%Paper: hep-th/9411126
%From: braval@math.tau.ac.il
%Date: Thu, 17 Nov 94 09:57:33+020

\baselineskip=18pt
\documentstyle{amsppt}
\TagsOnRight
\font\small=cmr8
\font\itsmall=cmti8
%\font\ref=cmr9
%\font\refit=cmti9
%\font\refbf=cmbx9

%\def\smallarea#1{\par\begingroup\baselineskip=10pt#1\endgroup\par}
%\def\abstract#1{\begingroup\leftskip=5mm \rightskip 5mm
%\baselineskip=10pt\par\small#1\par\endgroup}


\nopagenumbers

\input amstex
%\magnification=\magstep1








%------------------------------------------------------------


\def\today{\ifcase\month\or January\or February\or March\or
April\or May\or June\or July\or August\or September\or
October\or November\or December\fi \space\number\day,
\number\year}


%Greek letters
\def\alp{\alpha}		\def\Alp{\Alpha}
\def\bet{\beta}
\def\gam{\gamma}		\def\Gam{\Gamma}
\def\del{\delta}		\def\Del{\Delta}
\def\eps{\varepsilon}
\def\zet{\zeta}
\def\tet{\theta}		\def\Tet{\Theta}
\def\iot{\iota}
\def\kap{\kappa}
\def\lam{\lambda}		\def\Lam{\Lambda}
\def\sig{\sigma}		\def\Sig{\Sigma}
\def\vphi{\varphi}
\def\ome{\omega}		\def\Ome{\Omega}

%Caligraphic roman letters
\def\calA{{\Cal A}}
\def\calB{{\Cal B}}
\def\calC{{\Cal C}}
\def\calD{{\Cal D}}
\def\calE{{\Cal E}}
\def\calF{{\Cal F}}
\def\calG{{\Cal G}}
\def\calH{{\Cal H}}
\def\calI{{\Cal I}}
\def\calJ{{\Cal J}}
\def\calK{{\Cal K}}
\def\calL{{\Cal L}}
\def\calM{{\Cal M}}
\def\calN{{\Cal N}}
\def\calO{{\Cal O}}
\def\calQ{{\Cal Q}}
\def\calR{{\Cal R}}
\def\calS{{\Cal S}}
\def\calT{{\Cal T}}
\def\calU{{\Cal U}}
\def\calV{{\Cal V}}
\def\calW{{\Cal W}}
\def\calX{{\Cal X}}
\def\calY{{\Cal Y}}
\def\calZ{{\Cal Z}}

%Bold roman letters
\def\bfa{{\bf a}}		\def\bfA{{\bf A}}
\def\bfb{{\bf b}}		\def\bfB{{\bf B}}
\def\bfc{{\bf c}}		\def\bfC{{\bf C}}
\def\bfd{{\bf d}}		\def\bfD{{\bf D}}
\def\bfe{{\bf e}}		\def\bfE{{\bf E}}
\def\bff{{\bf f}}		\def\bfF{{\bf F}}
\def\bfg{{\bf g}}		\def\bfG{{\bf G}}
\def\bfh{{\bf h}}		\def\bfH{{\bf H}}
\def\bfi{{\bf i}}		\def\bfI{{\bf I}}
\def\bfj{{\bf j}}		\def\bfJ{{\bf J}}
\def\bfk{{\bf k}}		\def\bfK{{\bf K}}
\def\bfl{{\bf l}}		\def\bfL{{\bf L}}
\def\bfm{{\bf m}}		\def\bfM{{\bf M}}
\def\bfn{{\bf n}}		\def\bfN{{\bf N}}
\def\bfo{{\bf o}}		\def\bfO{{\bf O}}
\def\bfp{{\bf p}}		\def\bfP{{\bf P}}
\def\bfq{{\bf q}}		\def\bfQ{{\bf Q}}
\def\bfr{{\bf r}}		\def\bfR{{\bf R}}
\def\bfs{{\bf s}}		\def\bfS{{\bf S}}
\def\bft{{\bf t}}		\def\bfT{{\bf T}}
\def\bfu{{\bf u}}		\def\bfU{{\bf U}}
\def\bfv{{\bf v}}		\def\bfV{{\bf V}}
\def\bfw{{\bf w}}		\def\bfW{{\bf W}}
\def\bfx{{\bf x}}		\def\bfX{{\bf X}}
\def\bfy{{\bf y}}		\def\bfY{{\bf Y}}
\def\bfz{{\bf z}}		\def\bfZ{{\bf Z}}

%bold Greek capital letters
\def\bfTet{{\bf\Theta}}


%Capital roman double letters
%\def\CC{{\rlap {\raise 0.4ex \hbox{$\scriptscriptstyle |$}}
%\hskip -0.15em C}}
%\def\AA{{\mathchoice
%{I\hskip -3.7pt {\rm A}}
%{I\hskip -3.7pt {\rm A}}
%{I\hskip -3.1pt {\rm A}}
%{I\hskip -2.5pt {\rm A}}}}
%\def\BB{{I\!\!B}}
%\def\EE{{I\!\!E}}
%\def\FF{{I\!\!F}}
%\def\NN{{I\!\!N}}
%\def\PP{{I\hskip-2.5pt P}}
%\def\QQ{{\rlap {\raise 0.4ex \hbox{$\scriptscriptlsubspacesstyle |$}}
%\hskip -0.1em Q}}
%\def\RR{{I\!\!R}}
%\def\ZZ{{Z\!\!\! Z}}

%Special fonts
\font\tenboldgreek=cmmib10  \font\sevenboldgreek=cmmib10 at
7pt
\font\fiveboldgreek=cmmib10 at 7pt
\newfam\bgfam
\textfont\bgfam=\tenboldgreek \scriptfont\bgfam=\sevenboldgreek
\scriptscriptfont\bgfam=\fiveboldgreek
\def\bg{\fam\bgfam}
%\def\bg{\fam6}
%\mathchardef\alpha="700B
%\def\bfalp{{\fam=\bgfam\balp}}

