^{n}}{n!}{\rm e}^{{\rm i}kx},\qquad n\in\mathbf{N}_{0},$
$\displaystyle\left.\left(P^{j}\binom{D}{l}\triangleright{\rm e}^{{\rm
i}kx}\right)\right|_{x=0}=\left.\left(P^{j}\triangleright\frac{({\rm
i}kx)^{l}}{l!}{\rm e}^{{\rm
i}kx}\right)\right|_{x=0}=\binom{j}{l}(v_{\alpha}k_{\alpha})^{j}.$

Then we have

$\displaystyle\big{(}{\rm e}^{{\rm i}kx}*{\rm e}^{{\rm
i}qx}\big{)}\big{|}_{x=0}=\sum_{n=0}^{\infty}\left(\frac{u(u-1)}{\kappa^{2}}\right)^{n}(v\cdot
k)^{n}(v\cdot q)^{n}$ $\displaystyle\hphantom{\big{(}{\rm e}^{{\rm i}kx}*{\rm
e}^{{\rm
i}qx}\big{)}\big{|}_{x=0}}{}=\frac{1}{1-\frac{u(u-1)}{\kappa^{2}}(v\cdot
k)(v\cdot q)}={\rm e}^{\mathcal{A}(u;k,q,x)}\left.\right|_{x=0}.$ (2.5)

We now calculate the partial derivatives of the star product,

$\displaystyle\partial_{\mu}\big{(}{\rm e}^{{\rm i}kx}*{\rm e}^{{\rm
i}qx}\big{)}\big{|}_{x=0}=\sum_{j,l=0}^{\infty}\left(\frac{1}{\kappa}\right)^{j+l}\left\\{{\rm
i}k_{\mu}((u-1)P)^{j}\triangleright\left[\frac{(v\cdot
k)^{l}}{l!}+\frac{(v\cdot k)^{l-1}}{(l-1)!}\right]{\rm e}^{{\rm
i}kx}\right\\}$
$\displaystyle{}\times\left\\{(uP)^{l}\triangleright\frac{(v\cdot
q)^{j}}{j!}{\rm e}^{{\rm i}qx}\right\\}\bigg{|}_{x=0}$
$\displaystyle+\sum_{j,l=0}^{\infty}\left(\frac{1}{\kappa}\right)^{j+l}\left.\left\\{((u-1)P)^{j}\triangleright\frac{(v\cdot
k)^{l}}{l!}{\rm e}^{{\rm i}kx}\right\\}\left\\{{\rm
i}q_{\mu}(vP)^{l}\triangleright\left[\frac{(v\cdot q)^{j}}{j!}+\frac{(v\cdot
q)^{j-1}}{(j-1)!}{\rm e}^{{\rm i}qx}\right]\right\\}\right|_{x=0}$
$\displaystyle={\rm
i}(k_{\mu}+q_{\mu})\sum_{n=0}^{\infty}\left(\frac{u(u-1)}{\kappa^{2}}\right)^{n}(v\cdot
k)^{n}(v\cdot q)^{n}+{\rm
i}(k_{\mu}+q_{\mu})\sum_{n=0}^{\infty}\left(\frac{u(u-1)}{\kappa^{2}}\right)^{n}n(v\cdot
k)^{n}$ $\displaystyle+(v\cdot q)^{n}+{\rm
i}\left[k_{\mu}\frac{u}{\kappa}(v\cdot q)+q_{\mu}\frac{(u-1)}{\kappa}(v\cdot
k)\right]\sum_{n=0}^{\infty}\left(\frac{u(u-1)}{\kappa^{2}}\right)^{n}(n+1)(v\cdot
k)^{n}(v\cdot q)^{n}$ $\displaystyle={\rm
i}\mathcal{D}_{\mu}(k,q)\frac{1}{1-\frac{u(u-1)}{\kappa^{2}}(v\cdot k)(v\cdot
q)}$

and

$\displaystyle\partial_{\mu}\big{(}{\rm e}^{{\rm i}kx}*{\rm e}^{{\rm
i}qx}\big{)}\big{|}_{x=0}={\rm i}\left(k_{\mu}\left(1+\frac{u}{\kappa}(v\cdot
q)\right)+q_{\mu}\left(1+\frac{u-1}{\kappa}(v\cdot k)\right)\right)$
$\displaystyle\hphantom{\partial_{\mu}\big{(}{\rm e}^{{\rm i}kx}*{\rm e}^{{\rm
i}qx}\big{)}\big{|}_{x=0}=}{}\times\sum_{n=0}^{\infty}\left(\frac{u(u-1)}{\kappa^{2}}\right)^{n}(n+1)(v\cdot
k)^{n}(v\cdot q)^{n}$ $\displaystyle\hphantom{\partial_{\mu}\big{(}{\rm
e}^{{\rm i}kx}*{\rm e}^{{\rm i}qx}\big{)}\big{|}_{x=0}}{}={\rm
i}\mathcal{D}_{\mu}(k,q)\frac{1}{1-\frac{u(u-1)}{\kappa^{2}}(v\cdot k)(v\cdot
q)}={\rm i}\mathcal{D}_{\mu}(k,q)\big{(}{\rm e}^{{\rm i}kx}*{\rm e}^{{\rm
i}qx}\big{)}\big{|}_{x=0}.$

Note that

$\displaystyle{\rm
i}\mathcal{D}_{\mu}(k,q)=\left.\left(\frac{\partial\mathcal{A}(u;k,q,x)}{\partial
x_{\mu}}\right)\right|_{x=0}.$

It follows that

$\displaystyle\mathcal{D}_{\mu}(k,q)=\frac{k_{\mu}\big{(}1+\frac{u}{\kappa}(v\cdot
q)\big{)}+q_{\mu}\big{(}1+\frac{u-1}{\kappa}(v\cdot
k)\big{)}}{1-\frac{u(u-1)}{\kappa^{2}}(v\cdot k)(v\cdot q)}.$ (2.6)

###  2.3 Twisted coproduct $\boldsymbol{\Delta(p_{\mu})}$

Let now $p_{\mu}=-{\rm i}\partial_{\mu}$ be the momentum operator. Let us
define $\Delta p_{\mu}$ by

$\displaystyle\Delta p_{\mu}=\mathcal{D}_{\mu}(p\otimes 1,1\otimes
p)=\frac{p_{\mu}\otimes\big{(}1+\frac{u}{\kappa}P\big{)}+\big{(}1+\frac{u-1}{\kappa}P\big{)}\otimes
p_{\mu}}{1\otimes 1-\frac{u(u-1)}{\kappa^{2}}P\otimes P},\qquad
P=v_{\alpha}p_{\alpha}.$ (2.7)

We want to show that $\Delta p_{\mu}$ is the deformed coproduct with respect
to the twist $\mathcal{F}_{{\rm GZ},u}$,

$\displaystyle\Delta p_{\mu}=\mathcal{F}_{{\rm
GZ},u}\Delta_{0}p_{\mu}\mathcal{F}_{{\rm GZ},u}^{-1},$ (2.8)

where

$\displaystyle\Delta_{0}p_{\mu}=p_{\mu}\otimes 1+1\otimes p_{\mu}.$ (2.9)

Using (2.9) and (2.7), we may rewrite (2.8) as

$\displaystyle\mathcal{F}_{{\rm
GZ},u}^{-1}\frac{p_{\mu}\otimes\big{(}1+\frac{u}{\kappa}P\big{)}+\big{(}1+\frac{u-1}{\kappa}P\big{)}\otimes
p_{\mu}}{1\otimes 1-\frac{u(u-1)}{\kappa^{2}}P\otimes P}=(p_{\mu}\otimes
1+1\otimes p_{\mu})\mathcal{F}_{{\rm GZ},u}^{-1}$

and, after multiplying from the right by the denominator $1\otimes
1-\frac{u(u-1)}{\kappa^{2}}P\otimes P$, as

$\displaystyle\mathcal{F}_{{\rm
GZ},u}^{-1}\left(p_{\mu}\otimes\left(1+\frac{u}{\kappa}P\right)+\left(1+\frac{u-1}{\kappa}P\right)\otimes
p_{\mu}\right)$ $\displaystyle\qquad{}=(p_{\mu}\otimes 1+1\otimes
p_{\mu})\mathcal{F}_{{\rm GZ},u}^{-1}\left(1\otimes
1-\frac{u(u-1)}{\kappa^{2}}P\otimes P\right).$ (2.10)

We shall show the equality in (2.10) by splitting it into a sum of two
equalities, (2.11) and (2.12), which are then separately proved.
Descriptively, (2.11) involves all those summands in expanded (2.10) where, in
one of the factors, $p_{\mu}$ is at the left side from the tensor product,

$\displaystyle\mathcal{F}_{{\rm
GZ},u}^{-1}\left(p_{\mu}\otimes\left(1+\frac{u}{\kappa}P\right)\right)=(p_{\mu}\otimes
1)\mathcal{F}_{{\rm GZ},u}^{-1}\left(1\otimes
1-\frac{u(u-1)}{\kappa^{2}}P\otimes P\right).$ (2.11)

To prove this equality, we first observe that by induction the equality
$[P,D]=P$ implies the commutation relation

$\displaystyle p_{\mu}\binom{D}{k}=\binom{D+1}{k}p_{\mu}.$

Hence

$\displaystyle P\binom{D}{k}=\binom{D+1}{k}P,$

i.e.,

$\displaystyle\binom{D}{k}P=P\binom{D-1}{k}.$

We calculate the left-hand side of (2.11) as

$\displaystyle\mathcal{F}_{{\rm
GZ},u}^{-1}\left(p_{\mu}\otimes\left(1+\frac{u}{\kappa}P\right)\right)$
$\displaystyle\qquad{}=\sum_{k,l=0}^{\infty}\left(\left(\frac{u-1}{\kappa}P\right)^{k}\binom{D}{l}\otimes\left(\frac{uP}{\kappa}\right)^{l}\binom{D}{k}\right)\left(p_{\mu}\otimes
1+\frac{u}{\kappa}p_{\mu}\otimes P\right)$
$\displaystyle\qquad{}=\sum_{k,l=0}^{\infty}\left(\left(\frac{(u-1)P}{\kappa}\right)^{k}\binom{D}{l}\otimes\left(\frac{uP}{\kappa}\right)^{l}\binom{D}{k}\right.$
$\displaystyle\left.\qquad\quad{}+\left(\frac{(u-1)P}{\kappa}\right)^{k}\binom{D}{l}\otimes\left(\frac{uP}{\kappa}\right)^{l+1}\binom{D-1}{k}\right)(p_{\mu}\otimes
1)$

and the right-hand side of (2.11) as

$\displaystyle(p_{\mu}\otimes 1)\mathcal{F}_{{\rm GZ},u}^{-1}\left(1\otimes
1-\frac{u(u-1)}{\kappa^{2}}P\otimes P\right)$
$\displaystyle\qquad{}=\sum_{k,l=0}^{\infty}\left(\left(\frac{(u-1)P}{\kappa}\right)^{k}\binom{D+1}{l}\otimes\left(\frac{uP}{\kappa}\right)^{l}\binom{D}{k}\right.$
$\displaystyle\left.\qquad\quad{}-\left(\frac{(u-1)P}{\kappa}\right)^{k+1}\binom{D}{l}\otimes\left(\frac{uP}{\kappa}\right)^{l+1}\binom{D-1}{k}\right)(p_{\mu}\otimes
1).$

