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\begin{document}
\addtolength{\baselineskip}{1.5mm}
\begin{center}
{\Large \bf $R$-matrix formulation of deformed boson algebra}\\[2cm]
J.~Van der Jeugt\footnote{Research Associate N.F.W.O. (National Fund
for Scientific Research of Belgium)} \\[8mm]
Toegepaste Wiskunde en Informatica,
Universiteit Gent,\\
Krijgslaan 281--S9, B9000 Gent, Belgium
\end{center}
\vskip 1cm
\noindent Classification numbers : 02.20, 3.65F, 5.30J.
\vskip 2cm

\vskip 3cm
\begin{abstract}
Rewriting the Pusz-Woronowicz $q$-boson relations in terms of
the standard $R$-matrix for $SU_q(n)$, we give the definition
of a general deformed boson algebra ${\cal A}(R)$ depending upon
a matrix $R$. We investigate the conditions under which this algebra
is associative. For $n=2$, a set of matrices satisfying these
conditions is classified, and the corresponding ``twisted statistics''
is given.
\end{abstract}

\newpage

The interest in quantum groups~[1,2] and quantum enveloping algebras~[3]
(see also [4--6] for introductions to these topics) has led to the study
of $q$-deformations of the Heisenberg-Weyl algebra and the introduction
of so-called $q$-bosons~[7--9]. Two types of $q$-bosons for the
quantum enveloping algebra $su_q(n)$, or the quantum group $SU_q(n)$,
have been introduced. On the one hand, there are the Biedenharn-Macfarlane
$q$-bosonic operators~[7,8], which give rise to symmetric irreducible
representations of $su_q(n)$ and to a Jordan-Schwinger realisation of
the $su_q(n)$ Chevalley generators~[10]. On the other hand, there are the
Pusz-Woronowicz $q$-boson operators~[9], related to a covariant differential
calculus on the quantum group $SU_q(n)$. Here, the creation operators
transform as the components of the fundamental representation of
$su_q(n)$, and the annihilation operators as the components of the
dual representation. Also, the Pusz-Woronowicz operators transform
covariantly under the action of the quantum group $SU_q(n)$. As
operators acting in the $q$-Fock space, the Pusz-Woronowicz and the
Biedenharn-Macfarlane operators can be related to each other~[11--13].

In this Letter, we shall define the deformed boson algebra in terms
of the Pusz-Woronowicz operators. Their creation and annihilation
operators are related by expressions involving the fundamental $R$-matrix
of $SU_q(n)$. Taking these relations as the starting point for our
definition of the deformed boson algebra,
with an arbitrary matrix $R$, we investigate the conditions
under which an associative algebra with Hermitian conjugate is obtained.
Here, the technique is similar to that developed in [14,15], but
the ansatz is different.
This leads to three conditions for the matrix $R$ (a Hermiticity
condition, the Yang-Baxter Equation (YBE), and a Hecke condition), which are
clearly satisfied by the $SU_q(n)$ fundamental $R$-matrix. For
$n=2$, a particular set of matrices $R$ satisfying these conditions are
classified, showing that apart from the trivial solution and the usual
$SU_q(2)$ solution ($R_q$ and its double), there is a third matrix
$R$ satisfying the conditions required here.

The Pusz-Woronowicz $q$-bosonic creation and annihilation
operators $\Ad_i$ and $A_i$ ($i=1,\ldots,n$) satisfy the commutation
relations~[9,12,13]
\beq
 \begin{array}{l}
 A_iA_j-qA_jA_i=0, \qquad (i<j),\\[1mm]
 \Ad_i\Ad_j-q^{-1}\Ad_j\Ad_i=0, \qquad (i<j),\\[1mm]
 A_i\Ad_j-q\Ad_j A_i=0, \qquad(i\ne j),\\[1mm]
 \ds A_i\Ad_i-q^2\Ad_i A_i=1+(q^2-1)\sum_{j=1}^{i-1} \Ad_j A_j.
 \end{array}
\label{PW}
\eeq
These relations can be reexpressed in terms of the fundamental
$R$-matrix of $SU_q(n)$. This is an $n^2\times n^2$ matrix, and can be
written as~[6]
\beq
R=q\sum_i e_{ii}\otimes e_{ii} + \sum_{i\ne j} e_{ii}\otimes e_{jj}
 + (q-q^{-1}) \sum _{i<j} e_{ij}\otimes e_{ji},
\label{R}
\eeq
where $e_{ij}$ is the $n\times n$ matrix with entry 1 at position $(i,j)$
and 0 elsewhere. Note that
\beq
R= R_{ij,kl}\, e_{ik}\otimes e_{jl}\;\; ;
\label{Rcomp}
\eeq
in (\ref{Rcomp}), and in the rest of the paper, there is summation
over repeated indices.
Putting $V=\C^n$, $R$ can be seen as an element of ${\rm End}(V\otimes V)$.
The twist operator $P\in {\rm End}(V\otimes V)$ is defined as
$P(x\otimes y) = y\otimes x$, for all $x,y\in V$; in matrix notation
we have $P_{ij,kl}=\delta_{il}\delta_{jk}$.