\font\tengerman=eufm10 \font\sevengerman=eufm7 \font\fivegerman=eufm5
\font\tendouble=msym10 \font\sevendouble=msym7 \font\fivedouble=msym5
\font\smc=cmcsc10
\textfont4=\tengerman \scriptfont4=\sevengerman
\scriptscriptfont4=\fivegerman
\newfam\dbfam
\textfont\dbfam=\tendouble \scriptfont\dbfam=\sevendouble
\scriptscriptfont\dbfam=\fivedouble
%\def\gr{\fam4 \tengerman}
%\def\db{\fam5 \tendouble}
\def\gr{\fam4}
\def\db{\fam5}
\mathchardef\ng="702D
\mathchardef\dbA="7041
\mathchardef\sm="7072
\mathchardef\nvdash="7030
\mathchardef\nldash="7031
\mathchardef\lne="7008
\mathchardef\sneq="7024
\mathchardef\spneq="7025
\mathchardef\sne="7028
\mathchardef\spne="7029
\mathchardef\ltms="706E
\mathchardef\tmsl="706F
\def\notgiven{{\fam = \dbfam\ng}}
\mathchardef\dbA="7041
\def\ltimes{{\fam=\dbfam\ltms}}
\def\timesl{{\fam=\dbfam\tmsl}}
\def\AA{{\fam=\dbfam\dbA}}
\def\setmin{{\fam=\dbfam\sm}}
\def\notvdash{\,{\fam=\dbfam\nvdash}\,}
\def\notldash{\,{\fam=\dbfam\nldash}\,}
\def\lenoteq{{\fam=\dbfam\lne}\,}
\def\subsetneqq{\,{\fam=\dbfam\sneq}\,}
\def\supsetneqq{\,{\fam=\dbfam\spneq}\,}
\def\subsetne{\,{\fam=\dbfam\sne}\,}
\def\supsetne{\,{\fam=\dbfam\spne}\,}

%Euler Fraktur letters
\def\grA{{\gr A}}	\def\gra{{\gr a}}
\def\grB{{\gr B}}	\def\grb{{\gr b}}
\def\grC{{\gr C}}	\def\grc{{\gr c}}
\def\grD{{\gr D}}	\def\grd{{\gr d}}
\def\grE{{\gr E}}	\def\gre{{\gr e}}
\def\grF{{\gr F}}	\def\grf{{\gr f}}
\def\grG{{\gr G}}	\def\grg{{\gr g}}
\def\grH{{\gr H}}	\def\grh{{\gr h}}
\def\grI{{\gr I}}	\def\gri{{\gr i}}
\def\grJ{{\gr J}}	\def\grj{{\gr j}}
\def\grK{{\gr K}}	\def\grk{{\gr k}}
\def\grL{{\gr L}}	\def\grl{{\gr l}}
\def\grM{{\gr M}}	\def\grm{{\gr m}}
\def\grN{{\gr N}}	\def\grn{{\gr n}}
\def\grO{{\gr O}}	\def\gro{{\gr o}}
\def\grP{{\gr P}}	\def\grp{{\gr p}}
\def\grQ{{\gr Q}}	\def\grq{{\gr q}}
\def\grR{{\gr R}}	\def\grr{{\gr r}}
\def\grS{{\gr S}}	\def\grs{{\gr s}}
\def\grT{{\gr T}}	\def\grt{{\gr t}}
\def\grU{{\gr U}}	\def\gru{{\gr u}}
\def\grV{{\gr V}}	\def\grv{{\gr v}}
\def\grW{{\gr W}}	\def\grw{{\gr w}}
\def\grX{{\gr X}}	\def\grx{{\gr x}}
\def\grY{{\gr Y}}	\def\gry{{\gr y}}
\def\grZ{{\gr Z}}	\def\grz{{\gr z}}

%Capital roman double letters

\def\QQ{\Bbb Q}
\def\WW{\Bbb W}
\def\EE{\Bbb E}
\def\RR{\Bbb R}
\def\TT{\Bbb T}
\def\YY{\Bbb Y}
\def\UU{\Bbb U}
\def\II{\Bbb I}
\def\OO{\Bbb O}
\def\PP{\Bbb P}
\def\AA{\Bbb A}
\def\SS{\Bbb S}
\def\DD{\Bbb D}
\def\FF{\Bbb F}
\def\GG{\Bbb G}
\def\HH{\Bbb H}
\def\JJ{\Bbb J}
\def\KK{\Bbb K}
\def\LL{\Bbb L}
\def\ZZ{\Bbb Z}
\def\XX{\Bbb X}
\def\CC{\Bbb C}
\def\VV{\Bbb V}
\def\BB{\Bbb B}
\def\NN{\Bbb N}
\def\MM{\Bbb M}