Comparing the terms of type $P^{k}\otimes P^{l}$ for all $k$ and $l$, we find

$\displaystyle\left(\frac{(u-1)P}{\kappa}\right)^{k}\binom{D}{l}\otimes\left(\frac{uP}{\kappa}\right)^{l}\binom{D}{k}+\left(\frac{(u-1)P}{\kappa}\right)^{k}l\binom{D}{l-1}\otimes\left(\frac{uP}{\kappa}\right)^{l}\binom{D}{k}\frac{D-k}{D}$
$\displaystyle\qquad{}=\left(\frac{(u-1)P}{\kappa}\right)^{k}(D+1)\binom{D}{l-1}\otimes\left(\frac{uP}{\kappa}\right)^{l}\binom{D}{k}$
$\displaystyle\qquad\quad{}-\left(\frac{(u-1)P}{\kappa}\right)^{k}l\binom{D}{l-1}\otimes\left(\frac{uP}{\kappa}\right)^{l}\frac{k}{D}\binom{D}{k}$

and

$\displaystyle\left(\frac{(u-1)P}{\kappa}\right)^{k}\binom{D}{l}\otimes\left(\frac{uP}{\kappa}\right)^{l}\binom{D}{k}$
$\displaystyle\qquad{}=\left(\frac{(u-1)P}{\kappa}\right)^{k}l\binom{D}{l-1}\otimes\left(\frac{uP}{\kappa}\right)^{l}\binom{D}{k}$
$\displaystyle\qquad\quad{}+\left(\frac{(u-1)P}{\kappa}\right)^{k}\\!(D+1)\binom{D}{l-1}\otimes\left(\frac{uP}{\kappa}\right)^{l}\binom{D}{k}$
$\displaystyle\qquad{}=\left(\frac{u-1)P}{\kappa}\right)^{k}\binom{D}{l-1}(D-l+1)\otimes\left(\frac{uP}{\kappa}\right)^{l}\binom{D}{k}$
$\displaystyle\qquad{}=\left(\frac{(u-1)P}{\kappa}\right)^{k}\binom{D}{l}\otimes\left(\frac{uP}{\kappa}\right)^{l}\binom{D}{k},$

which proves (2.11).

Analogously, we prove the equality of the remaining summands in (2.10),

$\displaystyle\mathcal{F}_{{\rm
GZ},u}^{-1}\left(\left(1+(u-1)\frac{1}{\kappa}P\right)\otimes
p_{\mu}\right)=(1\otimes p_{\mu})\mathcal{F}_{{\rm GZ},u}^{-1}\left(1\otimes
1-\frac{u(u-1)}{\kappa^{2}}P\otimes P\right).$ (2.12)

Now (2.11) and (2.12) add to (2.10). Hence this proves (2.8), that is

$\displaystyle\Delta p_{\mu}=\mathcal{F}_{{\rm
GZ},u}\Delta_{0}(p_{\mu})\mathcal{F}_{{\rm GZ},u}^{-1}.$

The coproduct $\Delta p_{\mu}$ satisfies the coassociativity condition

$\displaystyle(\Delta\otimes 1)\Delta p_{\mu}=(1\otimes\Delta)\Delta p_{\mu}.$

Equation (2.8) can be rewritten as

$\displaystyle\Delta_{0}p_{\mu}\mathcal{F}_{{\rm GZ},u}^{-1}=\mathcal{F}_{{\rm
GZ},u}^{-1}\Delta p_{\mu}.$ (2.13)

This enables us to obtain explicit formulas for the derivatives of the star
product and for the star product from Section 2.2. Namely, for the partial
derivatives of the star product, we compute

$\displaystyle\partial_{\mu}\big{(}{\rm e}^{{\rm i}kx}*{\rm e}^{{\rm
i}qx}\big{)}\overset{(\ref{eq:expstarexp})}{=}\partial_{\mu}m\mathcal{F}_{{\rm
GZ},u}^{-1}\big{(}{\rm e}^{{\rm i}kx}\otimes{\rm e}^{{\rm
i}qx}\big{)}=m(\partial_{\mu}\otimes
1+1\otimes\partial_{\mu})\mathcal{F}_{{\rm GZ},u}^{-1}\big{(}{\rm e}^{{\rm
i}kx}\otimes{\rm e}^{{\rm i}qx}\big{)}$
$\displaystyle\hphantom{\partial_{\mu}\big{(}{\rm e}^{{\rm i}kx}*{\rm e}^{{\rm
i}qx}\big{)}}{}=m{\rm i}\Delta_{0}(p_{\mu})\mathcal{F}_{{\rm
GZ},u}^{-1}\big{(}{\rm e}^{{\rm i}kx}\otimes{\rm e}^{{\rm
i}qx}\big{)}\overset{(\ref{eq:D0pFeqFDp})}{=}{\rm i}m\mathcal{F}_{{\rm
GZ},u}^{-1}\Delta(p_{\mu})\big{(}{\rm e}^{{\rm i}kx}\otimes{\rm e}^{{\rm
i}qx}\big{)}$ $\displaystyle\hphantom{\partial_{\mu}\big{(}{\rm e}^{{\rm
i}kx}*{\rm e}^{{\rm i}qx}\big{)}}{}\overset{(\ref{eq:Deltapmu})}{=}m{\rm
i}\mathcal{F}_{{\rm GZ},u}^{-1}\mathcal{D}_{\mu}(p\otimes 1,1\otimes
p)\big{(}{\rm e}^{{\rm i}kx}\otimes{\rm e}^{{\rm i}qx}\big{)}={\rm
i}m\mathcal{F}_{{\rm GZ},u}^{-1}\mathcal{D}_{\mu}(k,q)\big{(}{\rm e}^{{\rm
i}kx}\otimes{\rm e}^{{\rm i}qx}\big{)}$
$\displaystyle\hphantom{\partial_{\mu}\big{(}{\rm e}^{{\rm i}kx}*{\rm e}^{{\rm
i}qx}\big{)}}{}={\rm i}\mathcal{D}_{\mu}(k,q)m\mathcal{F}_{{\rm
GZ},u}^{-1}\big{(}{\rm e}^{{\rm i}kx}\otimes{\rm e}^{{\rm
i}qx}\big{)}\overset{(\ref{eq:expstarexp})}{=}i\mathcal{D}_{\mu}(k,q)\big{(}{\rm
e}^{{\rm i}kx}*{\rm e}^{{\rm i}qx}\big{)},$ (2.14)

where $m$ denotes the multiplication map for usual functions. Knowing the
partial derivatives (2.14) and the initial value (2.5) of the star product at
$x=0$, we finally obtain

$\displaystyle{\rm e}^{{\rm i}kx}*{\rm e}^{{\rm i}qx}={\rm e}^{{\rm
i}\mathcal{D}_{\mu}(k,q)x^{\mu}}\frac{1}{1-\frac{u(u-1)}{\kappa^{2}}(v\cdot
k)(v\cdot q)},$ (2.15)

where $\mathcal{D}_{\mu}(k,q)$ is given in (2.6). This star product is
associative in agreement with the fact that twists $\mathcal{F}^{-1}_{{\rm
GZ},u}$ satisfy the 2-cocycle condition (2.2).

###  2.4 Noncommutative coordinates and realizations

Here we introduce noncommutative coordinates $\hat{x}_{\mu}$, the commutation
relations among them and their realizations. We use realizations of elements
of noncommutative algebras via a Heisenberg algebra with generators $x_{\mu}$,
$p_{\nu}$, $[x_{\mu},x_{\nu}]=0$, $[p_{\mu},p_{\nu}]=0$,
$[x_{\mu},p_{\nu}]=-{\rm i}\delta_{\mu,\nu}$. The following expression defines
noncommutative coordinates $\hat{x}_{\mu}$ [30],

$\displaystyle\hat{x}_{\mu}=m\big{(}\mathcal{F}_{{\rm
GZ},u}^{-1}(\triangleright\otimes 1)(x_{\mu}\otimes
1)\big{)}=x_{\mu}\left(1+\frac{u}{\kappa}P\right)+\frac{{\rm
i}}{\kappa}v_{\mu}(1-u)\left(1+\frac{u}{\kappa}P\right)D$
$\displaystyle\hphantom{\hat{x}_{\mu}}{}=\left(x_{\mu}+(1-u)\frac{{\rm
i}}{\kappa}v_{\mu}D\right)\left(1+\frac{u}{\kappa}P\right)+\frac{u(1-u)}{\kappa^{2}}{\rm
i}v_{\mu}P.$

Noncommutative coordinates $\hat{x}_{\mu}$ satisfy a $\kappa$-deformed
Heisenberg algebra that corresponds to the $\kappa$-Minkowski space [11, 15,
16, 24, 25, 29, 32, 34]

$\displaystyle[\hat{x}_{\mu},\hat{x}_{\nu}]=\frac{{\rm
i}}{\kappa}(v_{\mu}\hat{x}_{\nu}-v_{\nu}\hat{x}_{\mu}),$
$\displaystyle[p_{\mu},\hat{x}_{\nu}]=\left(-{\rm
i}\delta_{\mu,\nu}+\frac{{\rm
i}}{\kappa}v_{\nu}(1-u)p_{\mu}\right)\left(1+\frac{u}{\kappa}P\right).$

In the case $u=0$,

$\displaystyle\hat{x}_{\mu}=x_{\mu}+\frac{{\rm i}}{\kappa}v_{\mu}D.$

In the case $u=1$,

$\displaystyle\hat{x}_{\mu}=x_{\mu}\left(1+\frac{u}{\kappa}P\right).$

Using this realization of $\hat{x}_{\mu}$ and the method from [30] we obtain
the same star product (2.15).

