\magnification 1200 \def\[{[\![} \def\]{]\!]} \def\Z{{\bf Z}} \def\C{{\bf C}} \def\Ch{{\C[[h]]}} \def\l{\ldots} \def\n{\noindent} \def\h{\hat{h}} \def\e{\hat{e}} \def\f{\hat{f}} \def\n{\noindent} \def\t{\theta} \def\s{\smallskip} \def\b{\bigskip} \def\a{\hat{a}} \def\k{{\bar k}} \def\L{{\bar L}} \def\q{{\bar q}} \def\x{{\bar x}} \def\d{\delta} \def\v{\varepsilon} \def\ra{\rangle} \def\la{\langle} \def\bu{\bullet} %\def\o{\odot} \def\o{\circ} \def\c{\centerline} \b\b \baselineskip=16pt \font\twelvebf=cmbx12 \b\b \centerline{\twelvebf Fock representations of the superalgebra sl(n+1$|$m), its quantum } \centerline{\twelvebf analogue U$_{\bf q}$[sl(n+1$|$m)] and related quantum statistics} \leftskip 36pt \vskip 40pt \noindent T.D. Palev$^{\bf a)}$, N.I. Stoilova $^{\bf b)}$\footnote*{\rm Permanent address: Institute for Nuclear Research and Nuclear Energy, Boul. Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria, (e-mail: stoilova@inrne.bas.bg).} and J. Van der Jeugt$^{\bf c)}$\footnote{**}{\rm Research Associate of the Fund for Scientific Research - Flanders (Belgium); (e-mail: Joris.VanderJeugt@rug.ac.be).} \noindent $^{\bf a)}$Institute for Nuclear Research and Nuclear Energy, Boul. Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria, e-mail: tpalev@inrne.bas.bg \n$^{\bf b)}$Abdus Salam International Centre for Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy \n$^{\bf c)}$Department of Applied Mathematics and Computer Science, University of Ghent \n Krijgslaan 281-S9, B-9000 Gent, Belgium \vskip 54pt \n{\bf Abstract.} Fock space representations of the Lie superalgebra $sl(n+1|m)$ and of its quantum analogue $U_q[sl(n+1|m)]$ are written down. The results are based on a description of these superalgebras via creation and annihilation operators. The properties of the underlying statistics are shortly discussed. \b\b\b \n Short title~: Fock representations of $U_q[sl(n+1|m)]$ \vfill \eject \vskip 48pt \leftskip 0pt \n {\bf 1. Introduction} \bigskip\n The quantization of the simple Lie algebras [1] and the basic Lie superalgebras [2] is usually carried out in terms of their Chevalley generators $e_i, f_i, h_i, i=1,\ldots,n$ for an algebra of rank $n$. Recently it was realized that the quantum algebras $U_q[osp(1|2n)]~ [3],~ U_q[so(2n+1)] ~ [4]$,~ $ U_q[osp(2r+1|2m)], r+m=n ~[5],~ {\rm and~~also}~~U_q[sl(n+1)] $ ~[6] and $U_q[sl(n+1|m)]$ ~[7] admit a description in terms of an alternative set of generators $a_i^\pm,\; H_i,\;\;i=1,\ldots,n,$ referred to as (deformed) creation and annihilation operators (CAOs) or generators. This certainly holds also for the corresponding nondeformed Lie superalgebras. The concept of creation and annihilation operators of a simple Lie (super)algebra was introduced in [8]. Let ${\cal A}$ be such an algebra with a supercommutator $\[\;,\;\]$. The root vectors $a_1^{\xi},\l,a_n^{\xi}$ of ${\cal A}$ are said to be creation ($\xi=+$) and annihilation ($\xi=-$) operators of ${\cal A}$, if $$ {\cal A}=lin.env.