%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Latex file: % T.D. Palev and J. Van der Jeugt % The quantum superalgebra Uq[osp(1/2n)]: deformed para-Bose % operators and root of unity representations % J. Phys. A: Math. Gen. 28 (1995), 2605-2616. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentstyle[12pt]{article} \headheight=0mm \headsep=0mm \oddsidemargin=1mm \evensidemargin=1mm \textheight=230mm \textwidth=165mm % % definitions concerning automatic numbering of theorems, lemmas, etc. % \newtheorem{theo}{Theorem} \newtheorem{defi}[theo]{Definition} \newtheorem{lemm}[theo]{Lemma} \newtheorem{coro}[theo]{Corollary} \newtheorem{rema}[theo]{Remark} \newtheorem{prop}[theo]{Proposition} % % some definitions used in this paper % \def\beq{\begin{equation}} \def\eeq{\end{equation}} \def\bea{\begin{eqnarray}} \def\eea{\end{eqnarray}} \def\nn{\nonumber} \def\ba{\begin{array}} \def\ea{\end{array}} \def\lb{[\![} \def\rb{]\!]} \def\q{\hat q} \def\ep{\epsilon} \def\vep{\varepsilon} % % some font : % Blackboard boldface (\Bb) % This AMS-tex font is used for the symbols of Natural, Integer, Real and % Complex numbers. If the msbs-fonts are not available on your system, % delete the following 14 lines and replace them by the single line : % \def\C{\tt C} \def\Real{\tt R} % \font\twelvemsbm=msbm10 at 12 true pt \font\eightmsbm=msbm8 \font\sevenmsbm=msbm7 \newfam\msbmfam \textfont\msbmfam=\twelvemsbm \scriptfont\msbmfam=\eightmsbm \scriptscriptfont\msbmfam=\sevenmsbm \def\Bb#1{{\fam\msbmfam\relax#1}} % Blackboard boldface % % maths. symbols for real, complex numbers, etc., % \def\Real{\Bb R} \def\C{\Bb C} \def\Zah{\Bb Z} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \begin{document} %\addtolength{\baselineskip}{2mm} \renewcommand{\thefootnote}{\fnsymbol{footnote}} \setcounter{footnote}{1} \begin{center} %TITLE {\Large \bf The quantum superalgebra $U_q[osp(1/2n)]$~: deformed para-Bose operators and root of unity representations} \vspace{2cm} %AUTHORS {\bf T.D.\ Palev\footnote{Permanent address: Institute for Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria; E-mail~: palev@bgearn.bitnet} and J.\ Van der Jeugt\footnote{Senior Research Associate of N.F.W.O. (National Fund for Scientific Research of Belgium); E-mail address: Joris.VanderJeugt@rug.ac.be}} \vspace{0.5cm} %ADDRESS Department of Applied Mathematics and Computer Science \\ University of Ghent, Krijgslaan 281-S9, B-9000 Gent, Belgium \end{center} \vspace{2cm} \noindent {\bf Abstract}: We recall the relation between the Lie superalgebra $osp(1/2n)$ and para-Bose operators. The quantum superalgebra $U_q[osp(1/2n)]$, defined as usual in terms of its Chevalley generators, is shown to be isomorphic to an associative algebra generated by so-called pre-oscillator operators satisfying a number of relations. From these relations, and the analogue with the non-deformed case, one can interpret these pre-oscillator operators as deformed para-Bose operators. Some consequences for $U_q[osp(1/2n)]$ (Cartan-Weyl basis, Poincar\'e-Birkhoff-Witt basis) and its Hopf subalgebra $U_q[gl(n)]$ are pointed out. Finally, using a realization in terms of ``$q$-commuting'' $q$-bosons, we construct an irreducible finite-dimensional unitary Fock representation of $U_q[osp(1/2n)]$ and its decomposition in terms of $U_q[gl(n)]$ representations when $q$ is a root of unity. \vspace{1cm} \noindent Short title: Quantum superalgebra $U_q[osp(1/2n)]$ \vspace{1cm} \noindent Phys. Abs. Class. Nos.