%\mathchardef\dbA="7041 \def\AA{{\fam=\dbfam\dbA}}
%\mathchardef\dbB="7042 \def\BB{{\fam=\dbfam\dbB}}
%\mathchardef\dbC="7043 \def\CC{{\fam=\dbfam\dbC}}
%\mathchardef\dbD="7044 \def\DD{{\fam=\dbfam\dbD}}
%\mathchardef\dbE="7045 \def\EE{{\fam=\dbfam\dbE}}
%\mathchardef\dbF="7046 \def\FF{{\fam=\dbfam\dbF}}
%\mathchardef\dbG="7047 \def\GG{{\fam=\dbfam\dbG}}
%\mathchardef\dbH="7048 \def\HH{{\fam=\dbfam\dbH}}
%\mathchardef\dbI="7049 \def\II{{\fam=\dbfam\dbI}}
%\mathchardef\dbJ="704A \def\JJ{{\fam=\dbfam\dbJ}}
%\mathchardef\dbK="704B \def\KK{{\fam=\dbfam\dbK}}
%\mathchardef\dbL="704C \def\LL{{\fam=\dbfam\dbL}}
%\mathchardef\dbM="704D \def\MM{{\fam=\dbfam\dbM}}
%\mathchardef\dbN="704E \def\NN{{\fam=\dbfam\dbN}}
%\mathchardef\dbO="704F \def\OO{{\fam=\dbfam\dbO}}
%\mathchardef\dbP="7050 \def\PP{{\fam=\dbfam\dbP}}
%\mathchardef\dbQ="7051 \def\QQ{{\fam=\dbfam\dbQ}}
%\mathchardef\dbR="7052 \def\RR{{\fam=\dbfam\dbR}}
%\mathchardef\dbS="7053 \def\SS{{\fam=\dbfam\dbS}}
%\mathchardef\dbT="7054 \def\TT{{\fam=\dbfam\dbT}}
%\mathchardef\dbU="7055 \def\UU{{\fam=\dbfam\dbU}}
%\mathchardef\dbV="7056 \def\VV{{\fam=\dbfam\dbV}}
%\mathchardef\dbW="7057 \def\WW{{\fam=\dbfam\dbW}}
%\mathchardef\dbX="7058 \def\XX{{\fam=\dbfam\dbX}}
%\mathchardef\dbY="7059 \def\YY{{\fam=\dbfam\dbY}}
%\mathchardef\dbZ="705A \def\ZZ{{\fam=\dbfam\dbZ}}

\def\nek{,\ldots,}
\def\sdp{\times \hskip -0.3em {\raise 0.3ex
\hbox{$\scriptscriptstyle |$}}} % semidirect product

%words in roman font

\def\area{\operatorname{area}}
\def\Aug{\operatorname{Aug}}
\def\Aut{\operatorname{Aut}}
\def\Char{\operatorname{Char}}
\def\Cl{\operatorname{Cl}}
\def\cf{{\rm \,cf\,}}
\def\cont{\operatorname{cont}}
\def\codim{\operatorname{codim}}
\def\conv{\operatorname{conv}}
\def\Conv{\operatorname{Conv}}
\def\const{\operatorname{const}}
\def\Const{\operatorname{Const}}
\def\Def{\operatorname{Def}}
\def\Det{\operatorname{Det}}
\def\diag{\operatorname{diag}}
\def\diam{\operatorname{diam}}
\def\Diam{\operatorname{Diam}}
\def\dist{\operatorname{dist}}
\def\Dom{{\rm Dom}}
\def\dom{\operatorname{dom}}
\def\End{\operatorname{End\,}}
\def\Ext{\operatorname{Ext}}
\def\esssup{\operatorname{ess\ sup}}
\def\Ran{{\rm Ran}}
\def\RANK{\operatorname{rank}}
\def\Geo{\operatorname{Geo}}
\def\GL{\operatorname{GL}}
\def\Gr{\operatorname{Gr}}
\def\gl{\operatorname{gl}}
\def\grad{\mathop {\rm grad}}
\def\Hom{\operatorname {Hom}}
\def\im{\operatorname {im}}
\def\IM{\operatorname{Im}}
\def\Ind{\operatorname{Ind}}
\def\Inf{\operatorname{Inf}}
\def\Int{\operatorname{Int}}
\def\Min{\operatorname{Min}}
\def\min{\operatorname{min}}
\def\MOD{\operatorname{mod}}
\def\ord{\operatorname{ord}}
\def\Ka{\operatorname{Ka}}
\def\Ker{\operatorname{Ker}}
\def\PGL{{\rm PGL}}
\def\PGSp{{\rm PGSp}}
\def\Plt{\operatorname{Plt}}
\def\proj{\operatorname{proj}}
\def\Proj{\operatorname{Proj}}
\def\res{\rm res}
%\def\rank{\rm rank}
\def\Range{\operatorname{Range}}
\def\Re{\operatorname{Re}}
\def\Res{\operatorname{Res}}
\def\rot{\operatorname{rot}}
\def\Max{\operatorname{Max}}
\def\Maximum{\operatorname{Maximum}}
\def\Minimum{\operatorname{Minimum}}
\def\Minimize{\operatorname{Minimize}}
\def\Prob{\operatorname{Prob}}
\def\sech{\rm sech}
\def\sgn{\operatorname{sgn}}
\def\SL{{\rm SL}}
\def\Sbm{\operatorname{Sbm}}
\def\SO{{\rm SO}}
\def\Sp{{\rm Sp}}
\def\SPAN{\operatorname{span}}
\def\spec{{\rm spec}}
\def\supess{\operatorname{sup\ ess}}
\def\supp{\operatorname{supp}}
\def\Supp{\operatorname{Supp}}
\def\Sup{\operatorname{Sup}}
\def\Sym{\operatorname{Sym}}
\def\tr{\operatorname{tr}}
\def\Tor{\operatorname{Tor}}
\def\Var{\operatorname{Var}}
\def\Vol{\operatorname{Vol}}
%\def\vol{\operatorname{vol}}