##  3 Interpolation between Jordanian twists
induced by a 1-cochain

Another construction for a generalized Jordanian twist is possible [1]. This
twist, here denoted $\mathcal{F}_{R,u}$, has been introduced as a product of
three exponential factors,

$\displaystyle\mathcal{F}_{R,u}=\exp\left(\frac{u}{\kappa}(PD\otimes
1+1\otimes
PD)\right)\exp\left(\\!{-}\ln\left(1-\frac{1}{\kappa}P\right)\otimes
D\right)\exp\left(\Delta_{0}\left(\\!{-}\frac{u}{\kappa}PD\right)\right),\\!\\!\\!\\!$
(3.1)

where $u$ is a real parameter, $u\in\mathbb{R}$. The symbol $R$ in the
subscript refers to the position of the dilatation generator in the formula,
namely it is on the right with respect to $P$. The classical $r$-matrix
corresponding to twists $F_{{\rm GZ},u}$ (2.1) and $F_{R,u}$ (3.1) does not
depend on the parameter $u$, namely

$\displaystyle r=\frac{1}{\kappa}(D\otimes P-P\otimes D).$

The above form (3.1) of the family of twists $\mathcal{F}_{R,u}$ is obtained
from a simple Jordanian twist $\mathcal{F}_{0}$, using a transformation by a
1-cochain. Namely, according to Drinfeld [7, 27], if $\mathcal{F}$ is any
normalized Drinfeld twist and $\omega_{R}$ is any element in the Hopf algebra
satisfying the normalization $\epsilon(\omega_{R})=1$, then the formula
$\mathcal{F}_{\omega}:=\big{(}\omega^{-1}\otimes\omega^{-1}\big{)}\mathcal{F}\Delta(\omega)$
defines a normalized Drinfeld twist again (that is, the 2-cocycle and
counitality conditions are satisfied again). In particular, if
$\mathcal{F}=1\otimes 1$ we get a 2-coboundary twist
$\big{(}\omega^{-1}\otimes\omega^{-1}\big{)}\Delta(\omega)$. If the two
twists, $\mathcal{F}$ and $\mathcal{F}_{\omega}$, transform one into another
by a 1-cochain, we say that they are cohomologous in the sense of nonabelian
cohomology [27]. In this case, twisted Hopf algebras $H_{\mathcal{F}}$ and
$H_{\mathcal{F}_{\omega}}$ are isomorphic [27] and, for each $H$-module
algebra $M$, the corresponding twistings $M_{\mathcal{F}}$ and
$M_{\mathcal{F}_{\omega}}$ are also mutually isomorphic as algebras. If
$\omega$ is group like, $\mathcal{F}_{\omega}$ is evidently obtained from
$\mathcal{F}$ by an inner automorphism. Regarding that cohomologous twists
give isomorphic mathematical objects, one sometimes thinks of these twists as
gauge equivalent.

If $\mathcal{F}=\mathcal{F}_{0}$ is a simple Jordanian twist, and
$\omega=\omega_{R}=\exp\big{(}{-}\frac{u}{\kappa}PD\big{)}$, we obtain the
twist
$\mathcal{F}_{R,u}=\mathcal{F}_{\omega_{R}}=\big{(}\omega^{-1}_{R}\otimes\omega^{-1}_{R}\big{)}\mathcal{F}_{0}\Delta(\omega_{R})$,
see [1]. This also shows that, for any $u$, twist $\mathcal{F}_{R,u}$
satisfies the 2-cocycle and normalization conditions. Regarding that $u$
appeared by gauge transforming $\mathcal{F}_{0}$, we can view $u$ as a gauge
parameter (the reader should not confuse $u$ with a spectral parameter
involved in some other Jordanian deformations). For $u=0$, twist
$\mathcal{F}_{R,u}$ simplifies to $\mathcal{F}_{0}$ and for $u=1$ to
$\mathcal{F}_{1}$.

###  3.1 Hopf algebra

The coalgebra sector of the Hopf algebra $\mathcal{H}^{\mathcal{F}_{R,u}}$ for
the deformation with $\mathcal{F}_{R,u}$ is given by the formulas

$\displaystyle\Delta^{\mathcal{F}_{R,u}}p_{\mu}=\frac{p_{\mu}\otimes\big{(}1+u\frac{1}{\kappa}P\big{)}+\big{(}1-(1-u)\frac{1}{\kappa}P\big{)}\otimes
p_{\mu}}{1\otimes 1+u(1-u)\left(\frac{1}{\kappa}\right)^{2}P\otimes P},$
$\displaystyle\Delta^{\mathcal{F}_{R,u}}D=\left(1\otimes
1+\frac{u(1-u)}{\kappa^{2}}P\otimes
P\right)\left(D\otimes\frac{1}{1+\frac{u}{\kappa}P}+\frac{1}{1-\frac{1-u}{\kappa}P}\otimes
D\right),$ $\displaystyle
S^{\mathcal{F}_{R,u}}(p_{\mu})=-\frac{p_{\mu}}{1-(1-2u)\frac{1}{\kappa}P},$
$\displaystyle
S^{\mathcal{F}_{R,u}}(D)=-\left(1-\frac{1-u}{\kappa}P\right)D\left(\frac{1-\frac{1-2u}{\kappa}P}{1-\frac{1-u}{\kappa}P}\right).$

A similar analysis as in Section 2 for $\Delta^{\mathcal{F}_{{\rm
GZ},u}}p_{\mu}$ leads to the conclusion that $\Delta^{\mathcal{F}_{{\rm
GZ},u}}D=\Delta^{\mathcal{F}_{R,u}}D$.

###  3.2 Noncommutative coordinates and realizations

In general, we consider realizations of the form

$\displaystyle\hat{x}_{\mu}=x_{\alpha}\varphi_{\alpha\mu}(p)+\chi(p).$

We can obtain the appropriate realization via the twist as follows

$\displaystyle\hat{x}_{\mu}=m\big{(}\mathcal{F}_{R,u}^{-1}(\triangleright\otimes
1)(x_{\mu}\otimes 1)\big{)}=\left(x_{\mu}+\frac{{\rm
i}}{\kappa}v_{\mu}(1-u)D\right)\left(1+\frac{u}{\kappa}P\right)+u(1-u)\frac{{\rm
i}}{\kappa^{2}}v_{\mu}P.$

###  3.3 Star product

Using the above realization of $\hat{x}_{\mu}$ [30], we get

$\displaystyle{\rm e}^{{\rm i}kx}*{\rm e}^{{\rm i}qx}={\rm e}^{{\rm
i}\mathcal{D}_{\mu}(u;k,q)x_{\mu}+{\rm i}\mathcal{G}(u;k,q)}={\rm e}^{{\rm
i}\mathcal{D}_{\mu}(u;k,q)x_{\mu}}\frac{1}{1+\frac{u(1-u)}{\kappa^{2}}(v\cdot
k)(v\cdot q)},$

where $k$ and $q$ belong to the $n$-dimensional Minkowski spacetime
$\mathcal{M}_{1,n-1}$ and where

$\displaystyle\mathcal{D}_{\mu}(u;k,q)=\frac{k_{\mu}\big{(}1+\frac{u}{\kappa}(v\cdot
q)\big{)}+\big{(}1-\frac{1-u}{\kappa}(v\cdot
k)\big{)}q_{\mu}}{1+\frac{u(1-u)}{\kappa^{2}}(v\cdot k)(v\cdot q)}$

as in equation (2.6), and finally

$\displaystyle\mathcal{G}(u;k,q)={\rm
i}\ln\left(1+\frac{u(1-u)}{\kappa^{2}}(v\cdot k)(v\cdot q)\right).$

######  Remark 1.

Note that the corresponding quantum $R$-matrix is given by

$\displaystyle\mathcal{R}_{R,u}=\mathcal{F}_{R,u}^{21}\mathcal{F}_{R,u}^{-1}=\exp(u(PD\otimes
1+1\otimes PD)\mathcal{R}_{0}\exp(-u(PD\otimes 1+1\otimes PD)),$

where

$\displaystyle\mathcal{R}_{0}=\exp\left(-D\otimes\ln\left(1-\frac{P}{\kappa}\right)\right)\exp\left(\ln\left(1-\frac{P}{\kappa}\right)\otimes
D\right)$
$\displaystyle\hphantom{\mathcal{R}_{0}}{}=\sum_{k,l=0}^{\infty}\binom{-D}{l}\left(\frac{-P}{\kappa}\right)^{k}\otimes\left(\frac{-P}{\kappa}\right)^{l}\binom{D}{k}.$

Both twists, $\mathcal{F}_{{\rm GZ},u}^{-1}$ and $\mathcal{F}_{R,u}^{-1}$,
lead to the same Hopf algebra, the same realizations of noncommutative
coordinates $\hat{x}_{\mu}$ and likewise for the star product ${\rm e}^{{\rm
i}kx}*{\rm e}^{{\rm i}qx}$. This suggests that there must be a close relation
between the two twists, $\mathcal{F}_{{\rm GZ},u}^{-1}$ and
$\mathcal{F}_{R,u}^{-1}$. In the next section, we present a proof that indeed
$\mathcal{F}_{{\rm GZ},u}^{-1}=\mathcal{F}_{R,u}^{-1}$.