~: 0210, 0220, 0365 \newpage \renewcommand{\theequation}{\arabic{section}.{\arabic{equation}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setcounter{equation}{0} \section{Introduction} It has been established in a number of papers \cite{Palev78,Ohnuki,Palev82,Jag,Blank,Biswas,FlatoF,Okubo} that the proper mathematical framework of a system of $n$ para-Bose operators is the theory of (unitarizable) representations of the Lie superalgebra $osp(1/2n)$. This Lie superalgebra is $B(0/n)$ in Kac's notation, and its finite-dimensional irreducible representations (irreps) are completely classified (even their characters are known). Unfortunately, these representations are not unitarizable (see e.g.\ Ref.~\cite{Hughes}). The infinite-dimensional unitarizable representations have not been classified, and so far only certain special cases (corresponding to parastatistics of order $p$) have been considered. Following the recent interest in $q$-deformations, a number of papers have dealt with deformed parastatistics and in particular with $q$-deformed para-Bose operators from different points of view~\cite{Greenberg87}--\cite{Green94}. The definition of $q$-deformed para-Bose operators depends highly upon the framework one is working in, and most often one is inspired by considering $q$-analogues of ordinary Bose operators. Such approaches usually lead to deformed para-Bose operators that are incompatible with a Hopf algebra structure. From our point of view, the natural {\em ansatz} is the equivalence between $osp(1/2n)$ irreps and para-Bose operator representations. The $q$-deformed superalgebra $U_q[osp(1/2n)]$, a Hopf superalgebra, is by now a classical concept~\cite{Chaichian}--\cite{dHoker}. Inside $U_q[osp(1/2n)]$ one can in a natural way identify a set of elements with $q$-deformed para-Bose operators. This leads to two basic results~: an alternative definition of $U_q[osp(1/2n)]$ in terms of non-Chevalley generators (the deformed para-Bose operators) satisfying a number of relations; and the $q$-analogue of all triple relations defining para-Bose statistics. These relations, on their turn, imply the existence of a Poincar\'e-Birkhoff-Witt theorem for $U_q[osp(1/2n)]$ and thus a basis in terms of normally ordered monomials of Cartan-Weyl generators, the expressions of which become extremely simple in terms of the deformed para-Bose operators. Apart from these consequences, we also state a number of results for the Hopf subalgebra $U_q[gl(n)]\subset U_q[osp(1/2n)]$ and its realization in terms of the deformed para-Bose operators. Just as the defining relations for ordinary para-Bose operators are automatically satisfied by canonical Bose operators (leading to the oscillator representation of $osp(1/2n)$), we construct in the present paper a set of deformed Bose operators that satisfy, as a particular example, the deformed para-Bose operator relations. These deformed Bose operators are (up to some factor) equal to the usual $q$-bosons, but different modes ``$q$-commute'' instead of being commutative. The Fock space for these operators is constructed, and shown to be unitarizable only if $q$ is a primitive root of unity, in which case the representation becomes finite-dimensional. These finite-dimensional representations are given explicitly, and their decomposition into irreducible representation of $U_q[gl(n)]$ is considered. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The para-Bose algebra $pB_n$ and its relation to the Lie superalgebra $osp(1/2n)$} \setcounter{equation}{0} Let $A^\pm_i$ ($i=1,\ldots,n$) be a system of $n$ para-Bose operators. The defining relation for para-Bose operators (parabosons), introduced in quantum field theory by Green~\cite{Green} as a possible generalization of integer spin field statistics (see~\cite{Ohnuki} for a general introduction to parastatistics), is given by~: \beq [ \{A_i^\xi,A_j^\eta \},A_k^\ep ]= (\ep -\eta)\delta_{jk}A_i^\xi +(\ep -\xi )\delta_{ik}A_j^\eta,\quad(\xi,\eta,\ep=\pm 1). \label{pBose} \eeq The relations~(\ref{pBose}) generalize the canonical commutation relations of ordinary Bose operators (bosons) $a^\pm_i$~: \beq [a_i^-,a_j^+]=\delta_{ij}, \quad [a_i^-,a_j^-]=[a_i^+,a_j^+]=0, \label{Bose} \eeq and it is trivial to verify that the $a_i^\pm$ do indeed satisfy~(\ref{pBose}). The para-Bose algebra $pB_n$ is defined as the associative algebra with unity over $\C$ with generators $A^\pm_i$, subject to the relations~(\ref{pBose}). In fact, $pB_n$ is turned into a superalgebra (associative $\Zah_2$-graded algebra) by the requirement $\deg(A^\pm_i) = \bar 1$, $\forall i\in\{1,\ldots,n\}$, where $\Zah_2 =\{\bar 0, \bar 1\}$. By defining the supercommutator between any two homogeneous elements $a$ and $b$ of $pB_n$ by \beq \lb a,b \rb =ab-(-1)^{deg(a)deg(b)}ba , \label{supercomm} \eeq and extending it by bilinearity to the whole algebra, $pB_n$ is turned into a Lie superalgebra. Thus $n$ pairs of para-Bose operators generate a Lie superalgebra~\cite{Omote}, and we shall recall that in the present case this Lie superalgebra can be identified with $osp(1/2n)$~\cite{Palev78}. For this purpose, define the Lie superalgebra $osp(1/2n)$ (or $B(0/n)$ in Kac's notation~\cite{Kac}) as the set of $(2n+1)\times (2n+1)$ complex matrices of the form \beq \left(\begin{array}{ccc} 0 & x & y \\ y^T & d & e \\ -x^T & f &-d^T \end{array}\right), \label{matrix} \eeq where $T$ stands for transposition, $d,e$ and $f$ are $n\times n$ matrices with $e^T=e$ and $f^T=f$, and $x,y$ are $1\times n$ matrices. The even subalgebra $osp(1/2n)_{\bar 0}$ consists of all matrices with $x=y=0$ and is isomorphic to the symplectic Lie algebra $sp(2n)$. The odd subspace $osp(1/2n)_{\bar 1}$ consists of all matrices with $d=e=f=0$. The supercommutator between homogeneous elements is defined by means of~(\ref{supercomm}), and extended by bilinearity. Let $E_{kl}$ denote the $(2n+1)\times(2n+1)$ matrix with 1 at the intersection of row $k$ and column $l$, and zero elsewhere (where rows and columns are labelled from 0 to $2n$). The Cartan subalgebra $H$ of $osp(1/2n)$ is spanned by the elements $H_i=-E_{ii}+E_{n+i,n+i}$ ($i=1,\ldots,n$). With a suitable basis $\vep_i$ ($i=1,\ldots,n$) of the dual space $H^*$, the roots of $osp(1/2n)$ consist of $2n$ odd roots $\pm\vep_i$ and $2n^2$ even roots $\pm\vep_i\pm\vep_j$~\cite{Kac}. The Lie superalgebra $osp(1/2n)$ has the usual root space decomposition, with all root spaces one-dimensional. The root vectors corresponding to the odd roots can be written as follows~: \bea \vep_i \quad&:&\quad A_i^- =\sqrt{2} (E_{0,i}-E_{i+n,0}),\quad i=1,\ldots,n, \label{A-}\\ -\vep_i\quad&:&\quad A_i^+ =\sqrt{2} (E_{0,i+n}+E_{i,0}), \quad i=1,\ldots,n. \label{A+} \eea Therefore the anticommutator $\{A_i^{-\xi}, A_j^{-\eta}\}$ is a root vector with root $\xi\vep_i+\eta\vep_j$ ($\xi,\eta=\pm$), and one finds $\{A^-_i,A^+_i\}=-2H_i$ ($i,j=1,\ldots,n$). This implies that the odd root vectors $A^\pm_i$ generate the whole Lie superalgebra $osp(1/2n)$. Moreover, one can verify that the elements (\ref{A-})-(\ref{A+}) satisfy indeed~(\ref{pBose}). This gives rise to the following \begin{prop} The para-Bose algebra $pB_n$ is isomorphic to the universal enveloping algebra $U[osp(1/2n)]$ of $osp(1/2n)$. The finite dimensional subspace \beq {\rm span } \{\{A_i^\xi,A_j^\eta \}, A_k^\ep \vert i,j,k=1,\ldots,n;\; \xi,\eta,\ep=\pm\} \label{osp12n}, \eeq endowed with the supercommutator~(\ref{supercomm}), is a Lie superalgebra isomorphic to $osp(1/2n)$. \end{prop} Note that the even elements of $osp(1/2n)$ are spanned by the $\{A_i^\xi, A_j^\eta\}$, thus $sp(2n)={\rm span}\{ \{A_i^\xi, A_j^\eta\} | i, j = 1, \ldots,n;\; \xi,\eta=\pm\}$. In particular, from~(\ref{pBose}) one can derive a compact expression for the commutation relations of the $sp(2n)$ basis elements~: \bea [\{A_i^\xi,A_j^\eta\},\{A_k^\epsilon,A_l^\varphi\} ]& =& (\epsilon -\eta)\delta_{jk}\{A_i^\xi,A_l^\varphi\}+ (\epsilon -\xi )\delta_{ik}\{A_j^\eta,A_l^\varphi\} \nn\\ & &+(\varphi-\eta)\delta_{jl}\{A_i^\xi,A_k^\epsilon\}+ (\varphi-\xi)\delta_{il}\{A_j^\eta,A_k^\epsilon\}. \label{sp2n} \eea As a consequence from Proposition~1, determining all representations of the para-Bose operators is completely equivalent with finding all representations of the Lie superalgebra $osp(1/2n)$. For finite-dimensional irreducible representations of $osp(1/2n)$, there exists a character formula~\cite{Kac}, but explicit formulae for matrix elements are not available in the literature. In para-Bose statistics, one is rather interested in infinite dimensional unitarizable representations. The Fock space corresponding to the ordinary Bose operators~(\ref{Bose}) provides one such example; representations corresponding to a fixed order of parastatistics~\cite{Green,Greenberg} provide in principle others; but apart from that no general theory of unitarizable representations of $osp(1/2n)$ exists. Besides the definition of $U[osp(1/2n)]$ in terms of generators $A^\pm_i$ subject to the relations~(\ref{pBose}), there is an alternative definition in terms of Chevalley generators. This is perhaps the definition that most readers are more familiar with. Let $(\alpha_{ij})$ be a Cartan matrix chosen as an $n \times n $ symmetric matrix with \beq \alpha_{nn}=1,\quad \alpha_{ii}=2, \quad \alpha_{i,i+1}=\alpha_{i+1,i}=-1, \quad i=1,\ldots,n-1,\quad {\rm all \; other}\; \alpha_{ij}=0. \label{cartan} \eeq Again $U[osp(1/2n)]$ is defined as an associative superalgebra in terms of a number of generators subject to relations. The generators are the elements $h_i, e_i, f_i$ ($i=1,\ldots,n$); the relations are the Cartan-Kac relations \bea &&[h_i,h_j]=0,\nn \\ &&[h_i,e_j]=\alpha_{ij}e_j,\quad [h_i,f_j]=-\alpha_{ij}f_j,\quad \nn\\ &&[e_i,f_j]=\delta_{ij}h_i\quad{\rm except\ for}\quad i=j=n, \label{CK}\\ && \{ e_n,f_n\}=h_n ; \nn \eea the $e$-Serre relations \bea && [e_i,e_j]=0, \quad{\rm for}\quad |i-j|>1,\nn\\ && e_i^2e_{i+1}-2e_ie_{i+1}e_i+e_{i+1}e_i^2=0, \quad i=1,\ldots,n-1, \nn\\ && e_i^2e_{i-1}-2e_ie_{i-1}e_i+e_{i-1}e_i^2=0, \quad