In terms of (\ref{R}), the relations (\ref{PW}) can be rewritten as
\bea
A_i A_j &=& q^{-1} R_{ij,kl} A_l A_k , \label{AA}\\
\Ad_i \Ad_j & = & q^{-1} R_{lk,ij} \Ad_k \Ad_l ,\label{A+A+}\\
A_i \Ad_j &=& \de_{ij} + q R_{ki,jl} \Ad_k A_l . \label{AA+}
\eea
This form of the $q$-bosonic relations can be deduced from the
connection between the $q$-boson operators and the differential
calculus for $SU_q(n)$~[9,16--18]. In this paper, (\ref{AA}--\ref{AA+})
will be the starting point for our definition of the deformed
boson algebra ${\cal A}$. This will be a complex algebra generated
by $\C$ and the elements $\Ad_i$, $A_i$ ($i=1,\ldots,n$), equipped
with a Hermitian conjugation $^\dagger$ which is an antihomomorphism
(i.e.~$(ab)^\dagger = b^\dagger a^\dagger$) and such that
$(A_i)^\dagger=\Ad_i$, $(\Ad_i)^\dagger=A_i$, and $(\la)^\dagger
=\la^*$ (complex conjugate) for $\la\in\C$. Moreover, there will
be quadratic relations similar to (\ref{AA}--\ref{AA+}), but with
$q^{-1}$ and $q$ replaced by independent complex numbers $p$ and $p'$.
Note that from the invariance under Hermitian conjugation of the
relation
\beq
A_i \Ad_j = \de_{ij} + p' R_{ki,jl} \Ad_k A_l
\eeq
it follows that the matrix $R$ must satisfy
\beq
R_{ij,kl}=R^*_{lk,ji}.  \label{RR}
\eeq
Thus, we have the following definition.

\proclaim Definition.
Let $R$ be a complex $n^2\times n^2$ matrix satisfying (\ref{RR}).
The deformed boson algebra ${\cal A}(R)$ is the complex algebra
generated by $1$, $\Ad_i$ and $A_i$ ($i=1,\ldots,n$) subject to
the relations
\bea
A_i A_j &=& p R_{ij,kl} A_l A_k , \label{pAA}\\
\Ad_i \Ad_j & = & p^* R^*_{ji,kl} \Ad_k \Ad_l ,\label{pA+A+}\\
A_i \Ad_j &=& \de_{ij} + p' R_{ki,jl} \Ad_k A_l , \label{pAA+}
\eea
with antihomomorphism $^\dagger$ satisfying
$(A_i)^\dagger=\Ad_i$, $(\Ad_i)^\dagger=A_i$, and $(\la)^\dagger
=\la^*$ ($\la\in\C$).

Note that (\ref{pA+A+}) follows from applying $^\dagger$ to (\ref{pAA}).
In this definition, $p$ is a complex number, and $p'$ is a real
number.

Such an algebra would be of little interest if it were not associative.
In the following, we shall investigate the conditions under which
${\cal A}(R)$ is an associative algebra.
To express associativity, it is sufficient to require the braid
transposition schemes for triples of generators of ${\cal A}(R)$~[2].
This is by now a well-known technique. For the product $A_iA_jA_k$,
the braid transposition scheme is
\beq
\begin{array}{ccccccc}
 & & ikj & \rightarrow & kij & & \\
 & \nearrow& & & & \searrow& \\
ijk & & & & & & kji \\
 & \searrow& & & & \nearrow& \\
 & & jik & \rightarrow & jki & &
\end{array}
\label{braid}
\eeq
Applying this to $A_iA_jA_k$ , using (\ref{pAA}),
yields the YBE for the matrix $R$~:
\beq
\sum_{u,v,w}R_{ab,uv}R_{vw,cd}R_{ue,fw}=
 \sum_{u,v,w}R_{be,uv}R_{wu,fc}R_{av,wd}. \label{YBE}
\eeq
This relation between the braid scheme and the YBE is known; it
follows also from the fact that the $A_i$ satisfy the same relation~(\ref{pAA})
as the quantum plane coordinates~[2], and the equivalence of the associativity
of the quantum plane coordinates and the YBE for $R$.