\{a_i^{\xi},\;\[a_j^{\eta},a_k^{\varepsilon}\]\;|\; i,j,k=1,\l,n;\; \xi, \eta, \varepsilon =\pm \}, \eqno(1) $$ so that $a_1^+,\l,a_n^+~~~~ ({\rm resp.}~~ a_1^-,\l,a_n^-)$ are negative (resp. positive) root vectors of ${\cal A}$. The Fock representations of ${\cal A}$, defined in [8], are constructed in a much similar way as those of Bose or Fermi operators (or their generalizations, the parabosons and parafermions [9]). In a more mathematical terminology the Fock modules of ${\cal A}$ are induced from trivial one-dimensional modules of a subalgebra ${\cal B}$ of ${\cal A}$, namely $$ {\cal B}=lin.env.\{a_i^-,\;\[a_j^{\eta},a_k^{\varepsilon}\]\;|\; i,j,k=1,\l,n;\; \eta, \varepsilon =\pm \}. \eqno(2) $$ The reason for introducing such (more physical) terminology is based on the observation that the creation and the annihilation operators of the orthosymplectic Lie superalgebra (LS) $osp(2r+1|2m)$ have a direct physical significance: $a_1^\pm,\ldots,a_m^\pm~ ({\rm resp.}~~ a_{m+1}^\pm,\ldots,a_n^\pm)$ are para-Bose (resp. para-Fermi) operators [10], operators which generalize the statistics of tensor (resp. spinor) fields in quantum field theory (for $n=m=\infty$) [9]. Since $osp(2r+1|2m)$ is an algebra from the class $B$ in the classification of Kac [11] one may call the paraquantization a $B$-quantization. It was argued in [12] that to each class $A$, $B$, $C$ and $D$ of basic LSs there corresponds a quantum statistics, so that their CAOs can be interpreted as creation and annihilation operators of (quasi)particles, excitations, in the corresponding Fock space(s). This assumption holds for the classes $A$~, $B$~, $C$~, $D$ of simple Lie algebras [13]. It was studied in detail for the Lie algebras $sl(n+1)$ ($A-$statistics) [14], for the LSs $sl(1|m)$ ($A-$superstatistics) [8, 15] and recently for the classical Lie superalgebra $q(n+1)$ [16]. In the present note we report shortly on the properties of quantum statistics, related to the Lie superalgebra $sl(n+1|m)$ and its quantum analogue. These statistics are particular kinds of generalized quantum statistics. There are many publications on the subject especially in the part related to quantum groups (initiated in [17]). Other interesting approaches to generalized statistics and their Fock space representations are also available. We mention here the generalizations associated with the spectral quantum Yang-Baxter equations [18], the statistics based on Lie supertriple systems [19] and on triple operator algebras [20], or extended Haldane statistics [21]. In certain approaches, one starts with a deformation of the Fock space, and from here one deduces the properties of creation and annihilation operators and the related statistics, see e.g.