i=2,\ldots,n-1,\label{eSerre} \\ && e_n^3e_{n-1}- (e_n^2e_{n-1}e_n+e_ne_{n-1}e_n^2)+ e_{n-1}e_n^3=0; \nn \eea and the $f$-Serre relations obtained from (\ref{eSerre}) by replacing everywhere $e_i$ by $f_i$. The grading on the superalgebra is induced from the grading on its generators~: $\deg (e_n)=\deg(f_n)=\bar 1$ and $\deg(e_i)=\deg(f_i)=\bar 0$ for $i=1,\ldots,n-1$. Although the present definition of $U[osp(1/2n)]$ looks somewhat more complicated than the one given in terms of the para-Bose operators $A_i^\pm$, it cannot be avoided when turning to a Hopf superalgebra deformation $U_q[osp(1/2n)]$. Such deformations are well defined only in a Chevalley basis. In fact, relations like the ones given below, expressing the para-Bose operators in terms of the Chevalley generators, will be reconsidered in the following section to yield a definition of deformed para-Bose operators in terms of the Chevalley generators of $U_q[osp(1/2n)]$. Here, the relations read~: \bea &&A_i^-=-\sqrt{2} [e_i,[e_{i+1},[e_{i+2},[\ldots,[e_{n-2},[e_{n-1},e_n] \ldots ],\quad i=1,\ldots,n-1 \nn\\ &&A_i^+=\sqrt{2} [\ldots [f_n,f_{n-1}]_{q},f_{n-2}],\ldots], f_{i+2}],f_{i+1}],f_{i}], \quad i=1,\ldots,n-1, \\ &&A_n^-=-\sqrt{2}e_n, \qquad A_n^+=\sqrt{2}f_n. \nn \eea Clearly, the expressions of the Chevalley generators in terms of para-Bose operators also exist~: \bea &&e_i={1\over 2}\{A_i^-,A_{i+1}^+\},\quad f_i={1\over 2}\{A_i^+,A_{i+1}^-\},\quad i=1,\ldots,n-1,\label{ei}\\ &&h_i={1\over 2}\{A_{i+1}^-,A_{i+1}^+\}- \{A_{i}^-,A_{i}^+\}, \quad i=1,\ldots,n-1, \label{hi}\\ &&e_n=-{1\over \sqrt 2}A_n^-, \quad f_n={1\over \sqrt 2}A_n^+, \quad h_n=-{1\over 2}\{A_n^-,A_n^+\}. \eea An important statement can be deduced from these relations. From (\ref{CK})-(\ref{eSerre}) it follows that the enveloping algebra of $gl(n)$ is generated by $h_i$ ($i=1,\ldots,n$) and $e_i,f_i$ ($i=1,\ldots,n-1$). Then (\ref{ei})-(\ref{hi}) and (\ref{sp2n}) show that $U[gl(n)]$, a subalgebra of $U[osp(1/2n)]$, is generated by the elements $\{A_i^-,A_j^+\}$ ($i,j=1,\ldots,n$). In other words~: \beq gl(n)={\rm span }\{\{A_i^-,A_j^+\}| i,j=1,\ldots,n\}, \label{gln} \eeq endowed with the usual commutator product. Replacing in (\ref{gln}) the para-Bose operators with the ordinary Bose operators (\ref{Bose}) yields the familiar Schwinger realization of $gl(n)$; but observe that in the superalgebra grading the para-Bose and Bose operators are odd (``fermionic'') operators rather than even (``bosonic'') operators. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setcounter{equation}{0} \section{The quantum superalgebra $U_q[osp(1/2n)]$ and $q$-de\-for\-med para-Bose operators} In the present section we consider the well known quantum superalgebra $U_q[osp(1/2n)]$. A general procedure to construct a so-called Cartan-Weyl basis of $U_q[osp(1/2n)]$ (the analogue of a Cartan-Weyl basis for the Lie superalgebra $osp(1/2n)$) was given in Ref.