%overlined math alphabet
\def\oa{{\overline a}}
\def\oA{{\overline A}}
\def\ob{{\overline b}}
\def\oB{{\overline B}}
\def\oc{{\overline c}}
\def\oC{{\overline C}}
\def\oD{{\overline D}}
\def\od{{\overline d}}
\def\oE{{\overline E}}
\def\oe{{\overline e}}
\def\of{{\overline f}}
\def\oF{{\overline F}}
\def\og{{\overline g}}
\def\oG{{\overline G}}
\def\oh{{\overline h}}
\def\oH{{\overline H}}
\def\oI{{\overline I}}
\def\oj{{\overline j}}
\def\oJ{{\overline J}}
\def\ok{{\overline k}}
\def\oK{{\overline K}}
\def\oL{{\overline L}}
\def\om{{\overline m}}
\def\oM{{\overline M}}
\def\oN{{\overline N}}
\def\oO{{\overline O}}
\def\oo{{\overline o}}
\def\op{{\overline p}}
\def\oP{{\overline P}}
\def\oq{{\overline q}}
\def\oQ{{\overline Q}}
\def\OR{{\overline r}}
\def\oS{{\overline S}}
\def\os{{\overline s}}
\def\ot{{\overline t}}
\def\oT{{\overline T}}
\def\ou{{\overline u}}
\def\oU{{\overline U}}
\def\ov{{\overline v}}
\def\oV{{\overline V}}
\def\ow{{\overline w}}
\def\oW{{\overline W}}
\def\ox{{\overline x}}
\def\oX{{\overline X}}
\def\oy{{\overline y}}
\def\oY{{\overline Y}}
\def\oz{{\overline z}}
\def\oZ{{\overline Z}}

%overlined Greek alphabet
\def\oalp{{\overline\alpha}}
\def\obet{{\overline\bet}}
\def\ocup{{\overline\cup}}
\def\ovarphi{{\overline\varphi}}
\def\ochi{{\overline\chi}}
\def\oeps{{\overline\eps}}
\def\oeta{{\overline\eta}}
\def\ogam{{\overline\gam}}
\def\okap{{\overline\kap}}
\def\olam{{\overline\lambda}}
\def\oLam{{\overline\Lambda}}
\def\omu{{\overline\mu}}
\def\onu{{\overline\nu}}
\def\oOme{{\overline\Ome}}
\def\ophi{\overline\phi}
\def\oPhi{{\overline\Phi}}
\def\opi{{\overline\pi}}
\def\oPsi{{\overline\Psi}}
\def\opsi{{\overline\psi}}
\def\orho{{\overline\rho}}
\def\osig{{\overline\sig}}
\def\otau{{\overline\tau}}
\def\otet{{\overline\theta}}
\def\oxi{{\overline\xi}}
\def\oome{\overline\ome}
\def\opart{{\overline\partial}}


%underlined math alphabet
\def\ua{{\underline a}}
\def\ub{{\underline b}}
\def\uc{{\underline c}}
\def\uD{{\underline D}}
\def\uk{{\underline k}}
\def\ue{{\underline e}}
\def\uj{{\underline j}}
\def\ul{{\underline l}}
\def\uL{{\underline L}}
\def\uo{{\underline o}}
\def\uO{{\underline O}}
\def\uP{{\underline P}}
\def\uQ{{\underline Q}}
\def\um{{\underline m}}
\def\uM{{\underline M}}
\def\un{{\underline n}}
\def\us{{\underline s}}
\def\ut{{\underline t}}
\def\uu{{\underline u}}
\def\uv{{\underline v}}
\def\uV{{\underline V}}
\def\ux{{\underline x}}
\def\uX{{\underline X}}
\def\uy{{\underline y}}
\def\uz{{\underline z}}

%underline Greek alphabet
\def\ualp{{\underline\alp}}
\def\ubet{{\underline\bet}}
\def\uchi{{\underline\chi}}
\def\udel{{\underline\del}}
\def\uell{{\underline\ell}}
\def\ueps{{\underline\eps}}
\def\ueta{{\underline\eta}}
\def\uGam{{\underline\Gamma}}
\def\unu{{\underline\nu}}
\def\uome{{\underline\omega}}
\def\utet{{\underline\tet}}
\def\ulam{{\underline\lam}}

%math alphabet with hat
\def\hata{{\widehat a}}
\def\hatA{{\widehat A}}
\def\hatb{{\widehat b}}
\def\hatc{{\widehat c}}
\def\hatC{{\widehat C}}
\def\hatB{{\widehat B}}
\def\hatD{{\widehat D}}
\def\hate{{\widehat e}}
\def\hatf{{\widehat f}}
\def\hatg{{\widehat g}}
\def\hatG{{\widehat G}}
\def\hath{{\widehat h}}
\def\hatH{{\widehat H}}
\def\hati{{\hat i}}
\def\hatI{{\hat I}}
\def\hatj{{\widehat j}}
\def\hatJ{{\widehat J}}
\def\hatk{{\widehat k}}
\def\hatK{{\widehat K}}
\def\hatL{{\widehat L}}
\def\hatm{{\widehat m}}
\def\hatM{{\widehat M}}
\def\hatn{{\widehat n}}
\def\hatN{{\widehat N}}
\def\hato{{\widehat o}}
\def\hatp{{\widehat p}}
\def\hatP{{\widehat P}}
\def\hatr{{\widehat r}}
\def\hatR{{\widehat R}}
\def\hatq{{\widehat q}}
\def\hatQ{{\widehat Q}}
\def\hatT{{\widehat T}}
\def\hatu{{\widehat u}}
\def\hatU{{\widehat U}}
\def\hatV{{\widehat V}}
\def\hatv{{\widehat v}}
\def\hatw{{\widehat w}}
\def\hatW{{\widehat W}}
\def\hatx{{\widehat x}}
\def\hatX{{\widehat X}}
\def\haty{{\widehat y}}
\def\hatY{{\widehat Y}}
\def\hatZ{{\widehat Z}}
\def\hatz{{\widehat z}}