~Ref.~[22], or Ref.~[23] for orthobose and orthofermi statistics. For a selfconsistency of the exposition we recall in Sect. 2 the description of $sl(n+1|m)$ and $U_q[sl(n+1|m)]$ via (deformed) creation and annihilation operators as given in [7]. The transformations of the Fock modules under the action of the CAOs are written down in Sect. 3. In the last section we discuss shortly the underlying statistics of the creation and annihilation operators. Throughout the paper we use the notation: \smallskip LS, LS's - Lie superalgebra, Lie superalgebras; CAOs - creation and annihilation operators; lin.env. - linear envelope; $\Z$ - all integers; $\Z_+$ - all non-negative integers; $\Z_2=\{\bar{0},\bar{1}\}$ - the ring of all integers modulo 2; $\C$ - all complex numbers; $[p;q]=\{p,p+1,p+2,\l,q-1,q\}$, for $p\le q\in \Z $;\hfill (3) \smallskip $ \t_i=\cases {{\bar 0}, & if $\; i=0,1,2, \ldots , n$,\cr {\bar 1}, & if $\; i=n+1,n+2,\ldots ,n+m$,\cr }; \quad \t_{ij}=\t_i+\t_j; \hfill (4) $ \smallskip $ [a,b]=ab-ba,\;\; \{a,b\}=ab+ba, \;\;\[a,b\]=ab-(-1)^{\deg(a)\deg(b)}ba; \hfill (5) $ $ [a,b]_x=ab-xba,\;\; \{a,b\}_x=ab+xba, \;\;\[a,b\]_x=ab-(-1)^{\deg(a)\deg(b)}xba. \hfill (6) $ \smallskip \bigskip\n {\bf 2. The Lie superalgebra sl(n+1$|$m) and its quantization U$_q$[sl(n+1$|$m)]} \bigskip\n The universal enveloping algebra $U[gl(n+1|m)]$ of the general linear LS $gl(n+1|m)$ is a $\Z_2-$graded associative unital algebra generated by $(n+m+1)^2\;$ $\Z_2-$graded indeterminates $\{e_{ij}|i,j\in [0;n+m]\}$,\quad $\deg(e_{ij})=\t_{ij}\equiv \t_i+\t_j,$ subject to the relations $$ \[e_{ij},e_{kl}\]=\d_{jk}e_{il}-(- 1)^{\t_{ij}\t_{kl}}\d_{il}e_{kj}\quad i,j,k,l=1,\ldots , n+m. \eqno(7) $$ The LS $gl(n+1|m)$ is a subalgebra of $U[gl(n+1|m)]$, considered as a Lie superalgebra, with generators $\{e_{ij}|i,j\in [0;n+m]\}$ and supercommutation relations (7). The LS $sl(n+1|m)$ is a subalgebra of $gl(n+1|m)$: $$sl(n+1|m)=lin.env.\{e_{ij}, (-1)^{\t_k}e_{kk}-(-1)^{\t_l}e_{ll}| i\ne j;\; i,j,k,l\in [0; n+m]\}. \eqno(8)$$ \n The generators $e_{00},~e_{11},~\l,~e_{n+m,n+m}$ constitute a basis in the Cartan subalgebra of $gl(n+1|m)$. Denote by $\v_0,~\v_1,~\l,~\v_{n+m}$ the dual basis, $\v_i(e_{jj})=\d_{ij}$. The root vectors of both $gl(n+1|m)$ and $sl(n+1|m)$ are $e_{ij}, \; i\ne j,\;i,j\in [0;n+m]$. The root corresponding to $e_{ij}$ is $\v_i-\v_j$. With respect to the natural order of the basis in the Cartan subalgebra $e_{ij}$ is a positive (resp. a negative) root vector if $ij$). This description of $sl(n+1|m)$ is simple, but it is not appropriate for quantum deformations. Another definition is given in terms of the Chevalley generators $$ \h_i=e_{i-1,i-1}-(-1)^{\t_{i-1,i}}e_{ii},\quad \e_i=e_{i-1,i}, \quad \f_i=e_{i,i-1},\; i\in[1;n+m] \eqno(9) $$ \n and the $(n+m)\times (n+m)$ Cartan matrix $\{\alpha_{ij}\}$ with entries $$ \alpha_{ij}=(1+(-1)^{\t_{i-1,i}})\delta_{ij}- (-1)^{\t_{i-1,i}}\delta_{i,j-1}-\delta_{i-1,j},\;\; i,j\in [1;n+m].