~\cite{Khoroshkin}. We follow this procedure, and identify -- as in the non-deformed case of Section~1 -- the Cartan-Weyl basis elements corresponding to the odd roots as ``deformed para-Bose operators.'' The remaining questions that are treated in this section are~: the alternative definition of $U_q[osp(1/2n)]$ in terms of deformed para-Bose operators; the triple relations between deformed para-Bose operators (the analogue of~(\ref{pBose})); a Poincar\'e-Birkhoff-Witt (PBW) theorem for $U_q[osp(1/2n)]$; the Cartan-Weyl basis of the subalgebra $U_q[gl(n)]$ and its consequences. First, we introduce $U_{q}\equiv U_{q}[osp(1/2n)]$ by means of its classical definition in terms of the Cartan matrix~(\ref{cartan}). $U_{q}$ is the associative superalgebra with unity over $\C$, generated by the elements $k_i^\pm=q^{\pm h_i}$, $e_i, f_i$ ($i=1,\ldots,n$), subject to the following relations~: the Cartan-Kac relations \bea &&k_ik_i^{-1}=k_i^{-1}k_i=1, \quad k_ik_j=k_jk_i, \nn \\ &&k_ie_j=q^{\alpha_{ij}}e_jk_i, \quad k_if_j=q^{-\alpha_{ij}}f_jk_i, \nn\\ &&[e_i,f_j]=\delta_{ij}(k_i-k_i^{-1})/(q-q^{-1}) \quad {\rm except\ for}\quad i=j=n, \label{qCK}\\ && \{ e_n,f_n\} =(k_n-k_n^{-1})/(q-q^{-1}); \nn \eea the $e$-Serre relations \bea &&[e_i,e_j]=0, \quad {\rm for}\quad |i-j|>1, \nn\\ &&e_i^2e_{i+1}-(q+q^{-1})e_ie_{i+1}e_i+e_{i+1}e_i^2=0, \quad i=1,\ldots,n-1, \nn\\ &&e_i^2e_{i-1}-(q +q^{-1})e_ie_{i-1}e_i+e_{i-1}e_i^2=0, \quad i=2,\ldots,n-1, \label{qeSerre} \\ &&e_n^3e_{n-1}+(1-q -q^{-1})(e_n^2e_{n-1}e_n+e_ne_{n-1}e_n^2)+ e_{n-1}e_n^3=0;\nn \eea and the $f$-Serre relations obtained from above by replacing everywhere $e_i$ by $f_i$. The grading on $U_{q}$ is induced from the requirement that the generators $e_n,\; f_n$ are odd and all other generators are even. It is known that $U_{q}$ can be endowed with a comultiplication $\Delta$, a counit $\vep$ and an antipode $S$, turning it into a Hopf superalgebra; here we shall not be concerned with this additional structure. Following the procedure outlined in Ref.~\cite{Khoroshkin} one determines the Cartan-Weyl elements corresponding to the odd roots, and thus we define the deformed para-Bose operators as follows (see also~\cite{Palev93A})~: \bea && A_i^-=-\sqrt{2} [e_i,[e_{i+1},[e_{i+2},[\ldots,[e_{n-2},[e_{n-1},e_n]_{q^{-1}} ]_{q^{-1}}\ldots ]_{q^{-1}},\nn\\ && A_i^+=\sqrt{2} [\ldots [f_n,f_{n-1}]_{q},f_{n-2}]_{q},\ldots]_{q}, f_{i+2}]_{q},f_{i+1}]_{q},f_{i}]_{q}, \quad i=1,\ldots,n-1, \label{qAe}\\ && A_n^-=-\sqrt{2}e_n, \quad A_n^+=\sqrt{2}f_n, \nn \eea where $[u,v]_q=uv-qvu$. Besides these, we also introduce $n$ even ``Cartan" elements \beq L_i=k_ik_{i+1}\ldots k_n=q^{H_i} {\rm \ where\ } H_i=h_i+h_{i+1}+\ldots +h_n, \quad i=1,\ldots,n. \label{Li} \eeq We shall call the set of operators $A_i^\pm, L_i^\pm$ pre-oscillator operators, for reasons that will be obvious in the following section. Using their definition (\ref{qAe})-(\ref{Li}) and the defining relations (\ref{qCK})-(\ref{qeSerre}), one can also determine the Chevalley generators in terms of the pre-oscillator operators ($i\ne n$)~: \bea && e_i=-{q\over 2}\{A_i^-,A_{i+1}^+\}L_{i+1}^{-1},\quad f_i=-{1\over 2q}L_{i+1}\{A_i^+,A_{i+1}^-\}, \nn\\ && e_n=-(2)^{-1/2}A_n^-,\quad f_n=(2)^{-1/2}A_n^+ . \label{qeA} \eea Now, one can determine the relations between the pre-oscillator operators. A set of such relations were already obtained in Ref.