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\title{On the Casimir of the group $ISL(n,R)$ and its algebraic decomposition}
\author{J.N. Pecina-Cruz\footnotemark\ \\Bureau of Economic 
Geology\\J.J. Pickle Research Campus\\ 
The University of Texas at Austin\\University Station, Box X\\
Austin, Texas 78713-8294}
\addtocounter{footnote}{1}\footnotetext{Present address:\\Q-Chem, Inc., 317 
Wipple St.,Pittsburgh, PA 15218\\Theory Group, Department of Physics, Carnegie Mellon 
University\\5000 Forbes Avenue, Pittsburgh, PA 15213}
\date{Sept. 29, 1996}
\maketitle
\hyphenation{isomor-phism isomor-phic}
\hyphenation{sat-isfy def-f-ni-tion com-plete}
\hyphenation{de-ri-va-tion}
\hyphenation{pos-si-ble be-cause func-tions}
\hyphenation{ex-ist sy-mul-ta-ne-ous sys-tem to-tal dif-fer-en-tial 
equa-tions}
\begin{abstract}
In this article a formula for the Casimir of $ISL(n,R)$ is given. This 
algebraic expression is useful in the classification of particles in a 
theory based on $ISL(n,R)$. It is also proven that the Casimir 
of $ISL(n,R)$ can be decomposed in terms of the Casimirs of its little 
groups, a key point in the posterior construction of its irreducible 
representations.
\end{abstract}
{\bf1. Introduction}\\ \\
The special affine group $ISL(n,R)$ is the semidirect product of the Abelian 
group of translations in $n$ dimensions and the special linear, $SL(n,R).$ 
This group has been chosen as the gauge group in gauge theories of gravity
\cite{hehl}; therefore, the knowledge of its Casimirs will be necessary not only 
to investigate the irreducible representations of this group, but also to 
provide these theories with a wave equation.\cite{barut} The group $ISL(n,R)$ has 
a subgroup, the Poincare group, from which stems its importance in physics. 
The eigenvalues of the Casimir of the group $ISL(n,R)$ provide quantum numbers 
to classify the particles of these theories in the same way that the 
eigenvalues of the Casimirs of the Poincare groups allow us to classify the 
particles according to their mass and spin.  The eigenvalues 
of the Casimir of $ISL(n,R)$ label the irreducible 
representations of the group or of its Lie algebra. The invariants are also 
useful ingredients in the decomposition of reducible representations into 
irreducible ones. In the case of gauge theories of gravity based on
$ISL(n,R),$ it is important to decompose the unitary irreducible representations of 
the group $ISL(n,R)$ into the unitary irreducible representations of the Poincare 
subgroup. This would bring a physical insight into the behavior of the 
elementary particles of these theories. In section 2, we  
construct the formula for the Casimir of $ISL(n,R)$. In 
section 3, we discuss the induction proof used to guarantee the general 
validation of the formula for the Casimir of $ISL(n,R)$. Finally in section
4, the algebraic decomposition of $ISL(n,R)$ is achieved.\\ \\ \\ 
{\bf2. Construction of the Formula for the Casimir of $ISL(n,R)$}\\ \\
In Ref. 3 it is proved that the group $ISL(n,R)$ has one invariant. And in Ref. 4
it is proved that the order of this invariant is $\frac{1}{2}n(n+1)$. Based 
on this proof, the standard procedure for constructing invariants 
by contracting tensorial indices with the Levi Civita antisymmetric 
pseudo tensor and the generators of the Lie group \cite{mirman}, we 
found a formula for the invariant of $ISL(n,R).$ This expression is 
given by\\ \\