\eqno (10) $$ \n $U[sl(n+1|m)]$ is an associative unital algebra of the Chevalley generators, subject to the Cartan-Kac and the Serre relations: $$ \eqalignno{ & [\h_i,\h_j]=0,\quad [\h_i,\e_j]=\alpha_{ij}\e_j,&\cr & [\h_i,\f_j]=-\alpha_{ij}\f_j,\quad \[\e_i,\f_j\]=\delta _{ij}\h_i, & (11) \cr } $$ $$ \eqalignno{ & [\e_i, \e_j]=0, \quad [\f_i, \f_j]=0,\quad \hbox{if }\; |i-j|\neq 1;& (12a) \cr & \e^2_{n+1}=0,\quad \f^2_{n+1}=0; & (12b)\cr & [\e_i, [\e_i, \e_{i+1}]]=0,\quad [\f_i, [\f_i, \f_{i+1}]]=0, \quad i\ne n+1;& (12c) \cr & [\e_{i+1}, [\e_{i+1}, \e_{i}]]=0,\quad [\f_{i+1}, [\f_{i+1}, \f_{i}]]=0, \quad i\ne n;& (12d) \cr & \{[\e_{n+1}, \e_n],[\e_{n+1}, \e_{n+2}]\}=0,\quad \{[\f_{n+1}, \f_n],[\f_{n+1}, \f_{n+2}]\}=0. & (12e) \cr } $$ The grading on $U[sl(n+1|m)]$ is induced from the requirement that the only odd generators are $\e_{n+1}$ and $\f_{n+1}$. The LS $sl(n+1|m)$ is a subalgebra of $U[sl(n+1|m)]$, generated by the Chevalley generators in the sense of a Lie superalgebra. Consider the following root vectors from $sl(n+1|m)$: $$ \a_i^+=e_{i0}, \quad \a_i^-=e_{0i}, \;\;i\in [1;n+m], \eqno(13a) $$ or, equivalently $$ \eqalignno{ & \a_1^-=\e_1,\quad \a_i^-=[[[\l[[\e_1,\e_2],\e_3],\l],\e_{i-1}],\e_i]=[\a_{i-1}^-, e_i], \quad i\in [2;n+m], & \cr &\a_1^+=\f_1, \quad \a_i^+=[\f_i,[\f_{i-1},[\l ,[\f_3,[\f_2,\f_1]]\l]]]=[f_i,\a_{i-1}^+]. \quad i\in [2;n+m]. & (13b)\cr } $$ The root of $\a_i^-$ (resp. of $\a_i^+$) is $\v_0-\v_i$ (resp. $\v_i-\v_0$). Therefore (with respect to the natural order of the basis $\v_0,\v_1,\l,\v_{n+m}$) $\a_1^-,~\l,~\a_{n+m}^-$~~ are positive root vectors, and $\a_1^+,\l,~\a_{n+m}^+$~~ are negative root vectors. Moreover $$ sl(n+1|m)=lin.env.\{\a_i^{\xi},\;\[\a_j^{\eta},\a_k^{\varepsilon}\]\;| i,j,k\in[1;n];\; \xi, \eta, \varepsilon =\pm \}. \eqno(14) $$ Hence, the generators (13) are creation and annihilation operators of $sl(n+1|m)$. These generators satisfy the following triple relations: $$ \eqalignno{ & \[\a_i^\xi ,\a _j^\xi \]=0, \quad \xi=\pm, \quad i,j\in [1;n+m], & \cr %&\cr %&\cr & \[ \[ \a_i^+ ,\a _j^- \], \a _k^-\]=-(-1)^{\t_{ij}\t_k}\delta_{ik} \a _j^- - (-1)^{\t_i}\delta_{ij}\a _k^-, & (15)\cr & \[ \[ \a_i^+ ,\a _j^- \], \a _k^+\]=\delta_{jk}\a _i^++ (-1)^{\t_i}\delta_{ij}\a _k^+,\;i,j,k\in[1;n+m]. & \cr } $$ The CAOs (13) together with (15) define completely $sl(n+1|m)$. The outlined description via CAOs is somewhat similar to the Lie triple system description of Lie algebras, initiated by Jacobson [24] and generalized to Lie superalgebras by Okubo [19] (see also [20] for further development). The relations (15) are simple. They are however not convenient for a quantization. It turns out that one can take only a part of these relations, so that they