##  4 Proofs of the equality of the two twists

###  4.1 Differentiation with respect to parameter $\boldsymbol{u}$

Differentiating $\mathcal{F}_{R,u}^{-1}$ from equation (3.1) with respect to
the parameter $u$ gives

$\displaystyle\kappa\frac{{\rm d}\mathcal{F}_{R,u}^{-1}}{{\rm d}u}=(P\otimes
D+D\otimes P)\mathcal{F}_{R,u}^{-1}+\big{[}PD\otimes 1+1\otimes
PD,\mathcal{F}_{R,u}^{-1}\big{]}.$ (4.1)

Differentiating $\mathcal{F}_{{\rm GZ},u}^{-1}$ from equation (2.1) with
respect to $u$ gives

$\displaystyle\kappa\frac{{\rm d}\mathcal{F}_{{\rm GZ},u}^{-1}}{{\rm
d}u}=(P\otimes D+D\otimes P)\mathcal{F}_{{\rm GZ},u}^{-1}$
$\displaystyle\hphantom{\kappa\frac{{\rm d}\mathcal{F}_{{\rm GZ},u}^{-1}}{{\rm
d}u}=}{}+(P\otimes 1-1\otimes
P)\sum_{k,l=0}^{\infty}\frac{-k+l}{\kappa^{k+l}}(u-1)^{k}P^{k}\binom{D}{l}\otimes(uP)^{l}\binom{D}{k}.$
(4.2)

Using the commutation relations

$\displaystyle\left[PD,P^{k}\binom{D}{l}\right]=(l-k)P^{k+1}\binom{D}{l}$

and

$\displaystyle\left[PD,P^{l}\binom{D}{k}\right]=(k-l)P^{l+1}\binom{D}{k},$

we find that the right-hand sides of (4.1) and of (4.2) agree,

$\displaystyle{\rm r.h.s.}=(P\otimes D+D\otimes P)\mathcal{F}_{{\rm
GZ},u}^{-1}+\big{[}PD\otimes 1+1\otimes PD,\mathcal{F}_{{\rm
GZ},u}^{-1}\big{]}.$

This shows that $\mathcal{F}_{R,u}^{-1}$ and $\mathcal{F}_{{\rm GZ},u}^{-1}$
as functions of the parameter $u$ satisfy the same ordinary differential
equation, while the initial conditions agree. Indeed, at $u=0$,

$\displaystyle\mathcal{F}_{R,u=0}^{-1}=\mathcal{F}_{0}^{-1}=\mathcal{F}_{{\rm
GZ},u=0}^{-1}.$

Therefore $\mathcal{F}_{R,u}^{-1}\equiv\mathcal{F}_{{\rm GZ},u}^{-1}$.

###  4.2 Proof of the equality of the two twists

In the following proposition we state the conditions under which the two
twists are equal, along with a simple proof.

######  Proposition 4.1.

Let $\mathcal{P}$ be the Poincaré Weyl algebra generated with momenta
$p_{\mu}$, Lorentz generators $M_{\mu\nu}$ and dilatation $D$. Two twists
$\mathcal{F}_{1}\in\mathcal{U}(\mathcal{P})\otimes\mathcal{U}(\mathcal{P})$
and
$\mathcal{F}_{2}\in\mathcal{U}(\mathcal{P})\otimes\mathcal{U}(\mathcal{P})$
are identical if all the star products are identical, i.e., for all $f$ and
$g$ in the Minkowski space time algebra,

$\displaystyle
f*g=m\big{(}\mathcal{F}_{1}^{-1}(\triangleright\otimes\triangleright)(f\otimes
g)\big{)}=m\big{(}\mathcal{F}_{2}^{-1}(\triangleright\otimes\triangleright)(f\otimes
g)\big{)}.$

######  Proof 4.2.

If all star products are the same, $\mathcal{F}_{1}^{-1}$ and
$\mathcal{F}_{2}^{-1}$ could differ by an element in the right ideal
$\mathcal{J}_{0}$ generated by the elements $(x_{\mu}\otimes 1-1\otimes
x_{\mu})$ for all $\mu$ [14, 17, 18]. However,
$\mathcal{J}_{0}\cap\mathcal{U}(\mathcal{P})\otimes\mathcal{U}(\mathcal{P})=0$,
hence $\mathcal{F}_{1}=\mathcal{F}_{2}$.

Since we already proved that the twists $\mathcal{F}_{R,u}$ and
$\mathcal{F}_{{\rm GZ},u}$ give the same star products ${\rm e}^{{\rm
i}kx}*{\rm e}^{{\rm i}qx}$, the twists $\mathcal{F}_{R,u}$ and
$\mathcal{F}_{{\rm GZ},u}$ must be identical. Moreover, we have proved that
the noncommutative coordinates $\hat{x}_{\mu}$ and twisted coproducts $\Delta
p_{\mu}$ and $\Delta D$ from both twists are identical. Since
$\mathcal{F}_{R,u}$ satisfies the normalization and 2-cocycle conditions,
$\mathcal{F}_{{\rm GZ},u}$ also satisfies them.

##  5 Conclusion

We have constructed a 1-parameter family $\mathcal{F}_{{\rm GZ},u}$ (2.1) of
Jordanian twists that interpolates between the simple Jordanian twists
$\mathcal{F}_{0}$ and $\mathcal{F}_{1}$ defined in equation (1.1). We
explicitly proved that $\mathcal{F}_{{\rm GZ},u}^{-1}$ satisfies the 2-cocycle
condition (2.2). For $u=\frac{1}{2}$, $\mathcal{F}_{{\rm GZ},u=\frac{1}{2}}$
coincides with $\mathcal{F}_{{\rm GZ}}$ [10]. We have calculated the
corresponding star product ${\rm e}^{{\rm i}kx}*{\rm e}^{{\rm i}qx}$ (2.15)
and the corresponding deformed Hopf algebra structure. In Section 3, we have
presented another interpolation between Jordanian twists cohomologous to
$\mathcal{F}_{0}$ via a 1-cochain depending on $u$ [1]. It is pointed out that
$\mathcal{F}_{{\rm GZ},u}^{-1}$ and $\mathcal{F}_{R,u}^{-1}$ generate the same
star product and the same deformed Hopf algebra. In Section 4, a new result is
presented that $\mathcal{F}^{-1}_{{\rm GZ},u}=\mathcal{F}^{-1}_{R,u}$,
implying that $\mathcal{F}_{{\rm GZ},u}$ can be written in the form of a
product of three exponential factors. Twist $\mathcal{F}_{R,u}$ automatically
satisfies the 2-cocycle condition as it is obtained from a simple Jordanian
twist by twisting by a 1-cochain [27]. We note that for the twist
$\mathcal{F}^{-1}_{{\rm GZ}}$ [10], the star product, an explicit form of the
twist $\mathcal{F}_{{\rm GZ}}$ and the deformed Hopf algebra structure, were
not known in the literature so far. Jordanian twists have been of interest in
the recent literature [6, 13, 22, 28, 33, 36]. We note that our results could
be useful in future applications of Jordanian twists.

##  Appendix A Appendix

######  Lemma A.1.

If $x$, $y$, $z$ are mutually commuting variables, and $k$, $l$, $A$, $C$ with
$A\leq k$, $C\leq l$ nonnegative integers, then

$\displaystyle\binom{x}{l-C}\sum_{k_{1}=k-A}^{k}\binom{k_{1}}{k-A}\binom{z}{k_{1}}\binom{x+y-k-l+k_{1}+C}{C}\binom{y}{k-k_{1}}$
$\displaystyle\qquad{}=\binom{z}{k-A}\sum_{l_{1}=l-C}^{l}\binom{l_{1}}{l-C}\binom{x}{l_{1}}\binom{y+z-k-l+l_{1}+A}{A}\binom{y}{l-l_{1}}.$
(A.1)

######  Proof A.2.

To make the proof more transparent, we make a change of summation indices

$\displaystyle i=k_{1}-k+A,\qquad j=l_{1}-l+C,$

hence $k_{1}=i+k-A$ and $l_{1}=j+l-C$, to restate equation (A.1) as

$\displaystyle\binom{x}{l-C}\sum_{i=0}^{A}\binom{i+k-A}{k-A}\binom{z}{i+k-A}\binom{x+y-l+i-A+C}{C}\binom{y}{A-i}$
$\displaystyle\qquad{}=\binom{z}{k-A}\sum_{j=0}^{C}\binom{j+l-C}{l-C}\binom{x}{j+l-C}\binom{y+z-l+j-C+A}{A}\binom{y}{C-j}.$
(A.2)

We remind the reader of the simple identity

$\displaystyle\binom{r}{s}\binom{w}{r}=\binom{w}{s}\binom{w-s}{r-s},$

which we apply in (A.2) for $w=z$ on the left and for $w=x$ on the right, to
obtain an equivalent statement,

$\displaystyle\binom{x}{l-C}\binom{z}{k-A}\sum_{i=0}^{A}\binom{z-k+A}{i}\binom{x+y-l+i-A+C}{C}\binom{y}{A-i}$
$\displaystyle\qquad{}=\binom{z}{k-A}\binom{x}{l-C}\sum_{j=0}^{C}\binom{x}{j+l-C}\binom{y+z-k+j-C+A}{A}\binom{y}{C-j}.$
(A.3)

We expand

$\displaystyle\binom{x+y-l+i-A+C}{C}=\sum_{j=0}^{C}\binom{x-l+C}{j}\binom{y+i-A}{C-j}$

on the left-hand side, and

$\displaystyle\binom{y+z-k+j-C+A}{A}=\sum_{i=0}^{A}\binom{z-k+A}{i}\binom{y+j-C}{A-i}$

on the right-hand side of (A.3). Now both sides involve double summation over
$i$ and $j$. For each fixed pair $(i,j)$, compare the corresponding summands
on the two sides. The factors involving $x$ and $z$ are identical on both
sides. It remains to check that the factors involving $y$ agree. Indeed, by
definition,

$\displaystyle\binom{y+i-A}{C-j}\binom{y}{A-i}=\frac{y(y-1)\cdots(y+i-A+j-C+1)}{(C-j)!(A-i)!}=\binom{y+j-C}{A-i}\binom{y}{C-j}.\\!\\!\\!\\!$
∎

### Acknowledgements

We thank Anna Pachoł for useful discussions. Z.Š. has been partly supported by
the Croatian Science Foundation under the Project “New Geometries for Gravity
and Spacetime” (IP-2018-01-7615) and by the grant 18-00496S of the Czech
Science Foundation.