$CasimirISL(n,R)=\{ \zeta_{\xi_{0}\alpha_{1},\ldots,\alpha_{n-1}\beta_{1}[
\beta_{2}(\gamma_{11})],\ldots,[\beta_{n-1}(\gamma_{n-2,1},\ldots,\gamma_{n-2,
n-2})]}^{\rho_{1}[(\theta_{11})\rho_{2}]\,\ldots,[(\theta_{n-2,1},\ldots,
\theta_{n-2,n-2})\rho_{n-1}]}P^{\xi_{0}}P^{\alpha_{1}}$
\begin{equation}
\cdots P^{\alpha_{n-1}}E_{\rho_{1}}^{\beta_{1}}E_{[(\theta_{11})\rho_{2}]}^{[
\beta_{2}(\gamma_{11})]} \cdots E_{[(\theta_{n-2,1},\ldots,\theta_{n-2,n-2})
\rho_{n-1}]}^{[\beta_{n-1}(\gamma_{n-2,1},\ldots,\gamma_{n-2,n-2})]}\}_{
symmetrized}
\end{equation}\\
\mbox{\scriptsize$\xi_{0},\theta_{ij},\gamma_{ij},\alpha_{l},\rho_{k},
\beta_{m} = 0,1,\ldots,n-1$\hspace{.6cm} $i,j=1,2,\ldots,n-2$\hspace{.6cm}$l,
k,m=1,2,\ldots,n-1$} \\ \\
where\\ \\
$E_{[(\theta_{11})\rho_{2}]}^{[\beta_{2}(\gamma_{11})]}=E_{\theta_{11}}^{
\beta
_{2}}E_{\rho_{2}}^{\gamma_{11}}$\\ \\ 
and\\ \\
$E_{[(\theta_{n-2,1},\ldots,\theta_{n-2,n-2})\rho_{n-1}]}^{[\beta_{n-1}(\gamma_
{n-2,1},\ldots,\gamma_{n-2,n-2})]}=E_{\theta_{n-2,1}}^{\beta_{n-1}}E_{\theta_{
n-2,2}}^{\gamma_{n-2,
1}} \cdots E_{\theta_{n-2,n-2}}^{\gamma_{n-2,n-3}}E_{\rho_{n-1}}^{\gamma_{n-2,
n-2}},$\\ \\
where $\zeta_{\ldots\ldots}^{\ldots}$ is given by the expression\\ \\ 
$\zeta_{\ldots\ldots}^{\ldots}=\epsilon_{\xi_{0}\beta_{1}\cdots\beta_{n-1}}
(\delta_{\rho_{1}\alpha_{1}}\cdots\delta_{\rho_{n-1}\alpha_{n-1}})((\delta_{
\theta_{11}\gamma_{11}})(\delta_{\theta_{21}\gamma_{21}}\delta_{\theta_{22}
\gamma_{22}})\cdots$\\
\begin{equation}
(\delta_{\theta_{n-2,1}\gamma_{n-2,1}}\cdots\delta_{\theta_{n-2,n-2}\gamma_{
n-2,n-2}})).
\end{equation}\\
The $E_{\alpha}^{\beta}$ are the generators of the general linear group 
$GL(n,R)$ in the Weyl basis\cite{barut} and the $P^{\alpha}$ are the generators 
of the Abelian subgroup of $ISL(n,R)$. In formula 1, the following substitution
must be carried out\\
\begin{equation}
{\bf E}_{\alpha}^{\alpha}=E_{\alpha}^{\alpha}-E_{\alpha+1}^{\alpha+1}.
\end{equation}
The non-zero trace generators $E_{\alpha}^{\alpha}$ of the general linear 
group $GL(n,R)$ are substituted for the traceless generators ${\bf E}_{
\alpha}^{\alpha}$ of the special linear group $SL(n,R).$
In equation (2), we define $\epsilon_{\xi_{0}}=1$.\\ \\
{\bf3. The Induction Proof}\\ \\
In Refs. 6, it is proved that the invariant of $ISL(n,R)$ can 
be obtained by solving a system of linear first order partial differential 
equations (LFPDE). The system of (LFPDE) can trivially be solved for $n = 1$. 
Thus, the first part of the induction is proven.
In order to prove the second part of the induction method, we assume the 
formula is valid for $n=k$ and then prove that it is valid for $n=k+1.$\\
We can construct a scalar of the order required by Lemma 1, in Ref. 4, to 
be the invariant of $ISL(k+1,R).$ According to this Lemma, the order of the 
invariant in the generators of this group should be $\frac{1}{2}(k+1)(k+2).$ 
In this same reference, it is proven that the invariant for this group 
would be of $k+1$ order in the generators of the translations. Therefore 
the invariant for $ISL(k+1,R)$ must be of $k+1$ order in the translations 
and of $\frac{1}{2}k(k+1)$ order in the non-translations generators of 
the group $ISL(k+1,R).$ That is, the invariant of $ISL(k+1,R)$ must be 
different from the invariant for $ISL(k,R)$ by a factor given by\\
\begin{equation}
P^{\alpha_{k}}E_{[(\theta_{k-1,1},\ldots,\theta_{k-1,k-1})\rho_{k}]}^{[\beta_{
k}(\gamma_{k-1,1},\ldots,\gamma_{k-1,k-1})]}.