Similarly, the braid scheme for $\Ad_i\Ad_j\Ad_k$ gives rise to
the YBE equation for $R^*$, or using (\ref{RR}), to the YBE for $R$.
The triples of generators that remain to be investigated are
$A_iA_j\Ad_k$ and $A_i\Ad_j\Ad_k$.  Here, the calculations are
similar as in [14,15], but the starting point is different.
We shall give one calculation in more detail here. Using the
top half of (\ref{braid}) on $A_iA_j\Ad_k$, i.e.~first use
equation (\ref{pAA+}) on the last two components, then (\ref{pAA+})
on the first two components, and finally (\ref{pAA}) on the last
two components, one finds~:
\beq
A_iA_j\Ad_k=\de_{jk}A_i+p'R_{ij,kb}A_b+p'^2p
 R_{aj,kb}R_{ui,av}R_{vb,xy}\Ad_uA_yA_x. \label{lhs}
\eeq
Using the bottom half of (\ref{braid}) on $A_iA_j\Ad_k$, one finds~:
\beq
A_iA_j\Ad_k=pR_{ij,kb}A_b+pp'R_{ij,au}R_{ua,kv}A_v+p'^2p
 R_{ij,ab}R_{ua,kv}R_{xb,uy}\Ad_xA_yA_v. \label{rhs}
\eeq
The cubic terms in the right hand sides of (\ref{lhs}) and (\ref{rhs})
are equal if and only if $R$ satisfies the YBE. The linear terms
are equal if the following relation is satisfied~:
\beq
pp'R_{ij,ab}R_{ba,kl}+pR_{ij,kl}-p'R_{ij,kl}+\de_{jk}\de_{il}=0.
\eeq
Using the notation $\check R=PR$, this can be rewritten as follows~:
\beq
(p\check R -1)(p'\check R+1)=0.
\label{Hecke}
\eeq
This is the Hecke condition for $\check R$, and implies that $\check R$
has two eigenvalues $p^{-1}$ and $-p'^{-1}$. Similarly, if one considers
the braid scheme for $A_i\Ad_j\Ad_k$, using (\ref{RR}), one can show
that the cubic terms give again rise to the YBE for $R$, and that
the linear terms give the condition
\beq
(p^*\check R -1)(p'\check R+1)=0.
\eeq
It follows that $p$ has to be real. Thus we have the result~:

\proclaim Theorem.
The deformed boson algebra ${\cal A}(R)$ with $p$ and $p'$ real
is an associative algebra
provided $R$ satisfies the YBE (\ref{YBE}) and the Hecke condition
(\ref{Hecke}).

The following is a classical remark concerning the quantum group
and is worth repeating here in terms of the deformed boson algebra relations.
Consider the following transformations for the deformed bosons~:
\beq
 \begin{array}{l}
 B_i = M_{ij} A_j ,\\[1mm]
 B^\dagger_i = N_{ji} \Ad_j,
 \end{array}
\eeq
where the elements $M_{ij}$ and $N_{ji}$ are supposed to commute with
$A_k$ and $\Ad_k$, but not among themselves. Then (\ref{pAA}) holds
for the $B$-operators provided
\beq
R M_2M_1 = M_1M_2R,
\label{M}
\eeq
where $M_1=M\otimes 1$ and $M_2=1\otimes M$. Similarly, (\ref{pA+A+})
holds for the $B^\dagger$-operators provided
\beq
RN_1N_2 = N_2N_1R.
\label{N}
\eeq
Finally, relation (\ref{pAA+}) is valid for the $\{B,B^\dagger\}$-operators
if
\beq
M_{ia}N_{aj}=\de_{ij},\qquad N_{ia}M_{aj}=\de_{ij}.
\eeq
If this last relation holds, then (\ref{N}) and (\ref{M}) are
equivalent statements.