## References

  * [1] Borowiec A., Meljanac D., Meljanac S., Pachoł A., Interpolations between Jordanian twists induced by coboundary twists, SIGMA 15 (2019), 054, 22 pages, arXiv:1812.05535.
  * [2] Borowiec A., Pachoł A., $\kappa$-Minkowski spacetime as the result of Jordanian twist deformation, Phys. Rev. D 79 (2009), 045012, 11 pages, arXiv:0812.0576.
  * [3] Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
  * [4] Coll V., Gerstenhaber M., Giaquinto A., An explicit deformation formula with noncommuting derivations, in Ring Theory 1989 (Ramat Gan and Jerusalem, 1988/1989), Israel Math. Conf. Proc., Vol. 1, Weizmann, Jerusalem, 1989, 396–403.
  * [5] Demidov E.E., Manin Yu.I., Mukhin E.E., Zhdanovich D.V., Nonstandard quantum deformations of ${\rm GL}(n)$ and constant solutions of the Yang–Baxter equation, Progr. Theoret. Phys. Suppl. 102 (1990), 203–218.
  * [6] Dimitrijević M., Jonke L., Pachoł A., Gauge theory on twisted $\kappa$-Minkowski: old problems and possible solutions, SIGMA 10 (2014), 063, 22 pages, arXiv:1403.1857.
  * [7] Drinfel’d V.G., Hopf algebras and the quantum Yang–Baxter equation, Soviet Math. Dokl. 32 (1985), 254–258.
  * [8] Etingof P., Kazhdan D., Quantization of Lie bialgebras. I, Selecta Math. (N.S.) 2 (1996), 1–41, arXiv:q-alg/9506005.
  * [9] Etingof P., Schiffmann O., Lectures on quantum groups, 2nd ed., Lectures in Math. Phys., International Press, Somerville, MA, 2002.
  * [10] Giaquinto A., Zhang J.J., Bialgebra actions, twists, and universal deformation formulas, J. Pure Appl. Algebra 128 (1998), 133–151, arXiv:hep-th/9411140.
  * [11] Govindarajan T.R., Gupta K.S., Harikumar E., Meljanac S., Meljanac D., Twisted statistics in $\kappa$-Minkowski spacetime, Phys. Rev. D 77 (2008), 105010, 6 pages, arXiv:0802.1576.
  * [12] Gräbe H.-G., Vlassov A.T., On a formula of Coll–Gerstenhaber–Giaquinto, J. Geom. Phys. 28 (1998), 129–142.
  * [13] Hoare B., van Tongeren S.J., On jordanian deformations of ${\rm AdS}_{5}$ and supergravity, J. Phys. A: Math. Theor. 49 (2016), 434006, 22 pages, arXiv:1605.03554.
  * [14] Jurić T., Kovačević D., Meljanac S., $\kappa$-deformed phase space, Hopf algebroid and twisting, SIGMA 10 (2014), 106, 18 pages, arXiv:1402.0397.
  * [15] Jurić T., Meljanac S., Pikutić D., Realizations of $\kappa$-Minkowski space, Drinfeld twists and related symmetry algebra, Eur. Phys. J. C Part. Fields 75 (2015), 528, 16 pages, arXiv:1506.04955.
  * [16] Jurić T., Meljanac S., Pikutić D., Families of vector-like deformations of relativistic quantum phase spaces, twists and symmetries, Eur. Phys. J. C Part. Fields 77 (2017), 830, 12 pages, arXiv:1709.04745.
  * [17] Jurić T., Meljanac S., Štrajn R., $\kappa$-Poincaré–Hopf algebra and Hopf algebroid structure of phase space from twist, Phys. Lett. A 377 (2013), 2472–2476, arXiv:1303.0994.
  * [18] Jurić T., Meljanac S., Štrajn R., Twists, realizations and Hopf algebroid structure of $\kappa$-deformed phase space, Internat. J. Modern Phys. A 29 (2014), 1450022, 32 pages, arXiv:1305.3088.
  * [19] Khoroshkin S.M., Pop I.I., Samsonov M.E., Stolin A.A., Tolstoy V.N., On some Lie bialgebra structures on polynomial algebras and their quantization, Comm. Math. Phys. 282 (2008), 625–662, arXiv:0706.1651.
  * [20] Khoroshkin S.M., Stolin A.A., Tolstoy V.N., Deformation of Yangian $Y({\rm sl}_{2})$, Comm. Algebra 26 (1998), 1041–1055, arXiv:q-alg/9511005.
  * [21] Khoroshkin S.M., Stolin A.A., Tolstoy V.N., $q$-power function over $q$-commuting variables and deformed $XXX$ and $XXZ$ chains, Phys. Atomic Nuclei 64 (2001), 2173–2178, arXiv:math.QA/0012207.
  * [22] Kovačević D., Meljanac S., Pachoł A., Štrajn R., Generalized Poincaré algebras, Hopf algebras and $\kappa$-Minkowski spacetime, Phys. Lett. B 711 (2012), 122–127, arXiv:1202.3305.
  * [23] Kulish P.P., Stolin A.A., Deformed Yangians and integrable models, Czechoslovak J. Phys. 47 (1997), 1207–1212, arXiv:q-alg/9708024.
  * [24] Lukierski J., Nowicki A., Ruegg H., New quantum Poincaré algebra and $\kappa$-deformed field theory, Phys. Lett. B 293 (1992), 344–352.
  * [25] Lukierski J., Ruegg H., Nowicki A., Tolstoy V.N., $q$-deformation of Poincaré algebra, Phys. Lett. B 264 (1991), 331–338.
  * [26] Lyubashenko V.V., Hopf algebras and vector-symmetries, Russian Math. Surveys 41 (1986), no. 5, 153–154.
  * [27] Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
  * [28] Meljanac D., Meljanac S., Mignemi S., Štrajn R., $\kappa$-deformed phase spaces, Jordanian twists, Lorentz–Weyl algebra, Phys. Rev. D 99 (2019), 126012, 12 pages, arXiv:1903.08679.
  * [29] Meljanac S., Krešić-Jurić S., Differential structure on $\kappa$-Minkowski space, and $\kappa$-Poincaré algebra, Internat. J. Modern Phys. A 26 (2011), 3385–3402, arXiv:1004.4647.
  * [30] Meljanac S., Meljanac D., Mercati F., Pikutić D., Noncommutative spaces and Poincaré symmetry, Phys. Lett. B 766 (2017), 181–185, arXiv:1610.06716.
  * [31] Meljanac S., Meljanac D., Pachoł A., Pikutić D., Remarks on simple interpolation between Jordanian twists, J. Phys. A: Math. Theor. 50 (2017), 265201, 11 pages, arXiv:1612.07984.
  * [32] Meljanac S., Meljanac D., Samsarov A., Stojić M., $\kappa$-deformed Snyder spacetime, Modern Phys. Lett. A 25 (2010), 579–590, arXiv:0912.5087.
  * [33] Meljanac S., Pachoł A., Pikutić D., Twisted conformal algebra related to $\kappa$-Minkowski space, Phys. Rev. D 92 (2015), 105015, 8 pages, arXiv:1509.02115.
  * [34] Meljanac S., Stojić M., New realizations of Lie algebra kappa-deformed Euclidean space, Eur. Phys. J. C Part. Fields 47 (2006), 531–539, arXiv:hep-th/0605133.
  * [35] Ogievetsky O., Hopf structures on the Borel subalgebra of ${\rm sl}(2)$, Rend. Circ. Mat. Palermo (2) Suppl. (1994), 185–199.
  * [36] Pachoł A., Vitale P., $\kappa$-Minkowski star product in any dimension from symplectic realization, J. Phys. A: Math. Theor. 48 (2015), 445202, 16 pages, arXiv:1507.03523.
  * [37] Stolin A.A., Kulish P.P., New rational solutions of Yang–Baxter equation and deformed Yangians, Czechoslovak J. Phys. 47 (1997), 123–129, arXiv:q-alg/9608011.
  * [38] Tolstoy V.N., Quantum deformations of relativistic symmetries, arXiv:0704.0081.
  * [39] Tolstoy V.N., Twisted quantum deformations of Lorentz and Poincaré algebras, arXiv:0712.3962.


# Plasma heating and particle acceleration in collisionless shocks through
astrophysical observations

M Miceli1,2 1 Università degli Studi di Palermo, Dipartimento di Fisica e
Chimica E. Segrè, Piazza del Parlamento 1, 90134 Palermo, Italy  2 INAF-
Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, 90134 Palermo,
Italy<EMAIL_ADDRESS>
###### Abstract

Supernova remnants (SNRs), the products of stellar explosions, are powerful
astrophysical laboratories, which allow us to study the physics of
collisionless shocks, thanks to their bright electromagnetic emission. Blast
wave shocks generated by supernovae (SNe) provide us with an observational
window to study extreme conditions, characterized by high Mach (and Alfvénic
Mach) numbers, together with powerful nonthermal processes. In collisionless
shocks, temperature equilibration between different species may not be reached
at the shock front. In this framework, different particle species might be
heated at different temperatures (depending on their mass) in the post-shock
medium of SNRs. SNRs are also characterized by a broadband nonthermal emission
stemming at the shock front as a result of nonthermal populations of leptons
and hadrons. These particles, known as cosmic rays, are accelerated up to
ultrarelativistic energies via diffusive shock acceleration. If SNRs lose a
significant fraction of their ram energy to accelerate cosmic rays, the shock
dynamics should be altered with respect to the adiabatic case. This shock
modification should result in an increase of the total shock compression ratio
with respect to the Rankine-Hugoniot value of 4. Here I show that the
combination of X-ray high resolution spectroscopy (to measure ion
temperatures) and moderate resolution spectroscopy (for a detailed diagnostic
of the post-shock density) can be exploited to study both the heating
mechanism and the particle acceleration in collisionless shocks. I report on
new results in the temperatures measured for different ion species in the
remnant of the SN observed in 1987 in the Large Magellanic Cloud (SN 1987A). I
also discuss evidence of shock modification recently obtained in the remnant
of SN 1006 a. D., where the shock compression ratio increases significantly as
the angle between the shock velocity and the ambient magnetic field is
reduced.

††: Plasma Phys. Control. Fusion

Keywords: Plasmas, Shock waves, Acceleration of particles

##  1 Introduction

Astrophysical shocks are ubiquitous in the universe, showing a wide variety of
Mach numbers and characteristic length-scales. Astrophysical shocks are
typically collisionless, because they expand in a rarefied medium, where
Coulomb collisions cannot provide the viscous dissipation at the shock front.
Collisionless shocks have been indeed observed at all scales, ranging from
small, “local” shocks in the solar wind ([1]) up to giant, cosmological shock
waves ([2]).

Supernova remnants (SNRs) are the leftovers of supernova (SN) explosions, and
are characterized by powerful collisionless shocks. Their birth is associated
with a sudden (less than one second) release of a huge amount of energy, which
is of the order of $E=10^{53}$ erg for core-collapse SNe and $E=10^{51}$ erg
for Type Ia SNe (associated with the explosion of a white dwarf). Both core-
collapse and Type Ia supernova explosions determine the violent ejection of
supersonic stellar fragments (ejecta), having a characteristic kinetic energy
of $10^{51}$ erg (which correspond to the energy that the Sun would radiate if
it would maintain its current luminosity for almost 16 billion years, i.e.
more than the age of the universe). The velocity of the ejecta can reach
values as high as a few $10^{4}$ km s-1 and their expansion and deceleration
in the interstellar medium (ISM) drive strong shocks, which compress and heat
the ambient medium (forward shocks) and the ejecta themselves (reverse shocks)
up to X-ray emitting temperatures.