%Greek alphabet with hat
\def\hatalp{{\widehat\alpha}}
\def\hatdel{{\widehat\delta}}
\def\hatDel{{\widehat\Delta}}
\def\hatbet{{\widehat\beta}}
\def\hateps{{\hat\eps}}
\def\hatgam{{\widehat\gamma}}
\def\hatGam{{\widehat\Gamma}}
\def\hatlam{{\widehat\lambda}}
\def\hatmu{{\widehat\mu}}
\def\hatnu{{\widehat\nu}}
\def\hatOme{{\widehat\Ome}}
\def\hatphi{{\widehat\phi}}
\def\hatPhi{{\widehat\Phi}}
\def\hatpi{{\widehat\pi}}
\def\hatpsi{{\widehat\psi}}
\def\hatPsi{{\widehat\Psi}}
\def\hatrho{{\widehat\rho}}
\def\hatsig{{\widehat\sig}}
\def\hatSig{{\widehat\Sig}}
\def\hattau{{\widehat\tau}}
\def\hattet{{\widehat\theta}}
\def\hatvarphi{{\widehat\varphi}}
\def\hatZZ{{\widehat\ZZ}}


%roman with widetilde

\def\tilA{{\widetilde A}}
\def\tila{{\widetilde a}}
\def\tilB{{\widetilde B}}
\def\tilb{{\widetilde b}}
\def\tilc{{\widetilde c}}
\def\tilC{{\widetilde C}}
\def\tilD{{\widetilde D}}
\def\tilE{{\widetilde E}}
\def\tilf{{\widetilde f}}
\def\tilF{{\widetilde F}}
\def\tilg{{\widetilde g}}
\def\tilG{{\widetilde G}}
\def\tilh{{\widetilde h}}
\def\tili{{\widetilde i}}
\def\tilk{{\widetilde k}}
\def\tilK{{\widetilde K}}
\def\tilj{{\widetilde j}}
\def\tilm{{\widetilde m}}
\def\tilM{{\widetilde M}}
\def\tilH{{\widetilde H}}
\def\tilL{{\widetilde L}}
\def\tilN{{\widetilde N}}
\def\tiln{{\widetilde n}}
\def\tilO{{\widetilde O}}
\def\tilP{{\widetilde P}}
\def\tilp{{\widetilde p}}
\def\tilq{{\widetilde q}}
\def\tilQ{{\widetilde Q}}
\def\tilR{{\widetilde R}}
\def\tilr{{\widetilde r}}
\def\tilS{{\widetilde S}}
\def\tils{{\widetilde s}}
\def\tilT{{\widetilde T}}
\def\tilt{{\widetilde t}}
\def\tilu{{\widetilde u}}
\def\tilU{{\widetilde U}}
\def\tilv{{\widetilde v}}
\def\tilV{{\widetilde V}}
\def\tilw{{\widetilde w}}
\def\tilW{{\widetilde W}}
\def\tilX{{\widetilde X}}
\def\tilx{{\widetilde x}}
\def\tily{{\widetilde y}}
\def\tilY{{\widetilde Y}}
\def\tilZ{{\widetilde Z}}
\def\tilz{{\widetilde z}}

%Greek alphabet with widetilde
\def\tilalp{{\widetilde\alpha}}
\def\tilbet{{\widetilde\beta}}
\def\tildel{{\widetilde\delta}}
\def\tileta{{\widetilde\eta}}
\def\tilgam{{\widetilde\gamma}}
\def\tilGam{{\widetilde\Gamma}}
\def\tilome{{\widetilde\ome}}
\def\tillam{{\widetilde\lam}}
\def\tilmu{{\widetilde\mu}}
\def\tilphi{{\widetilde\phi}}
\def\tilpi{{\widetilde\pi}}
\def\tilpsi{{\widetilde\psi}}
\def\tilome{{\widetilde\ome}}
\def\tilOme{{\widetilde\Ome}}
\def\tilPhi{{\widetilde\Phi}}
\def\tilQQ{{\widetilde\QQ}}
\def\tilrho{{\widetilde\rho}}
\def\tilsig{{\widetilde\sig}}
\def\tiltau{{\widetilde\tau}}
\def\tiltet{{\widetilde\theta}}
\def\tilvarphi{{\widetilde\varphi}}
\def\tilxi{{\widetilde\xi}}
\def\twolongrightarrow{\ \hbox{$\longrightarrow\hskip -17pt
\longrightarrow$}\ }
\def\sqr#1#2{{\vcenter{\hrule height.#2pt\hbox{\vrule
width.#2pt height#1pt \kern#1pt \vrule width.#2pt}\hrule
height.#2pt}}}
%\def\square{\mathchoice{\sqr34}{\sqr34}{\sqr{2.1}3}{\sqr{1.5}3}}
%\def\bsquare{\mathchoice{\sqr{4.1}5}{\sqr{4.1}5}{\sqr{2.1}3}
%{\sqr{1.5}3}}

\def\buildrul#1\under#2{\mathrel{\mathop{\null#2}\limits_{#1}}}
\def\wtx{{\widetilde x}}

\def\boxit#1{\vbox{\hrule\hbox{\vrule\kern3pt\vbox{\kern3pt#1
\kern3pt}\kern3pt\vrule}\hrule}}
\def\bigucdot{\mathop{\bigcup\hskip -7.5pt\cdot}}
\def\bigucirc{\mathop{\bigcup\hskip -7.5pt\circ}}

\def\lesim{\mathrel{\vcenter{\offinterlineskip{\hbox{$<$}
\hbox{$\sim$}}}}}
\def\gesim{\mathrel{\vcenter{\offinterlineskip{\hbox{$>$}
\hbox{$\sim$}}}}}
\def\subsetsim{\mathrel{\vcenter{\baselineskip 6pt{\hbox{$
\subset$}\hbox{$\sim$}}}}}
\def\suml{\sum\limits}
\def\prodl{\prod\limits}
\def\bigcupl{\bigcup\limits}
\def\intl{\int\limits}
\def\half{{\textstyle{1\over 2}}}
\def\quarter{{\textstyle{1\over 4}}}
\def\threequarters{{\textstyle{3\over 4}}}
\def\third{{\textstyle{1\over 3}}}
\def\iint{\mathop{\int\!\!\!\int}}