\end{equation}
If we take into account the form of the invariants of $ISL(n,R)$ for n=1,2,3,4,\cite{pecina}
the formula we are assuming valid for $n=k$ and the factor given above, 
we can construct a scalar given by\\ \\
$CasimirISL(k+1,R)=$ \\ \\$ \{ 
\zeta_{\xi_{0}\alpha_{1},\ldots,\alpha_{k-1}\alpha_{k}\beta_{1}[\beta_{2}(
\gamma_{11})],\ldots,[\beta_{k-1}(\gamma_{k-2,1},\ldots,\gamma_{k-2,k-2})][
\beta_{k}(\gamma_{k-1,1},\ldots,\gamma_{k-1,k-1})]}^{\rho_{1}[(\theta_{11})
\rho_{2}],\ldots,[(\theta_{k-2,1},\ldots,\theta_{k-2,k-2})\rho_{k-1}][(
\theta_{k-1,1,\ldots,\theta_{k-1,k-1})\rho_{k}]}}P^{\xi_{0}}P^{\alpha_{1}}
\cdots P^{\alpha_{k-1}}P^{\alpha_{k}}$\\ 
\begin{equation}
E_{\rho_{1}}^{\beta_{1}}E_{[(\theta_{11})\rho_{2}]}^{[\beta_{2}(\gamma_{11})]}
\cdots E_{[(\theta_{k-2,1},\ldots,\theta_{k-2,k-2})\rho_{k-1}]}^{[\beta_{k-1}(
\gamma_{k-2,1},\ldots,\gamma_{k-2,k-2})]}E_{[(\theta_{k-1,1},\ldots,\theta_{
k-1,k-1})\rho_{k}]}^{[\beta_{k}(\gamma_{k-1,1},\ldots,\gamma_{k-1,k-1})]}\}_{
symmetrized}.
\end{equation}\\ \\
This formula coincides with the eqn. (1) for $n=k+1.$ However, the proof 
is not yet complete, since the scalar that we have constructed to be the 
invariant of $ISL(k+1,R)$ could be zero. Therefore, we must prove that 
the scalar given by eqn. (5) is not zero.\\ \\
{\bf4. The Algebraic Decomposition of $ISL(N,R)$}\\ \\
The proof that the scalar given by eqn. (5) does not vanish is based on an
algebraic decomposition of the Casimir of $ISL(n,R)$ in terms of the Casimirs of its
little groups. This decomposition allows an immediate classification of the existent particles 
in a theory based on $ISL(n_{1},R)$, with $n_{1}$ any number.\cite{smolin}\\
Then making all the translations equal zero except $P^{0}$ in eqn. (5). 
That is,\\ \\
$\xi_{0}=\alpha_{1}=\ldots=\alpha_{k}=0,$\\ \\
therefore\\ \\$\rho_{1}=\rho_{2}=\ldots=\rho_{k}=0.$\\ \\
Hence, the Casimir with all the translations zero except $P^{0}$ is 
given by\\ \\
$CasimirISL(k+1,R)=$ \\ \\
$\{ (P^{0})^{k+1}\zeta_{00,\ldots,0\beta_{1}[\beta_{2}(\gamma_{11})],\ldots,[
\beta_{k}(\gamma_{k-1,1},\ldots,\gamma_{k-1,k-1})]}^{0[(\theta_{11})0],\ldots,
[(\theta_{k-1,1},\ldots,\theta_{k-1,k-1})0]}E_{0}^{\beta{1}}E_{0}^{
\gamma_{11}}\cdots E_{0}^{\gamma_{k-1,k-1}}$\\ \\
\begin{equation}
E_{\theta_{11}}^{\beta_{2}}E_{[(\theta_{21})\theta_{22}]}^{[\beta_{3}(\gamma_{
21})]}\cdots E_{[(\theta_{k-1,1},\ldots,\theta_{k-1,k-2})\theta_{k-1,k-1}]}^{
[\beta_{k}(\gamma_{k-1,1},\ldots,\gamma_{k-1,k-2})]}\}_{symmetrized}
\end{equation}
where\\ \\
$\zeta_{00,\ldots,0\beta_{1}[\beta_{2}(\gamma_{11})],\ldots,[\beta_{k}(
\gamma_{k-1,1},\ldots,\gamma_{k-1,k-1})]}^{0[(\theta_{11})0],\ldots,[(
\theta_{k-1,1},\ldots,\theta_{k-1,k-1})0]}=\epsilon_{0\beta_{1}\cdots
\beta_{k}}(\delta_{\theta_{11}\gamma_{11}}\delta_{\theta_{22}\gamma_{22}}
\cdots\delta_{\theta_{k-1,k-1}\gamma_{k-1,k-1}})$\\ \\
\begin{equation}
((\delta_{\theta_{21}\gamma_{21}})(\delta_{\theta_{31}\gamma_{31}}\delta_{
\theta_{32}\gamma_{32}})\cdots(\delta_{\theta_{k-1,1}\gamma_{k-1,1}}\cdots
\delta_{\theta_{k-1,k-2}\gamma_{k-1,k-2}})).