Let us now turn to the study of matrices $R$ satisfying the three
properties (\ref{RR}), (\ref{YBE}) and (\ref{Hecke}) in the case $n=2$.
It would still be a formidable task to find all matrices satisfying
these three conditions. Therefore, we make one further assumption.
We shall assume in (\ref{pAA}) that in the relations between
$A_1A_2$ and $A_2A_1$ no $A_1A_1$ and $A_2A_2$ appear, and vice versa.
Concretely, this means that the matrix $R$ now takes the special form
\beq
 \left(
 \begin{array}{cccc} \times&0&0&\times\\ 0&\times&\times&0 \\
  0&\times&\times&0 \\ \times&0&0&\times \end{array}
 \right) .
\eeq
Let us choose the following labelling for $R$ ($p$ is real)~:
\beq
pR= \left(
 \begin{array}{cccc} a&0&0&d\\ 0&b&c&0 \\ 0&c'&b^*&0 \\ d^*&0&0&a' \end{array}
 \right),
\eeq
where it follows from (\ref{RR}) that $a,a',c,c'\in\Real$ and $b,d\in\C$.
Putting $B=pPR$ and using (\ref{Hecke}), the matrix $B$ must satisfy
$(B-1)(B+\al)=0$, with $\alpha=p/p'$. This leads to the following
conditions~:
\beq
 \begin{array}{l}
 (a+a'+\al-1)d=0,\\[1mm]
 dd^*=(a+\al)(1-a)=(a'+\al)(1-a'),\\[1mm]
 (c+c'+\al-1)b=0,\\[1mm]
 bb^*=(c+\al)(1-c)=(c'+\al)(1-c').
 \end{array}
\label{cond1}
\eeq
Finally, there are the conditions following from the YBE for $R$ (or
for $pR$); these are rather numerous and we will not write them here
explicitly. It is, however, possible to solve the system of equations
completely. The following solutions emerge~:
\def\rn{\romannumeral}
\begin{itemize}

\item[({\rn 1})] $d=0$, $b=0$.\\
The YBE leads to $a=a'=c=c'=1$ or $a=a'=c=c'=-\al$. Thus the solutions
are
\beq
pR = P \qquad\hbox{or}\qquad pR=-\al P.
\eeq

\item[({\rn 2})] $d=0$, $b\ne 0$.\\
The YBE implies $cc'=0$. There are two cases to distinguish. If $c=0$,
then $c'=1-\al$ and $bb^*=\al$. Thus $\al$ must be positive. Putting
$\al=q^2$ ($q$ real), the most general solution in this case is~:
\beq
pR= \left(
 \begin{array}{cccc} \{1,-q^2\}&0&0&0\\ 0&\pm q&0&0 \\ 0&1-q^2&\pm q&0 \\
  0&0&0&\{1,-q^2\} \end{array}
 \right),
\label{lower}
\eeq
where $\{1,-q^2\}$ indicates that for this entry one can choose either
$1$ or else $-q^2$.

In the second case, $c'=0$ and $c=1-\al$. Using the same notation,
this leads to
\beq
pR= \left(
 \begin{array}{cccc} \{1,-q^2\}&0&0&0\\ 0&\pm q&1-q^2&0 \\ 0&0&\pm q&0 \\
  0&0&0&\{1,-q^2\} \end{array}
 \right).
\label{upper}
\eeq
The classical $R$-matrix, and its quantum double, belong to this class
of solutions.

\item[({\rn 3})] $d\ne 0$.\\
In this case, the YBE implies $c=c'=(1-\al)/2$. The last equation
of (\ref{cond1}) implies $bb^*=\left((\al+1)/2\right)^2$, and the
YBE implies that $dd^*=cc'=\left((1-\al)/2\right)^2$. From a number
of conditions following from the explicit form of the YBE, one
deduces that $\al\geq 0$, so again we put $\al=q^2$; then
$a=(1-q^2)/2\pm q$ and $a'=(1-q^2)/2\mp q$, and $b$ and $d$
must be real. The most general solution in this case reads~:
\beq
pR= \left(
 \begin{array}{cccc}
 {1-q^2\over 2}+\ep q&0&0&\ep'{1-q^2\over 2}\\
 0&\ep''{1+q^2\over 2}&{1-q^2\over 2}&0 \\[1mm]
 0&{1-q^2\over 2}&\ep''{1+q^2\over 2}&0 \\
 \ep'{1-q^2\over 2}&0&0&{1-q^2\over 2}-\ep q
 \end{array}
 \right).
\label{sol}
\eeq
Herein, $\ep,\ep'$ and $\ep''$ are three independent signs~:
$\ep,\ep',\ep''\in\{-1,+1\}$.
\end{itemize}