X-ray observations of SNRs show spectacular extended nebulae, which reveal the
complex morphology of shocked ISM and ejecta, and convey a wealth of
information on many aspects of the remnant origin and evolution, and on the
shock physics. Figure 1 shows a composite image of the remnant of the
supernova exploded in 1006 A. D. (hereafter SN 1006). Soft X-rays ($0.5-1$
keV, in red) show a complex pattern, with ripples of emission revealing knots
of shocked ejecta ([3, 4]), hard X-rays ($2.5-7$ keV, in blue) are associated
with synchrotron radiation in the northeastern and southwestern limbs ([5]),
while optical Hα emission (in green) mark the northwestern shock front in
great detail. By considering characteristic values for the post-shock
temperature and density ($T=10^{7}$ K [3], and $n=0.1$ cm-3 [6, 7],
respectively), the Coulomb collision mean free path is of the order of
$10^{19}$ cm, i. e. comparable with the radius of the remnant and much larger
than the width of shock front. Since the viscous dissipation at the shock
front is localized in an extremely narrow region, it cannot be provided by
collisions and collective effects, as electromagnetic fluctuations and plasma
waves, instead, are thought to be responsible of the plasma heating. Indeed,
in this case the shock width might be of the order of $10^{8}$ cm, given that
the ion inertial length for SN 1006 is of approximately $10^{8}$ cm, and the
proton gyroradius for protons with the same speed as the shock ($5000$ km/s
[4]) in a magnetic field of $90$ $\mu$G ([8, 9]) is of approximately $5\times
10^{8}$ cm.

The sudden heating and compression within the thin shock front is followed by
a slow relaxation toward the equilibrium in the downstream flow. Therein, the
almost neutral optically thin coronal plasma is slowly ionized through
electron-ion collisions, and reaches the collisional ionization equilibrium
only after a timescale of approximately $t_{CIE}=10^{12}/n$ s (where $n$ is
the post shock density in cm-3, [10]), which corresponds to more than $10^{5}$
yr for the typical densities of SN 1006, i. e., more than ten times the age of
the remnant. It is then unsurprising that the X-ray spectra of SNRs typically
show signatures of an underionized plasma, though with some notable exception
(e. g., W49B [11, 12, 13], IC 443 [14, 15, 16] and W44 [17]).

Moreover, different particle species are expected to be heated at different
temperature in collisionless shocks, and their thermalization proceeds slowly
in the post shock medium. For example, the time-scale for electron-proton
temperature equilibration is $t_{ep}\sim
10^{4}n^{-1}(kT/\rm{1~{}keV})^{3/2}(\ln{\Lambda}/31)$ yr, which can be as well
larger (much larger for historical SNRs) than the age of a remnant. For
example, in SN 1006, $t_{ep}\sim 10^{5}$ yr. In general, therefore, the
shocked plasma in SNRs is _not_ in equilibrium.

Shock fronts of SNRs are known to be powerful particle accelerators, with
particles gaining energy by diffusively crossing back and forth the shock
front through the first-order Fermi mechanism (or diffusive shock
acceleration, DSA). Radio synchrotron radiation from ultrarelativistic (GeV)
electrons is systematically observed in the shell of SNRs [18], and X-ray
synchrotron emission from electrons with energies reaching $\sim 10^{13}$ eV
can be observed at the shock front of young SNRs ([19]), including SN 1006
(see Fig. 1). Evidence for high energy hadrons in SNRs has been also obtained
in a few cases ([20, 21, 22, 23]).

Here, I first focus on the shock heating mechanism and on the evolution of the
ion temperature in the shocked plasma in the remnant of the supernova observed
on 1987 February 23 in the Large Magellanic Cloud (hereafter SN 1987A). I will
then show the effects of the back-reaction of hadron acceleration in the shock
front of SN 1006.

The paper is organized as follows: Section 2 is dedicated to the collisionless
shock heating in SN 1987A, Sect. 3 shows the shock modification induced by
efficient particle acceleration in SN 1006, while my conclusions are
summarized in Sect. 4

Figure 1: _Left panel_ : Flux images of SN 1006 in the $0.5-1$ keV (red) and
$2.5-7$ keV (blue) bands, obtained with the _Chandra X-ray telescope_ ,
together with Balmer H$\alpha$ emission (green). _Right panels:_ close-up
views of the H$\alpha$ filament marking the shock front. In all panels North
is up and East is to the left. Physical scales (indicated by white arrows)
were obtained for a distance of $2.2$ kpc ([24]).

##  2 Collisionless shock heating in SN 1987A

In a collisionless shock propagating in a medium with different particle
species, by adopting a multi-fluid approach, one may be tempted to write down
the Rankine-Hugoniot conditions for each particle species. In this framework,
the post-shock temperature $T_{i}$ for the i-th species might be obtained from
the relation:

$kT_{i}=\frac{3}{16}m_{i}v_{sh}^{2}$ (1)

where $m_{i}$ is the particle mass for the i-th species and $v_{sh}$ is the
shock velocity. A mass-proportional post-shock temperature is expected in the
case of scattering isotropization of the incoming particles by plasma waves.
However, this approach does not consider that (partial) equilibration between
different species can also occur within the shock front (e.g., by wave-
particle interactions).

It is not straightforward to test the validity of eq. 1 by considering protons
and electrons. This is because the processes responsible for electron heating
in collisionless shocks are expected to be different from those acting for
protons and ions ([2, 25, 26]). Indeed, the electron to proton temperature
ratio in SNRs is lower than 1, though being typically larger than that
predicted by eq. 1 [27, 28, 29, 30], and showing a dependence on the shock
velocity which can be described by $T_{e}/T_{p}\propto v_{sh}^{-2}$ ([31]).

To test whether in collisionless shocks the post-shock temperature scales
linearly with the particle mass, it is then crucial to measure the temperature
of ions with different masses in the shocked plasma. Early UV observations of
SN 1006 indicated an Oxygen to proton temperature ratio larger than 1, but
lower than that predicted by eq. 1 ([32]). On the other hand, more recent
measurements have shown a mass-proportional heating for He, C and N ions in
the same remnant. Since bright emission line complexes of heavy ions are
typically observed in the X-ray band, the analysis of X-ray spectra is crucial
to extend the study of collisionless particle heating to a wider range of
masses.

The temperature of ions can be deduced by the thermal broadening of their
X-ray emission lines ([33, 34]). This is a delicate measurement, since
different effects contribute to the line broadening of an emission line
stemming in the plasma of a SNR, namely i) the intrinsic resolution of the
X-ray spectrometers, ii) the bulk Doppler motion (if approaching and receding
plasmas lie along the same line of sight); iii) the angular extension of the
X-ray emission (current high resolution spectrometers are typically slitless,
so the spectrum is convolved with the profile of the remnant emission in the
dispersion direction), and iv) thermal broadening. It is then crucial to
disentangle the contribution of thermal broadening from other effects to get
an accurate measurement of the ion temperatures.

To this end, SN 1987A is a privileged target: its X-ray emission has been
monitored regularly, thus providing a unique dataset for the evolution of the
X-ray spectra in a young SNR. Moreover, X-ray data are complemented by a
wealth of observations in an extremely wide range of wavelengths (from radio
to $\gamma-$rays), which provided us with an accurate diagnostics of the
complex interaction of the shock front with the surrounding inhomogeneous
medium ([35, 36]). The constraints derived from the multi-epochs and multi-
wavelengths observations were adopted as setup parameters to develop a
thorough 3-D magnetohydrodynamic (MHD) model, which is able to describe in
great detail both the origin and the evolution of the remnant and of its
thermal and nonthermal emission self-consistently ([37, 38, 39, 40]).

A novel approach combining data and MHD simulations was developed to analyze
the high resolution X-ray spectra of SN 1987A collected with the _Chandra_
X-ray telescope ([41]). The work was focused on the 2007 and 2011 data, i. e.,
the two deepest observations performed with the _Chandra_ High Energy
Transmission Grating (HETG) available at that time. The accurate comparison of
the actual spectra with those synthesized from the MHD model, made it possible
to isolate the contribution of thermal broadening to the observed line widths
of several emission lines, thus deriving the post shock temperature of
different heavy elements, namely of Ne, Mg, Si and Fe. Results clearly show
that, for these species, the ion to proton temperature ratio is always
significantly higher than one and is consistent with increasing linearly with
the ion mass, as predicted by eq. 1, for a wide range shock parameters ([41]).
A similar methodology was later adopted to analyze the Chandra HETG
observation taken in 2018 ([42]), confirming that, on average, the observed
line widths are well described when thermal broadening associated with a mass
proportional heating is taken into account.

I here combine the results obtained for the 2007-2011 ([41]) spectra with
those obtained for the 2018 observation ([42]). By adopting the procedure
described in [41], I compare the actual line widths with those predicted by
the MHD simulation with and without including the effects of thermal
broadening, thus determining the ion temperature. Figure 2 shows the ion to
proton temperature ratio for Ne, Mg, Si and Fe, together with the mass-
proportional relation predicted by eq. 1. The general trend clearly confirms
previous results and is in remarkable agreement with predictions of hybrid
simulations of collisionless shocks, which shows that the post shock
temperature scales linearly with the atomic mass $A$ (with simulations
performed up to A=8, [43]).

Figure 2: Ion to proton temperature ratios for Ne, Mg, Si and Fe in SN 1987A
obtained by combining the 2007 and 2011 results ([41]) with the 2018 results
([42]). The red line marks the mass-proportional trend predicted by eq. 1.

On the other hand, Fig. 2 shows that the post-shock temperature of Mg is
slightly lower than expected. The measure of the Mg temperature mainly depends
on the width of its brightest emission line, namely the Ly$\alpha$ for the
ionization state Mg XII at $\sim 8.4$ Å. A possible explanation for the
relatively low temperature of Mg is that this emission line stems well behind
the shock front, in a region where ions are approaching thermal equilibrium
with other particles and the temperature is expected to be lower than that
achieved at the shock front. Temperature equilibration between ions and
protons is expected to be reached when the time integral of the electron
density reaches $\tau\sim 5-10\times 10^{10}$ s cm-3 ([44]). On the other
hand, it must be considered that the thermalization timescale increases by
about a factor of 5 ([45]) in a very turbulent magnetic field, which is
expected to be present in the SN 1987A shocked medium [46]. This suggests that
significant temperature variations with respect to the immediate post-shock
region are expected for those ions whose emission lines originate in a plasma
with $\tau$ larger than $\sim 2-5\times 10^{11}$ s cm-3. The MHD simulations
adopted to synthesize the spectra include Coulomb collisions between electrons
and protons and follow their temporal evolution downstream, while collisions
between ions and protons are not included. However, thanks to the synthetic
spectra, it is possible to derive the value of $\tau$ in the plasma which
mainly contribute to the line emission of a given element. I here focus on the
synthetic spectra extracted from model B18.3 ([40], see also [47] for a
similar approach for the Fe K line emission) describing SN 1987A 31 years
after the explosion (corresponding to 2018 a. D.).