\end{equation}\\ \\
The terms of eqn. (6) with the $\theta's=0$ generated by the contraction 
of the Levi-Civita pseudo tensor cancel out by antisymmetry.\\ 
Therefore, the indices  $\beta, \theta, \gamma$ can be shifted; instead  of 
running from $0,1,\ldots,k$ they will run from $0,1,\ldots,k-1$. {\em This 
defines an isomorphism} between the subset of the generators, belonging to 
the factor which multiply $P^{k+1}$ in eqn. (6), of the Lie algebra of 
$ISL(k+1,R)$ and the Lie algebra of $ISL(k,R).$\\
Hence eqn. (7) can be given by\\ \\
$\zeta_{\beta_{1}\gamma_{11}\gamma_{22},\ldots,\gamma_{k-1,k-1}\beta_{2}[
\beta_{3}(\gamma_{21})],\ldots,[\beta_{k}(\gamma_{k-1,1},\ldots,\gamma_{
k-1,k-2})]}^{\theta_{11}[(\theta_{21})\theta_{22}],\ldots,[(\theta_{k-1,1},
\ldots,\theta_{k-1,k-2})\theta_{k-1,k-1}]}=\epsilon_{\beta_{1}\cdots\beta_{k}}
(\delta_{\theta_{11}\gamma_{11}}\delta_{\theta_{22}\gamma_{22}}\cdots\delta_{
\theta_{k-1,k-1}\gamma_{k-1,k-1}})$\\ \\
\begin{equation}
((\delta_{\theta_{21}\gamma_{21}})(\delta_{\theta_{31}\gamma_{31}}\delta_{
\theta_{32}\gamma_{32}})\cdots(\delta_{\theta_{k-1,1}\gamma_{k-1,1}}\cdots
\delta_{\theta_{k-1,k-2}\gamma_{k-1,k-2}})).
\end{equation}\\ 
The basis elements of the Lie algebra of the group $ISL(n,R)$ can be 
represented by the $n+1$ by $n+1$ matrices given below:
\[ \left( \begin{array}{cc}
SL(n,R)&P\\ \\
      0&0\\  \\
\end{array}  \right) \]\\
where $P$ are the generators of the group of translations, and $SL(n,R)$ are 
the generators of the special linear group in $n$ dimensions. Therefore, 
the $E_{0}^{\alpha}$ generators of the eqn. (6) can 
be considered as the translations $P^{\alpha}$ generators of 
the little group $ISL(k,R)$ of $(p^{0},0,0,\ldots,0_{k}).$\cite{elliott}\\
Using eqn. (8), eqn. (6) can be written in the following form:\\ \\
$CasimirISL(k+1,R)=$\\ \\
$\{(P^{0})^{k+1}\zeta_{\beta_{1}\gamma_{11}\gamma_{22},\ldots,\gamma_{k-1,k-1}
\beta_{2}[\beta_{3}(\gamma_{21})],\ldots,[\beta_{k}(\gamma_{k-1,1},\ldots,
\gamma_{k-1,k-2})]}^{\theta_{11}[(\theta_{21})\theta_{22}],\ldots,[(\theta_{
k-1,1},\ldots,\theta_{k-1,k-2})\theta_{k-1,k-1}]}E_{0}^{\beta{1}}E_{0}^{
\gamma_{11}}\cdots E_{0}^{\gamma_{k-1,k-1}}$\\ \\
\begin{equation}
E_{\theta_{11}}^{\beta_{2}}E_{[(\theta_{21})\theta_{22}]}^{[\beta_{3}(
\gamma_{21})]}\cdots E_{[(\theta_{k-1,1},\ldots,\theta_{k-1,k-2})\theta_{
k-1,k-1}]}^{[\beta_{k}(\gamma_{k-1,1},\ldots,\gamma_{k-1,k-2})]}\}_{
symmetrized}.