\def\sucsim
{\mathrel{\vcenter{\offinterlineskip\hbox{$\scriptstyle\succ$}\hbox
{$\scriptstyle\sim$}}}}
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%\define\rg{R_q[G]}
\def\mqg{\calM _q(\grg)}
\def\mog{\calM _{q_1}(\grg _1)}
\def\mtg{\calM _{q_2}(\grg _2)}
\def\msn{\calM _q(sl_n)}
\def\mst{\calM _{q_1}(sl_2)}
\def\mg{\calM (\grg)}
\def\msn{\calM _q(sl_n)}
\def\rgo{R_{q_1}[G]}
\def\rgoo{R_{q_1}[G_1]}
\define\rgt{R_{q_2}[G_2]}
\redefine\rs{R_q[SL(2)]}
\redefine\rso{R_{q_1}[SL(2)]}
\redefine\rst{R_{q_2}[SL(2)]}
\define\riso{R_{q_{1_i}}[SL_2]}
\define\ris{R_{q_i}[SL(2)]}
\define\smat{\pmatrix a & b\\ c & d\endpmatrix }
\def\b{\la b\ra}
\def\c{la b\ra}
\def\ug{U(\grg)}
\def\us{U_q(sl_2)}
\def\uqg{U_q(\grg)}
\def\uhg{U_h(\grg)}
\def\ve{{\gr Vec}}
\def\vl{V(\lam)}
\def\un{U_q(sl_n)}
\def\vnl{V(n\lam)}
\def\vmu{V(\mu)}
\def\vtn{V^{\otimes n}}
\def\qh{quasi-Hopf algebra\ }

\topmatter

\title On quantum flag algebras
\endtitle

\author
Alexander Braverman
\endauthor

\address
School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv,
Israel
\endaddress

\email
braval\@math.tau.ac.il
\endemail


\abstract
Let $G$ be a semisimple simply connected algebraic group over an algebraically
closed field of characteristic $0$. Let $V$ be a simple finite-dimensional
$G$-module and let $y$ be its highest weight vector. It is a classical result
of B.~Kostant that the algebra of functions on the closure of $G\cdot y$ is
quadratic.
In this paper we generalize
this result to the case of the quantum group $\uqg$. The proof uses
information about $R$-matrix due to Drinfeld and Reshetikhin.
\endabstract
\endtopmatter

\heading 0. Introduction
\endheading
\subheading{0.1. Notations}Let $\grg$ be a semisimple Lie algebra over an
algebraically closed field $k$ of characteristic $0$. Let us choose a Borel
subalgebra $\grb\subset \grg$ and a Cartan subalgebra $\grh\subset\grb$.
Let $\Pi\subset \grh^*$ be the set of simple roots of $\grh$ with respect
to these choises. We shall denote by $P(\Pi)$ the corresponding weight lattice
and by $P(\Pi)^+$ the set of dominant weights in $P(\Pi)$. We shall denote by
$\rho\in P(\Pi)^+$ the half sum of all positive roots. For any
$\lam,\mu\in\grh ^*$ the symbol $\lam>\mu$ will mean that $\lam-\mu$ is
a sum of positive roots.  Choose an invariant bilinear form on $\grg$. We shall
denote by $(\cdot,\cdot)$ its restriction to $\grh$ (and to $\grh^*$ by
transport of structure) and by $\|\cdot\|$ the corresponding
norm on $\grh^*$.
%------------------------------------------------------------------------------
\subheading{0.2}Let $K=k(q)$ where $q$ is transcendental over $k$. Let
$\uqg$ denote the quantized universal enveloping algebra of $\grg$
(cf. for example [D1], [L]) constructed using the invariant form of 0.1
(as in [D1]). For any
$\lam\in P(\Pi)^+$ we shall denote by $\vl$ the simple $\uqg$-module with
highest weight $\lam$ (cf. [L]). We shall also denote by $\mqg$ the tensor
category of locally finite integrable (cf. [L]) $\uqg$-modules.
%------------------------------------------------------------------------------
\subheading{0.3. Quadratic algebras}For any $K$-vector space $V$ we shall
denote by $T(V)$ the tensor algebra of $V$. Let $A=\oplus _{n=0}^{\infty}A_n$
be a graded $K$-algebra.
\proclaim{Definition}We say that $A$ is {\it quadratic} if
\roster
\item $A_0=K$ and $A$ is generated by $A_1$
\item the ideal of relations of $A$ is quadratic, i.e. if we let $i$ denote the
natural map $T(A_1)\to A$ then
\endroster
$$
\ker i=\sum_{i,j\in\ZZ ^+}V^{\ten i}\ten W\ten V^{\ten j}
$$
where $W=\ker i\bigcap V^{\ten 2}$.
\endproclaim

%-----------------------------------------------------------------------------
\subheading{0.4. The result} Let $\lam\in P(\Pi)^+$
and let $V=\vl$. Let us define a graded algebra $A(V)=\oplus_{n=0}^{\infty}
A_n(V)$
by
$$
A_n(V):=V^{\ten n}/\sum_{\vmu\subset\vtn, \mu <n\lam}\vmu
$$
and multiplication coming from the tensor algebra of $V$.
In the classical ($q$=1) case the algebra $A(V)$ is isomorphic to the algebra
of functions on the closure of $G\cdot y$ where $G$ is the simply connected
algebraic group which corresponds to $\grg$ and $y$
is a highest weight vector in the $\grg$-module
$V^*$, dual to $V$ and hence can be identified with the coordinate ring
of the flag variety of $\grg$ with respect to the invertible sheaf defined
by $V$. This is why in our situation the algebra $A(V)$ deserves the name
of a quantum flag algebra.
\proclaim{Theorem}The algebra $A(V)$ is quadratic.
\endproclaim
In the classical case this the analogous theorem was proven by B.~Kostant
(cf. for example [FH]).