Figure 3 shows how the different values of temperature and ionization
parameter of the X-ray-emitting plasma in SN 1987A contribute to the line
emission. In particular, left panel of Fig. 3 shows the normalized continuum-
subtracted line flux of the Mg XII emission line as a function of the plasma
temperature and $\tau$. The figure clearly shows that the bulk of the emission
line originates in a region where $\tau$ is slightly larger than $10^{11}$ s
cm-3, i. e. where thermalization between Mg and colder protons starts to be
relevant. This is in nice agreement with the Mg showing a lower temperature
than that expected at the shock front. As a comparison, I also report the
distribution of the continuum-subtracted line flux for the Si XIII emission
line (He$\alpha$ at $6.65$ Å, right panel of Fig. 3), which is the brightest
Si line (i.e., the estimate of the Si temperature mainly depends on its
width): in this case the bulk of the line emission originates from a plasma
with $\tau<10^{11}$ s cm-3, where we expect the ion temperature to be almost
the same as that achieved at the shock front.

Figure 3: Normalized continuum-subtracted line flux of the Mg XII Ly$\alpha$
(_left panel_) emission line and of the Si XIII He$\alpha$ (_right panel_)
emission line as a function of the plasma temperature and time integral of the
electron density ($\tau$). Contours mark 25%, 50% and 75% of the maximum.

Further investigations require the inclusion of ion-proton collisions in the
model. However, this approach already shows that the combination of data
analysis and MHD simulations provides a powerful diagnostic tool to study both
the immediate post-shock conditions and the evolution toward the equilibrium
in the post shock flow.

##  3 Shock modification in SN 1006

As explained in Sect. 1, collisionless shock heating is associated with
particle acceleration and SNRs are known to be sites of efficient
acceleration. Self consistent hybrid (kinetic ions-fluid electrons)
simulations show that the acceleration efficiency is expected to increase by
reducing the angle between the shock velocity and the ambient magnetic field
and that magnetic turbulences are associated with efficient ion acceleration
(quasi-parallel scenario, e. g., [48]). This is in agreement with measurement
of radio polarization. For example, the radio polarization study of SN 1006
shows efficient particle acceleration and higher magnetic turbulence for
quasi-parallel shocks ([49]). Similarly, radio polarization measurements in
Kepler’s SNR indicate an almost radial (i. e. parallel to the shock velocity)
magnetic field and a lower level of polarization ([50]), which is indicative
of high magnetic turbulence in the acceleration site ([51]). On the other
hand, re-acceleration of ambient cosmic-ray seeds is nearly independent on the
shock inclination ([52])

Indeed, SNRs can sustain the observed flux of Galactic cosmic rays provided
that they inject $10-20\%$ of their kinetic energy to the particles (e. g.,
[53]). Such effect, if present, is expected to alter the shock dynamics by
increasing the shock compression ratio, and decreasing of the post-shock
temperature with respect to the Rankine-Hugoniot values ([54, 55, 56, 57,
23]). In particular, recent hybrid simulations show the formation of a shock
postcursor associated with efficient particle acceleration. This postcursor
moves downstream (approximately at the Afvén speed), acting as an additional
energy sink, which allows non-linear magnetic fluctuations, and particles, to
drift away from the shock front ([58]). When the effects of the postcursor are
taken into account, a shock compression ratio $r_{s}\sim 6-7$ (to be compared
with the canonical value of 4 for strong shocks) can be achieved when the
cosmic ray pressure is of the order of 10% of the bulk ram pressure.

The presence of shock modification has been recently revealed in SN 1006 [7].
In this remnant, thanks to the the almost uniform magnetic field,
approximately aligned in the southwest-northeast direction, it is possible to
observe, in the same object, regions with efficient particle acceleration
(i.e. regions in quasi-parallel conditions, indicated by the blue limbs in
Fig. 1) and regions where we do not expect shock modification ([6]). A careful
spatially resolved spectral analysis of the X-ray spectra stemming from the
shocked interstellar medium at different positions of the shell reveals a
regular azimuthal modulation of $r_{s}$, with a minimum $r_{s}=4$ in quasi-
perpendicular conditions and a maximum which reaches values as high as
$r_{s}=7$ in quasi-parallel conditions ([7]). The accurate comparison of the
azimuthal profile of the shock compression ratio with state of the art
theoretical models of modified shocks including the effect of the postcursor
shows a nice agreement between model and observations and provide important
constraints on the acceleration efficiency and on its dependence on the angle
between the ambient magnetic field and the shock velocity. In particular, in
quasi-parallel conditions, the cosmic rays pressure is $\sim 12\%$ of the
total, while the normalized magnetic pressure is $\sim 5\%$. Moreover, the
$r_{s}$ azimuthal profile shows a sharp minimum, which is strongly indicative
of efficient (normalized pressure $\sim 6\%$) re-acceleration of pre-existing
Galactic cosmic rays ([7]).

The shock modification is expected to affect the spectrum of the accelerated
particles, by modifying their spectral index ([59]). Remarkably, when
considering the model parameters that best reproduce the azimuthal profile of
$r_{s}$, a spectral index of $\sim 2.2$ is obtained for the energy spectrum in
the regions with maximum acceleration efficiency (the nonthermal limbs). When
considering the synchrotron emission from the accelerated electrons, this
corresponds to a photon index $\alpha\sim 0.6$, which is in nice agreement
with that observed for SN 1006 ([18]).

##  4 Conclusion

Astrophysical shocks are natural laboratories which allow us to access the
shock physics at extreme conditions. X-ray observations of SNRs are a powerful
diagnostic tool to study the abrupt collisionless shock heating and the
relaxation of the post shock plasma towards the equilibrium. Also, X-ray
observations can allow us to observe synchrotron radiation from
ultrarelativistic electrons accelerated at the shock front and to probe the
back-reaction of cosmic rays on the shock dynamics ([60]).

To extract all the information stored in the X-ray data, a detailed comparison
with state of the art models is necessary.

The physical processes localized at the shock front affect the whole remnant
structure and its global evolution. Therefore, both hybrid simulations
investigating the microphysics at the shock front and MHD simulations
describing the evolution of the system on larger spatial and time scales
provide powerful tools to get a higher level of diagnostics.

Moreover, the upcoming generation of X-ray telescopes (as _XRISM_ and
_Athena_) will be equipped with high resolution spectrometers, based on
microcalorimeters, which will provide us with an unprecedented level of
details. The development of novel advanced tools for data analysis and for
quantitative comparison with theoretical models is a challenging task which
will allow us to exploit the quality of the data in the near future.

## Acknowledgements

This work was partially supported by the INAF mini-grant “X-raying shock
modification in supernova remnants”.