\end{equation}\\ 
To take advantage of the isomorphism given above, we make the following
substitution:\\ \\
$\beta_{1}\rightarrow\xi_{0}, \beta_{2}\rightarrow\beta_{1}, \beta_{3}
\rightarrow\beta_{2},\ldots,\beta_{k}\rightarrow\beta_{k-1}$\\ \\$\theta_{11}
\rightarrow\rho_{1}, \theta_{22}\rightarrow\rho_{2},\ldots,\theta_{k-1,k-1}
\rightarrow\rho_{k-1}$\\ \\$\gamma_{11}\rightarrow\alpha_{1},\gamma_{22}
\rightarrow\alpha_{2},\ldots,\gamma_{k-1,k-1}\rightarrow\alpha_{k-1}$\\ \\
$\theta_{21}\rightarrow\theta_{11}, (\theta_{31}\rightarrow\theta_{21}, 
\theta_{32}\rightarrow\theta_{22}),\ldots,(\theta_{k-1,1}\rightarrow\theta_{
k-2,1}, \theta_{k-1,2}\rightarrow\theta_{k-2,2},\ldots,\theta_{k-1,k-2}
\rightarrow\\ \theta_{k-2,k-2})$\\ \\$\gamma_{21}\rightarrow\gamma_{11}, (
\gamma_{31}\rightarrow\gamma_{21}, \gamma_{32}\rightarrow\gamma_{22}),\ldots,(
\gamma_{k-1,1}\rightarrow\gamma_{k-2,1}, \gamma_{k-1,2}\rightarrow\gamma_{
k-2,2},\ldots,\gamma_{k-1,k-2}\rightarrow\\ \gamma_{k-2,k-2})$\\ \\
and substituting it into eqn. (9), we obtain\\ 
\begin{equation}
CasimirISL(k+1,R)=\{(P^{0})^{k+1}(CasimirISL(k,R))\}_{symmetrized}
\end{equation}\\
We have obtained the Casimir of the little group $ISL(k,R)$ from the Casimir of
the group $ISL(k+1,R).$ We arrive at the same result if we take any of the 
other translations.\\
From the above discussion, it is clear that the Casimir of $ISL(k+1,R)$ 
given by equation (5) does not vanish, as claimed. This completes the 
induction proof. We conclude that the formula  given by equation (1) is 
valid for any integer $n.$\\ \\ 
{\bf Conclusion}\\ \\
Although the formula for the Casimir of $ISL(n,R)$ has been written in the Weyl
basis, this does not limit its application range. The advantage of our formula
for $ISL(n,R),$ over other possible formulation, is its immediate physical 
and mathematical application as shown above in the little group Casimir 
decomposition of $ISL(n,R).$\\
In gauge theories of gravity based on the group $ISL(4,R),$ it should be 
verified the correct usage of the Casimir operator. The reason is that in 
these theories, the group $ISO(1,3)$ must be a subgroup of the gauge group. 
This group has a different Lie algebra than that of the group $ISO(4)$ 
which is a subgroup of $ISL(4,R)$. The applications of a deunitarizing inner 
automorphism,\cite{sijacki} which changes some of the generators of the group 
$ISL(n,R)$ for a factor  $\sqrt -1$, is necessary to extend the range of 
application of our formula. To avoid confusion we suggest using the notation 
$ISL(1,n-1,R)$ for the group that has as a  subgroup $ISO(1,n-1).$\\ \\
{\bf Acknowledgments}\\ \\
 I am indebted to Dr. Bob Hardage for his 
constant encouragement during the preparation of this manuscript. I thank Prof. Greg 
Plaxton, from the Computer Sciences Department of The University of Texas at 
Austin, for many useful dicussions and suggestions. I am also grateful to Prof. L. Wolfenstein 
and Prof. F. Gildman for their support at Carnegie Mellon University while this article 
was concluded.\\ \\
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\end{document}