%-----------------------------------------------------------------------------
\subheading{0.5. Remarks}1) In fact one can easily see that the algebra
$A(V)$ is defined as an algebra in the category $\mqg$ and theorem 0.4
is equivalent to the statement that $A(V)$ is quadratic as an $\mqg$-algebra
(cf [HS] for the relevant definitions).

2) In the case $\grg=sl_n$ theorem 0.4 can be deduced from [TT].

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%-------------------------------------------------------------------------------
\heading{1. Proof of the theorem}
\endheading
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%-------------------------------------------------------------------------------
\subheading{1.1. The braiding}Let $(\calC, \ten)$ be a monoidal category (cf.
[SS]). A {\it braiding} on $\calC$ is an isomorphism $B$ of two functors
$\calC\x\calC\to \calC$ (namely, $(V,W)\to V\ten W$ and $(V,W)\to W\ten V$ for
any $(V,W)\in Ob~\calC\x\calC$), which satisfies the hexagon identities (cf.
[SS]). Drinfeld in [D1] has shown that $\mqg$ has a braiding and it was shown
by
D.~Gaitsgory ([G]) that this braiding was essentially unique.
For any $V,W\in Ob~\mqg$ we shall denote by $B_{V,W}$ the corresponding
morphism $B_{V,W}:\, V\ten W\to W\ten V$. Let also
$S_{V,W}=B_{W,V}\circ B_{V,W}\in \End V\ten W$.
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%-------------------------------------------------------------------------------
\subheading{1.2}For any $\lam\in\grh ^*$ define $c(\lam)=(\lam, \lam +2\rho)=
\|\lam + \rho\|^2-\|\rho\|^2$. Let $\lam,\mu\in P(\Pi)^+$. For any
$\gam\in P(\Pi)^+$ denote by $P_{\lam ,\mu}^{\gam}\in \End (V(\lam)\ten
V(\mu))$
the projector from $\vl\ten \vmu$ to its $V(\gam)$-isotypical component.
The following statement is due to N.~Reshetikhin ([R], cf. also [D2, $\S$ 5]).
\proclaim{Lemma}
$$
S_{\vl,\vmu}=\sum _{\gam\in
P(\Pi)^+}q^{2(c(\gam)-c(\lam)-c(\mu))}P_{\lam,\mu}^{\gam}
$$
\endproclaim
%------------------------------------------------------------------------------
\subheading{1.3}Let $V=V(\lam)$ and $S=S_{V,V^{\ten n-1}}$ (for some $n\in
\NN$).
\proclaim{Corollary}
$$
\{ x\in \vtn| Sx=q^{2(c(n\lam)-c(\lam)-c((n-1)\lam))}x\} =\vnl\subset \vtn
$$
i.e. $\vnl$ is a full eigenspace of $S$ with eigenvalue
$q^{2(c(n\lam)-c(\lam)-c((n-1)\lam))}$.
\endproclaim
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%-------------------------------------------------------------------------------
\demo{Proof}Let $P_{n-1}=\{ \mu\in P(\Pi)^+ |\vmu\subset \vtn\} $. We know from
1.2 that $\vnl$ is an eigenspace of $S$ with eigenvalue
$q^{2(c(n\lam)-c(\lam)-c((n-1)\lam))}$.
On the other hand 1.2 again implies that any other eigenvalue of $S$ takes the
form $q^{2(c(\gam)-c(\lam)-c(\mu))}$ where $\mu\in P_{n-1}$ and
$V(\gam)\in V\ten \vmu$. Let us prove that if $\gam\neq n\lam$ or
$\mu\neq (n-1)\lam$ then
$$
c(\gam)-c(\mu)<c(n\lam)-c((n-1)\lam) \tag{$*$}
$$
(this clearly implies the corollary). The left hand side of ($*$) is equal to
$\|\gam+\rho\|^2-\|\mu+\rho\|^2$. Clearly $\gam\leq\mu +\lam$, so, it is enough
to prove ($*$) only in the case when $\gam=\mu +\lam$. In this case
$$\align
[c(n\lam)-c((n-1)\lam)]&-[c(\gam)-c(\mu)]=\\
(\|n\lam +\rho\|^2-\|(n-1)\lam +\rho\|^2)&-(\|\mu+\lam
+\rho\|^2-\|\mu+\rho\|^2)=\\
2((n-1)\lam+\rho,\lam)-2(\mu+\rho,\lam&)=2((n-1)\lam-\mu,\lam)
\endalign
$$
Let $\nu =(n-1)\lam-\mu$.
Since $\lam$ is dominant and $\mu \leq (n-1)\lam$, we have
$(\nu,\lam)\geq 0$. Suppose that $(\nu,\lam)=0$. Then
$$
0\leq (\nu,\mu)=(\nu,(n-1)\lam-\nu)=(n-1)(\nu,\lam)-\|\nu\|^2=
-\|\nu\|^2<0
$$
and therefore $\nu=0$ and $\gam =n\lam $ and $\mu =(n-1)\lam$ which is a
contradiction to our assumption. $\square$
\enddemo