~\cite{Palev93A}, but with the purpose of giving an alternative definition of $U_{q}[osp(1/2n)]$ in terms of deformed para-Bose or pre-oscillator operators it would be interesting to find the direct analogue of (\ref{qCK})-(\ref{qeSerre}), and thus present a minimal set of relations. This is given in the following~: \begin{prop} The relations of $U_{q}[osp(1/2n)]$ in terms of its Chevalley generators hold if and only if the pre-oscillator operators satisfy ($i,j=1,\ldots,n$, $\xi=\pm$)~: \bea && L_iL_i^{-1}=L_i^{-1}L_i=1, \quad L_iL_j=L_jL_i, \nn\\ && L_iA_j^{\pm}={q}^{\mp\delta_{ij}}A_j^{\pm}L_i, \nn\\ && \{A_i^-,A_i^+\}=-2(L_i- L_i^{-1})/(q-q^{-1}), \label{minim}\\ && [\{A_i^{-\xi},A_{i\pm 1}^{\xi}\},A_j^{-\xi}]_{{q}^{\pm \delta _{ij}}}=-2\xi \delta _{j,i \pm 1}L_j^{\pm \xi}A_i^{-\xi},\nn\\ && [\{A_{n-1}^\xi,A_n^\xi\},A_n^\xi]]_{q}=0 . \nn \eea \end{prop} We shall refer to (\ref{minim}) as the pre-oscillator realization of $U_{q}$. Although the definition of $U_{q}[osp(1/2n)]$ in terms of deformed para-Bose operators is more complicated than in the non-deformed case, where only one relation~(\ref{pBose}) was necessary, it should be observed that the present definition by means of (\ref{minim}) is certainly not more involved than the list of classical relations, i.e.\ the Cartan-Kac, the $e$-Serre and the $f$-Serre relations. The remaining advantage of the definition by means of Chevalley generators is the simplicity of the other Hopf superalgebra functions $\Delta, \vep$ and $S$ which become very complicated expressions on $A_i^\pm$ and $L_i^{\pm 1}$, although they are certainly well defined. Next, we are concerned with deriving the analogue of (\ref{pBose}), i.e.\ all triple relations between deformed para-Bose operators. These relations are derived using (\ref{minim}), and they are far more complicated than the classical relation (\ref{pBose}). In fact, it would be quite impossible to cast all of them in a single expression. Nevertheless, we think they are of importance since they constitute the direct Hopf algebra $q$-deformation of para-Bose statistics. In the following list, we use the abbreviation~: \beq \tau_{i_1,i_2,\ldots,i_k}= \left\{ \begin{array}{rl} -1, & {\rm if\ }i_1>i_2>\ldots >i_k,\\ 1, & {\rm if\ }i_1i_2>\ldots >i_k,\\ 0 & {\rm otherwise.} \end{array}\right. \eeq A set of Cartan-Weyl elements of $U_{q}[gl(n)]$ has been considered before~\cite{Palev91}, and consists of $n$ ``Cartan" elements $L_1, L_2,\ldots,L_n$ and $n(n-1)$ root vectors $e_{ij}$, $i\neq j=1,\ldots,n$. Remember that $e_{ij}$ is positive if $ij$. Among the {\em prv}, the normal order induced from~(\ref{ord1})-(\ref{ord2}) yields~: \beq e_{ij}0$ and $e_{kl}<0$~: \bea [e_{ij},e_{kl}]&=&\left( (q-q^{-1}) \theta_{jkil}e_{kj}e_{il} -\delta_{il}\theta_{jk}e_{kj}+ \delta_{jk}\theta_{il}e_{il}\right) L_k L_i^{-1} \nn\\ &&+L_l L_j^{-1}\left(-(q-q^{-1})\theta_{kjli}e_{il}e_{kj} -\delta_{il}\theta_{kj}e_{kj}+\delta_{jk}\theta_{li}e_{il} \right) \nn\\ && + \delta_{il}\delta_{jk} (L_i L_j^{-1} - L_i^{-1} L_j)/(q-q^{-1}); \label{g2} \eea \item Set $\xi=1$ if $0e_{ij}>e_{kl}$. Then \beq e_{ij}e_{kl}-q^{\xi(\delta_{ik}-\delta_{il}-\delta_{jk}+\delta_{jl})} e_{kl}e_{ij} =\delta_{jk}e_{il}+(q-q^{-1})\tau_{ljki}e_{kj}e_{il}. \label{g3} \eeq \end{enumerate} To obtain the above relations we have considered Equations (3.10)-(3.14) from Ref.~\cite{Palev91} only for the even generators, first replacing $q$ by $-q$ and then $q^{e_{ii}}$ by $L_i$. The link with the present operators is now given by~: \beq e_{ij}=-{1\over 2}L_j^{-1} \{A_i^-,A_j^+\} {\rm \ for\ } ij. \label{eij} \eeq It is tedious but straightforward to verify that (\ref{eij}) satisfy indeed the relations (\ref{g1})-(\ref{g3}). To conclude this section, observe that for $U_{q}[osp(1/2n)]$, resp.