## References

## References

  * [1] Tsurutani B T and Stone R G 1985 Washington DC American Geophysical Union Geophysical Monograph Series 35
  * [2] Bykov A M, Dolag K and Durret F 2008 Space Science Reviews 134 119–140 (Preprint 0801.0995)
  * [3] Miceli M, Bocchino F, Iakubovskyi D, Orlando S, Telezhinsky I, Kirsch M G F, Petruk O, Dubner G and Castelletti G 2009 Astron. & Astrophys. 501 239–249 (Preprint 0903.3392)
  * [4] Williams B J, Borkowski K J, Ghavamian P, Hewitt J W, Mao S A, Petre R, Reynolds S P and Blondin J M 2013 Astrophys. J. 770 129 (Preprint 1305.0567)
  * [5] Koyama K, Petre R, Gotthelf E V, Hwang U, Matsura M, Ozaki M and Holt S S 1995 Nature 378 255
  * [6] Miceli M, Bocchino F, Decourchelle A, Maurin G, Vink J, Orlando S, Reale F and Broersen S 2012 Astron. & Astrophys. 546 A66 (Preprint 1208.5966)
  * [7] Giuffrida R, Miceli M, Caprioli D, Decourchelle A, Vink J, Orlando S, Bocchino F, Greco E and Peres G 2022 Nature Communications 13 5098 (Preprint 2208.14491)
  * [8] Morlino G, Amato E, Blasi P and Caprioli D 2010 Mon. Not. R. Astron. Soc. 405 L21–L25 (Preprint 0912.2972)
  * [9] Ballet J 2006 Advances in Space Research 37 1902–1908 (Preprint arXiv:astro-ph/0503309)
  * [10] Smith R K and Hughes J P 2010 Astrophys. J. 718 583–585 (Preprint 1006.0254)
  * [11] Ozawa M, Koyama K, Yamaguchi H, Masai K and Tamagawa T 2009 Astrophys. J. Lett. 706 L71–L75 (Preprint 0910.3302)
  * [12] Miceli M, Bocchino F, Decourchelle A, Ballet J and Reale F 2010 Astron. & Astrophys. 514 L2+ (Preprint 1004.1604)
  * [13] Yamaguchi H, Tanaka T, Wik D R, Rho J, Bamba A, Castro D, Smith R K, Foster A R, Uchida H, Petre R and Williams B J 2018 Astrophys. J. Lett. 868 L35 (Preprint 1811.04426)
  * [14] Kawasaki M, Ozaki M, Nagase F, Inoue H and Petre R 2005 Astrophys. J. 631 935–946 (Preprint astro-ph/0507348)
  * [15] Greco E, Miceli M, Orlando S, Peres G, Troja E and Bocchino F 2018 Astron. & Astrophys. 615 A157 (Preprint 1804.06714)
  * [16] Okon H, Tanaka T, Uchida H, Tsuru T G, Seta M, Kokusho T and Smith R K 2021 Astrophys. J. 921 99 (Preprint 2108.05944)
  * [17] Okon H, Tanaka T, Uchida H, Yamaguchi H, Tsuru T G, Seta M, Smith R K, Yoshiike S, Orlando S, Bocchino F and Miceli M 2020 Astrophys. J. 890 62 (Preprint 1912.08129)
  * [18] Green D A 2019 Journal of Astrophysics and Astronomy 40 36 (Preprint 1907.02638)
  * [19] Vink J 2012 Astron. & Astrophys. Rev. 20 49 (Preprint 1112.0576)
  * [20] Tavani M, Giuliani A, Chen A W, Argan A, Barbiellini G, Bulgarelli A, Caraveo P, Cattaneo P W, Cocco V, Contessi T, D’Ammando F, Costa E, De Paris G, Del Monte E, Di Cocco G, Donnarumma I, Evangelista Y, Ferrari A, Feroci M, Fuschino F, Galli M, Gianotti F, Labanti C, Lapshov I, Lazzarotto F, Lipari P, Longo F, Marisaldi M, Mastropietro M, Mereghetti S, Morelli E, Moretti E, Morselli A, Pacciani L, Pellizzoni A, Perotti F, Piano G, Picozza P, Pilia M, Pucella G, Prest M, Rapisarda M, Rappoldi A, Scalise E, Rubini A, Sabatini S, Striani E, Soffitta P, Trifoglio M, Trois A, Vallazza E, Vercellone S, Vittorini V, Zambra A, Zanello D, Pittori C, Verrecchia F, Santolamazza P, Giommi P, Colafrancesco S, Antonelli L A and Salotti L 2010 Astrophys. J. Lett. 710 L151–L155 (Preprint 1001.5150)
  * [21] Morlino G and Caprioli D 2012 Astron. & Astrophys. 538 A81 (Preprint 1105.6342)
  * [22] Ackermann M, Ajello M, Allafort A, Baldini L, Ballet J, Barbiellini G, Baring M G, Bastieri D, Bechtol K, Bellazzini R, Bland ford R D, Bloom E D, Bonamente E, Borgland A W, Bottacini E, Brandt T J, Bregeon J, Brigida M, Bruel P, Buehler R, Busetto G, Buson S, Caliandro G A, Cameron R A, Caraveo P A, Casandjian J M, Cecchi C, Çelik Ö, Charles E, Chaty S, Chaves R C G, Chekhtman A, Cheung C C, Chiang J, Chiaro G, Cillis A N, Ciprini S, Claus R, Cohen-Tanugi J, Cominsky L R, Conrad J, Corbel S, Cutini S, D’Ammando F, de Angelis A, de Palma F, Dermer C D, do Couto e Silva E, Drell P S, Drlica-Wagner A, Falletti L, Favuzzi C, Ferrara E C, Franckowiak A, Fukazawa Y, Funk S, Fusco P, Gargano F, Germani S, Giglietto N, Giommi P, Giordano F, Giroletti M, Glanzman T, Godfrey G, Grenier I A, Grondin M H, Grove J E, Guiriec S, Hadasch D, Hanabata Y, Harding A K, Hayashida M, Hayashi K, Hays E, Hewitt J W, Hill A B, Hughes R E, Jackson M S, Jogler T, Jóhannesson G, Johnson A S, Kamae T, Kataoka J, Katsuta J, Knödlseder J, Kuss M, Lande J, Larsson S, Latronico L, Lemoine-Goumard M, Longo F, Loparco F, Lovellette M N, Lubrano P, Madejski G M, Massaro F, Mayer M, Mazziotta M N, McEnery J E, Mehault J, Michelson P F, Mignani R P, Mitthumsiri W, Mizuno T, Moiseev A A, Monzani M E, Morselli A, Moskalenko I V, Murgia S, Nakamori T, Nemmen R, Nuss E, Ohno M, Ohsugi T, Omodei N, Orienti M, Orlando E, Ormes J F, Paneque D, Perkins J S, Pesce-Rollins M, Piron F, Pivato G, Rainò S, Rando R, Razzano M, Razzaque S, Reimer A, Reimer O, Ritz S, Romoli C, Sánchez-Conde M, Schulz A, Sgrò C, Simeon P E, Siskind E J, Smith D A, Spand re G, Spinelli P, Stecker F W, Strong A W, Suson D J, Tajima H, Takahashi H, Takahashi T, Tanaka T, Thayer J G, Thayer J B, Thompson D J, Thorsett S E, Tibaldo L, Tibolla O, Tinivella M, Troja E, Uchiyama Y, Usher T L, Vandenbroucke J, Vasileiou V, Vianello G, Vitale V, Waite A P, Werner M, Winer B L, Wood K S, Wood M, Yamazaki R, Yang Z and Zimmer S 2013 Science 339 807–811 (Preprint 1302.3307)
  * [23] Slane P, Lee S H, Ellison D C, Patnaude D J, Hughes J P, Eriksen K A, Castro D and Nagataki S 2014 Astrophys. J. 783 33 (Preprint 1401.2556)
  * [24] Winkler P F, Gupta G and Long K S 2003 Astrophys. J. 585 324–335 (Preprint arXiv:astro-ph/0208415)
  * [25] Shimada N and Hoshino M 2000 Astrophys. J. Lett. 543 L67–L71
  * [26] Park J, Caprioli D and Spitkovsky A 2015 Physical Review Letters 114 085003 (Preprint 1412.0672)
  * [27] Rakowski C E, Ghavamian P and Hughes J P 2003 Astrophys. J. 590 846–857 (Preprint astro-ph/0303069)
  * [28] Ghavamian P, Schwartz S J, Mitchell J, Masters A and Laming J M 2013 Space Science Reviews 178 633–663 (Preprint 1305.6617)
  * [29] Raymond J C 2018 Space Science Reviews 214 28
  * [30] van Adelsberg M, Heng K, McCray R and Raymond J C 2008 Astrophys. J. 689 1089-1104 (Preprint 0803.2521)
  * [31] Ghavamian P, Laming J M and Rakowski C E 2007 Astrophys. J. 654 L69–L72 (Preprint arXiv:astro-ph/0611306)
  * [32] Korreck K E, Raymond J C, Zurbuchen T H and Ghavamian P 2004 Astrophys. J. 615 280–285
  * [33] Vink J, Laming J M, Gu M F, Rasmussen A and Kaastra J S 2003 Astrophys. J. Lett. 587 L31–L34 (Preprint arXiv:astro-ph/0303051)
  * [34] Broersen S, Vink J, Miceli M, Bocchino F, Maurin G and Decourchelle A 2013 Astron. & Astrophys. 552 A9 (Preprint 1210.7249)
  * [35] McCray R 1993 Annu. Rev. Astron. Astrophys. 31 175–216
  * [36] McCray R and Fransson C 2016 Annu. Rev. Astron. Astrophys. 54 19–52
  * [37] Orlando S, Miceli M, Pumo M L and Bocchino F 2016 Astrophys. J. 822 22 (Preprint 1603.03690)
  * [38] Orlando S, Miceli M, Petruk O, Ono M, Nagataki S, Aloy M A, Mimica P, Lee S H, Bocchino F, Peres G and Guarrasi M 2019 Astron. & Astrophys. 622 A73 (Preprint 1812.00021)
  * [39] Ono M, Nagataki S, Ferrand G, Takahashi K, Umeda H, Yoshida T, Orlando S and Miceli M 2020 Astrophys. J. 888 111 (Preprint 1912.02234)
  * [40] Orlando S, Ono M, Nagataki S, Miceli M, Umeda H, Ferrand G, Bocchino F, Petruk O, Peres G, Takahashi K and Yoshida T 2020 Astron. & Astrophys. 636 A22 (Preprint 1912.03070)
  * [41] Miceli M, Orlando S, Burrows D N, Frank K A, Argiroffi C, Reale F, Peres G, Petruk O and Bocchino F 2019 Nature Astronomy 3 236–241 (Preprint 1901.10336)
  * [42] Ravi A P, Park S, Zhekov S A, Miceli M, Orlando S, Frank K A and Burrows D N 2021 Astrophys. J. 922 140 (Preprint 2109.02881)
  * [43] Caprioli D, Yi D T and Spitkovsky A 2017 Phys. Rev. Lett. 119 171101 (Preprint 1704.08252)
  * [44] Spitzer L 1962 Physics of Fully Ionized Gases (Physics of Fully Ionized Gases, New York: Interscience (2nd edition), 1962)
  * [45] Narayan R and Medvedev M V 2001 Astrophys. J. Lett. 562 L129–L132 (Preprint astro-ph/0110567)
  * [46] Zanardo G, Staveley-Smith L, Gaensler B M, Indebetouw R, Ng C Y, Matsuura M and Tzioumis A K 2018 Astrophys. J. Lett. 861 L9 (Preprint 1806.04741)
  * [47] Greco E, Miceli M, Orlando S, Olmi B, Bocchino F, Nagataki S, Sun L, Vink J, Sapienza V, Ono M, Dohi A and Peres G 2022 Astrophys. J. 931 132 (Preprint 2204.06804)
  * [48] Caprioli D and Spitkovsky A 2014 Astrophys. J. 783 91 (Preprint 1310.2943)
  * [49] Reynoso E M, Hughes J P and Moffett D A 2013 Astron. J. 145 104 (Preprint 1302.4678)
  * [50] DeLaney T, Koralesky B, Rudnick L and Dickel J R 2002 Astrophys. J. 580 914–927 (Preprint astro-ph/0210355)
  * [51] Sapienza V, Miceli M, Bamba A, Katsuda S, Nagayoshi T, Terada Y, Bocchino F, Orlando S and Peres G 2022 Astrophys. J. 935 152 (Preprint 2207.06804)
  * [52] Caprioli D, Zhang H and Spitkovsky A 2018 J. of Plasma Phys. 84 715840301 (Preprint 1801.01510)
  * [53] Hillas A M 2005 Journal of Physics G Nuclear Physics 31 R95–R131
  * [54] Drury L O 1983 Reports on Progress in Physics 46 973–1027
  * [55] Decourchelle A, Ellison D C and Ballet J 2000 Astrophys. J. Lett. 543 L57–L60 (Preprint arXiv:astro-ph/0008344)
  * [56] Blasi P 2002 Astroparticle Physics 16 429–439 (Preprint arXiv:astro-ph/0104064)
  * [57] Vink J, Yamazaki R, Helder E A and Schure K M 2010 Astrophys. J. 722 1727–1734 (Preprint 1008.4367)
  * [58] Haggerty C C and Caprioli D 2020 Astrophys. J. 905 1 (Preprint 2008.12308)
  * [59] Caprioli D, Haggerty C C and Blasi P 2020 Astrophys. J. 905 2 (Preprint 2009.00007)
  * [60] Shimoda J, Ohira Y, Bamba A, Terada Y, Yamazaki R, Inoue T and Tanaka S J 2022 Publ. Astron. Soc. Jpn. (Preprint 2201.07607)