%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%-------------------------------------------------------------------------------
\proclaim{1.4. Lemma}$B_{V,V}|_{V(2\lam)\subset V\ten V}=\kap \circ
id_{V(2\lam)}$ for some $\kap\in K$.
\endproclaim
\demo{Proof}This follows from the fact that $B_{V,V}$ is a morphism of modules
and that $V(2\lam)$ has multiplicity one in $V\ten V$.
\enddemo
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%-------------------------------------------------------------------------------
\subheading{1.5}Now we are ready to prove theorem 0.4. Let $\hatlam$ denote the
highest weight of the module $V^*$, dual to $V$. We are going to prove
the quadraticity of the algebra $A(V^*)$. Let
$W=\sum_{\vmu\subset V^*\ten V^*,\mu<2\hatlam}\vmu\subset V^*\ten V^*$. Then
the quadraticity of the
algebra $A(V^*)$ is equivalent to
$$
\sum_{i=1}^{n-1}V^{*\ten i-1}\ten W\ten V^{*\ten n-i-1}=
\sum_{\vmu\subset V^{*\ten n},\mu<n\hatlam}\vmu\subset V^{*\ten n}\tag{$**$}
$$
Let $V^n_i=V^{\ten i-1}\ten V(2\lam)\ten V^{\ten n-i-1}\subset \vtn$. Then
the orthogonal complement to the left hand side of ($**$) in $\vtn$ is equal to
$\bigcap _{i=1}^n V^n_i$ and the orthogonal complement to the right hand side
of ($**$) is equal to $\vnl$. Hence ($**$) is equivalent to the following
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%-------------------------------------------------------------------------------
\proclaim{Proposition}$\bigcap _{i=1}^n V^n_i=\vnl$
\endproclaim
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%-------------------------------------------------------------------------------
\demo{Proof}After corollary 1.3 it is enough to check that the left hand side
is
an eigenspace of $S$ (since we have obvious embedding $\vnl\subset\bigcap
_{i=1}^n V^n_i$).
For any $1\leq i\leq n-1$ denote by $B^{i,i+1}$ the morphism
$id_{V^{\ten i-1}}\ten B_{V,V}\ten id_{V^{\ten n-i-1}}$. Then the hexagon
identities imply that $B_{V,V^{\ten n-1}}=B^{n-1,n}\circ ...\circ B^{1,2}$
and $B_{V^{\ten n-1},V}=B^{1,2}\circ ...\circ B^{n-1,n}$. Let $\kap$ be as
in 1.4. Then
$$
\align
S|_{\bigcap _{i=1}^n V^n_i}=
B_{V^{\ten  n-1},V}\circ B_{V,V^{\ten n-1}}|_{\bigcap _{i=1}^n V^n_i}=&\\
B^{1,2}\circ B^{2,3}\circ ...\circ B^{n-1,n}\circ B^{n-1,n}\circ ...
\circ B^{2,3}\circ B^{1,2}|_{\bigcap _{i=1}^n V^n_i}&=
\kap ^{2(n-1)}\circ id _{\bigcap _{i=1}^n V^n_i}
\endalign
$$
which finishes the proof. $\square$
\enddemo


%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%-------------------------------------------------------------------------------
\subheading{1.6. Concluding remarks}In [Be] R.~Bezrukavnikov extended Kostant's
result, showing that for $q=1$ not only is the algebra $A(V)$ quadratic, but it
is also a  Koszul algebra (cf. for example [BG]). It would be interesting to
extend this result to the quantum case. One idea in this direction could be the
following. It is remarked in [BG] that if $V$ is a vector space and
$B\in \End V$ is a Yang-Baxter operator (cf. [D1] or [BG] ) which is
unitary (i.e. $B^2=id$) then the subspace
$\{ x\in V\ten V|\ Bx=x\} $ of $V\ten V$ defines a Koszul algebra. In our
situation one can show that in fact $V(2\lam)\in V\ten V$ is an eigenspace
of $B_{V,V}$. However $B_{V,V}^2\neq id$, but one may try to prove the
statement
of [BG] holds for a broader class of Yang-Baxter operators, which will include
$B_{V,V}$. This will prove the Koszulity of the dual algebra of $A(V)$
and hence the Koszulity of $A(V)$ itself (we refer to [BG] for the definition
of duality for quadratic algebras and its relation to Koszulity).
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%-------------------------------------------------------------------------------
\subheading{1.7. Acknowledgements}I am grateful to A.~Joseph for posing tis
problem and stimulating discussions. I also would like to thank J.~Bernstein,
D.~Gaitsgory and S.~Shnider for valuable conversations concerning the subject.


%\heading 1.Quasi-Hopf algebras and Drinfeld's model
%\endheading
%All definitions and results of this section are due to V.~G.~Drinfeld.
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%-------------------------------------------------------------------------------
%\subheading{A.1}Let $\CC [[h]]$ denote the ring of formal power series in a
%parameter $h$. We have an embedding of rings $i:\,\CC (q)\hookrightarrow \CC
%%%[[h]]$ such that $i(q)=exp(h/2)$. We shall denote by $\uhg$ the $\CC [[h]]$
%-Hopf algebra $\uqg\ten _\CC (q)\ten \CC [[h]]$. We shall denote by $\grh
%%%[[h]]$ the sublagebra $U_0\ten \CC [[h]]$ of $\uhg$.
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%-------------------------------------------------------------------------------
%\subheading{A.2. Quasi-Hopf algebras}
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%-------------------------------------------------------------------------------
%\subheading{1.3. Drinfeld's model}The following theorem is due to Drinfeld.
%\proclaim{Theorem}As a quasitriangular \qh the algebra $\uhg$ is equivalent to
%%%a universal algebra $A=(\ug[[h]],\Del,exp(ht),\Phi$ where
%$\Del$ is the usual comultiplication on $\ug[h]]$, $t\in Sym^2(\grg)$ is a
%%%symmetric invariant 2-tensor (corresponding to the Killing form) and
%$\Phi=\Sigma_{i=0}^{\infty}\Phi _i h^i,\ \Phi _i\in \ug ^{\ten 3}$ such
%that $\Phi _0=1\ten 1\ten 1$ and $\Phi _1 =0$. Moreover the equivalence can be
%%%made identical on $\grh [[h]]$.
%\endproclaim
%% FOLLOWING LINE CANNOT BE BROKEN BEFORE 80 CHAR
%-------------------------------------------------------------------------------
%\subheading{A.4. Proo
%f of ...}We shall keep here the notations of 1.3. Accordin to theorem A.3 in
%%%the




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\enddocument