still completely define $sl(n+1|m)$ and are appropriate for Hopf algebra deformations. \smallskip\n {\it Proposition 1} [7]. $U[sl(n+1|m)]$ is an associative unital superalgebra with generators $\a_i^{\pm}, \;\; i\in [1;n+m]$ and relations: $$ \eqalignno{ & \[ \a_1^\xi ,\a _2^\xi \]=0, \quad \[ a_1^\xi , a_1^\xi\]=0, \quad \xi=\pm, & \cr %&\cr & \[ \[ \a_i^+ ,\a _j^- \], \a _k^+\]=\delta_{jk}\a _i^++ (-1)^{\t_i}\delta_{ij}\a _k^+,\quad & \cr %&\cr & \[ \[ \a_i^+ ,\a _j^- \], \a _k^-\]=-(-1)^{\t_{ij}\t_k}\delta_{ik} \a _j^- - (-1)^{\t_i}\delta_{ij}\a _k^-,\; & (16) \cr & |i-j|\leq 1, \;\; i,j,k\in[1;n+m]. & \cr } $$ The $\Z_2-$grading in $U[sl(n+1|m)]$ is induced from $$ \deg(\a_i^\pm)=\t_i. \quad \eqno(17) $$ Passing to the quantum case, we skip the description of $U_q[sl(n+1|m)]$ via Chevalley generators. We write down directly the analogue of the relations (16). To this end introduce first the following Cartan generators: $$ H_i=h_1+(-1)^{\t_1}h_2+(-1)^{\t_2}h_3+\l+(-1)^{\t_{i-1}}h_i. \eqno(18) $$ \bigskip\n {\it Theorem} [7]. $U_q[sl(n+1|m)]$ is a unital associative algebra with generators \hfill\break $\{H_i,\; a_i^\pm\}_{i\in [1;n+m]}$ and relations $$ \eqalignno{ & [H_i,H_j]=0, & (19a) \cr %&&\cr & [H_i,a_j^{\pm}]=\mp(1+(-1)^{\t_i}\delta_{ij})a_j^{\pm}, &(19b)\cr %&&\cr & \[ a_i^-, a_i^+\]={L_i-\bar{L}_i\over{q-\bar{q}}},\quad L_i=q^{H_i},\;\bar{L}_i\equiv L_i^{-1}=q^{-H_i},\; \bar{q}\equiv q^{-1}, & (19c)\cr & \[\[a_i^{\eta}, a_{i+\xi}^{-\eta}\], a_k^{\eta} \]_{q^{\xi (1+(-1)^{\t_i}\delta_{ik})}}= \eta^{\t_k}\delta_{k,i+\xi}L_k^{-\xi\eta} a_i^{\eta}, & (19d)\cr & \[ a_1^\xi , a_2^\xi \]_q =0, \quad \[ a_1^\xi , a_1^\xi \] =0, \quad \xi, \eta =\pm \;\;\hbox{or}\;\; \pm 1. & (19e) \cr } $$ \bigskip\bigskip\n {\bf 3. Fock representations} \bigskip\n We proceed to describe the Fock representations of $sl(n+1|m)$ and $U_q[sl(n+1|m)]$. The derivation, which is lengthy and nontrivial (especially in the quantum case), will be given elsewhere. The irreducible Fock representations are labelled by one non-negative integer $p=1,2,\ldots$, called an order of the statistics. To construct them assume that the corresponding representation space $W_p$ contains (up to a multiple) a cyclic vector $|0\ra$, such that $$ \eqalignno{ & a_i^-|0\ra =0,\quad i=1,2,\ldots ,n+m;&\cr & \[ a_{ i}^-,a_{ j}^+\]|0\ra=0,\quad i\neq j=1,2, \ldots ,n+m; & (20) \cr & H_i|0\ra =p|0\ra, \quad i=1,2, \ldots ,n+m.&\cr } $$ Note that the above relations determine one-dimensional representations (labelled by $p$) of a subalgebra, which in the nondeformed case reduces to ${\cal B}$, Eq. (2). {}From (19) one derives that the deformed creation (resp. annihilation) operators \hfill\break $q$-supercommute $$ \[ a_i^\xi , a_j^\xi \]_{q} =0,\quad \; i0, \; \xi,\;\eta =\pm & \cr } $$ where $$ \epsilon(j,k,i)=\cases {\;\;\;1, & if $j>k>i$;\cr -1, & if $j