Let us now consider explicitly the ``statistics'' of the deformed
boson operators in these situations. For case ({\rn 1}), the relations
are easy to work out and rather trivial. For case ({\rn 2}), (\ref{lower})
and (\ref{upper}) are similar; here, we only consider
(\ref{upper}). Note that the matrix (\ref{upper}) can in its most
general form be rewritten as follows~:
\beq
pR= \left(
 \begin{array}{cccc} {1-q^2\over 2}+\ep{1+q^2\over 2}&0&0&0\\
  0&\ep' q&1-q^2&0 \\
  0&0&\ep' q&0 \\
  0&0&0&{1-q^2\over 2}+\ep''{1+q^2\over 2}
 \end{array}
 \right),
\eeq
with three independent signs~: $\ep,\ep',\ep''\in\{-1,+1\}$. The
relations following from (\ref{pAA}) read~:
\beq
 \begin{array}{l}
 (\ep-1)A_1A_1 = (\ep''-1)A_2A_2 =0 ,\\[1mm]
 A_1A_2 = \ep' q^{-1} A_2 A_1 .
 \end{array}
\label{49}
\eeq
The relations among $\Ad_i$ follow by applying the antihomomorphism,
and the relations (\ref{pAA+}) become~:
\beq
 \begin{array}{l}
 A_1\Ad_1-\left({q^{-2}-1\over 2}+\ep{q^{-2}+1\over 2}\right)\Ad_1A_1=1,\\[1mm]
 A_i\Ad_j=\ep'q^{-1}\Ad_jA_i,\qquad(i\ne j),\\[1mm]
 A_2\Ad_2-\left({q^{-2}-1\over 2}+\ep''{q^{-2}+1\over 2}\right)\Ad_2A_2=1+
  (q^{-2}-1) \Ad_1 A_1 .
 \end{array}
\label{410}
\eeq
With $\ep=\ep'=\ep''=1$ and $q$ replaced by $q^{-1}$, these relations
coincide with (\ref{PW}). When $q=1$ in (\ref{49}) and (\ref{410}), this
becomes
\beq
 \begin{array}{ll}
 (\ep-1)A_1^2 = 0, & A_1\Ad_1-\ep\Ad_1A_1=1,\\[1mm]
 A_1A_2-\ep'A_2A_1=0,\qquad & A_i\Ad_j-\ep'q^{-1}\Ad_jA_i=0,\qquad(i\ne j),\\[1mm]
 (\ep''-1)A_2^2=0, & A_2\Ad_2-\ep''\Ad_2A_2=1 .
 \end{array}
\eeq
For $\ep=1$ (resp.\ $-1$), $(A_1,\Ad_1)$ is a boson (resp.\ a fermion)
annihilation and creation operator pair. Similarly, for
$\ep''=1$ (resp.\ $-1$), $(A_2,\Ad_2)$ is a boson (resp.\ a fermion)
pair. For $\ep'=1$ (resp.\ $-1$), the two modes commute (resp.\
anticommute).

Finally, consider the peculiar statistics implied by case ({\rn 3}).
From (\ref{pAA}), the following two relations are obtained~:
\beq
 \begin{array}{l}
 (1-\ep q)A_1A_1 = \ep' (1+\ep q) A_2A_2 , \\[1mm]
 A_1A_2 = \ep'' A_2 A_1 ,
 \end{array}
\label{411}
\eeq
and similarly for the quadratic relations in $\Ad_i$. From (\ref{pAA+}),
one finds~:
\beq
 \begin{array}{rcl}
 A_1\Ad_1+\Ad_1A_1&=&1+{\ep q^{-1}+1\over 2}\left( (\ep q^{-1}+1)\Ad_1A_1
  + (\ep q^{-1}-1)\Ad_2A_2 \right), \\[1mm]
 A_1\Ad_2-\ep''\Ad_2A_1&=&{q^{-2}-1\over 2}(\ep'\Ad_1A_2+\ep''\Ad_2A_1),\\[1mm]
 A_2\Ad_1-\ep''\Ad_1A_2&=&{q^{-2}-1\over 2}(\ep'\Ad_2A_1+\ep''\Ad_1A_2),\\[1mm]
 A_2\Ad_2+\Ad_2A_2&=&1+{\ep q^{-1}-1\over 2}\left( (\ep q^{-1}+1)\Ad_1A_1
  + (\ep q^{-1}-1)\Ad_2A_2 \right).
 \end{array}
\label{412}
\eeq
When $q=1$ in (\ref{411}) and (\ref{412}), one can verify that it is a system
of one boson pair and one fermion pair which commute or anticommute (depending
on whether $\ep''$ is $1$ or $-1$).

The relations (\ref{49})--(\ref{410}) and (\ref{411})--(\ref{412})
can be considered as a further generalisation of the Pusz-Woronowicz
relations (\ref{PW}) in terms of an $R$-matrix which is still
compatible with an associative algebra.
All the classical situations, such as a system of two bosons or of two
fermions, are easily seen to be special limits of the above deformed cases.

\section*{Acknowledgements}

The author would like to thank Dr.\ C.\ Quesne (Universit\'e Libre de
Bruxelles) for useful discussions.

\newpage
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\end{enumerate}
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