\ for $U_{q}[gl(n)]$, the expressions of the Cartan-Weyl root vectors in terms of the deformed para-Bose operators are the same as the expressions of the Cartan-Weyl basis elements in terms of the non-deformed para-Bose operators for $osp(1/2n)$, resp.\ for $gl(n)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \setcounter{equation}{0} \section{Realization of $U_{q}[osp(1/2n)]$ in terms of deformed Bose oscillators} We have seen that the canonical Bose operators~(\ref{Bose}) satisfy the para-Bose relations~(\ref{pBose}). As a consequence the Fock space built on the Bose operators forms a representation of the para-Bose algebra $pB_n\equiv U[osp(1/2n)]$, usually referred to as the oscillator representation of $osp(1/2n)$. In the present section we shall define a new set of operators which can be interpreted as deformed Bose operators. They satisfy the deformed para-Bose relations~(\ref{minim}) and hence their Fock space forms a representation of $U_{q}[osp(1/2n)]$, the analogue of the usual oscillator representation. Surprisingly, the unitarity conditions lead to the condition that $q$ should be a root of unity, yielding a finite-dimensional Fock space. This Fock representation is studied in detail, and in particular its decomposition with respect to $U_{q}[gl(n)]$ is considered, giving rise to certain root of unity representations for this quantum algebra. Define a set of operators \beq a_i^\pm, \; \kappa_i=q^{N_i},\quad i=1,\ldots,n, \eeq satisfying the relations \bea && a_i^-a_i^+ -q^{\pm 1}a_i^+a_i^- ={2\over {q^{1/2}+q^{-1/2}}}\kappa_i^{\mp 1}, \nn\\ && \kappa_i a^{\pm}_j = q^{\pm\delta_{ij}} a^\pm_j \kappa_i, \label{a-t}\\ && a_i^\xi a_j^\eta =q^{\xi \eta} a_j^\eta a_i^\xi , \quad {\rm for\ all\ }ij. \eea Then, the actual matrix elements follow from~(\ref{action}). Note that the subspace spanned by vectors $|m_1,\ldots,m_n\rangle$ with $m_1+\cdots+m_n=m$ ($m$ constant) forms in fact a submodule for the $U_{q}[gl(n)]$ action. Since the matrix elements~(\ref{action}) of $a_i^-$, resp.\ $a_i^+$, are nonzero for $m_i\ne 0$, resp.\ $m_i\ne k$, one deduces that this $U_{q}[gl(n)]$ module is also irreducible. Thus, the $k^n$-dimensional irreducible $U_{q}[osp(1/2n)]$ Fock representation splits into $nk-n+1$ irreducible $U_{q}[gl(n)]$ representation $(m)$ characterized by a ``total number'' $m$ taking values from $0$ up to $n(k-1)$. The dimension of this irreducible representation $(m)$ is equal to the coefficient of $x^m$ in the expansion of \beq (1+x+\cdots+x^{k-1})^n = \left({1-x^k\over 1-x}\right)^n, \eeq or, more explicitly, \beq \dim(m) = \sum_{j_0,j_1,\ldots,j_{k-1}} {n!\over j_0!j_1!\cdots j_{k-1}!}, \eeq where the $j_i$ assume all nonnegative integer values such that $j_0+j_1 + \cdots + j_{k-1}=n$ and $j_1+2j_2 + \cdots + (k-1)j_{k-1}=m$. Of course, the representations $(m)$ obtained by means of deformed Bose operators are only a small part of the so-called type~1 representations of $U_q[gl(n)]$ (see, e.g., Chapter~11 of Ref.~\cite{Chari}, and references therein). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Acknowledgements} We have pleasure in thanking Prof.\ R.\ Jagannathan and Dr.\ N.I.\ Stoilova for stimulating discussions. One of us (TDP) is thankful to Prof.\ G.\ Vanden Berghe for the kind hospitality at the Department of Applied Mathematics and Computer Science, University of Ghent. 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