%revised 5/13/99
%Definitions for input files of invariant book
% Use \b* for l.c. bold 
% Use \b*\ to have space after character
\newcommand{\bg}{\mbox{${\bf g}$}}
\newcommand{\bk}{\mbox{${\bf k}$}}
\newcommand{\ba}{\mbox{${\bf a}$}}
\newcommand{\bm}{\mbox{${\bf m}$}}
\newcommand{\bn}{\mbox{${\bf n}$}}


\newfont{\largefrak}{eufm10 }
\newfont{\smallfrak}{eufm7 }
\newcommand{\frak}[1]{\mbox{\largefrak #1}}
\newcommand{\sfrak}[1]{\mbox{\smallfrak #1}}

% Use \fr* for fracture font 
%No $ needed, but use \fr*\ for space after character
\newcommand{\fra}{\mbox{${\frak a}$}}
\newcommand{\frb}{\mbox{${\frak b}$}}
\newcommand{\frc}{\mbox{${\frak c}$}}
\newcommand{\frd}{\mbox{${\frak d}$}}
\newcommand{\fre}{\mbox{${\frak e}$}}
\newcommand{\frg}{\mbox{${\frak g}$}}
\newcommand{\frh}{\mbox{${\frak h}$}}
\newcommand{\frk}{\mbox{${\frak k}$}}
\newcommand{\frm}{\mbox{${\frak m}$}}
\newcommand{\frn}{\mbox{${\frak n}$}}
\newcommand{\frp}{\mbox{${\frak p}$}}
\newcommand{\frs}{\mbox{${\frak s}$}}
\newcommand{\fru}{\mbox{${\frak u}$}}
\newcommand{\frv}{\mbox{${\frak v}$}}
\newcommand{\frz}{\mbox{${\frak z}$}}
\newcommand{\fgl}{\mbox{${\frak g}{\frak l}$}}
\newcommand{\fsl}{\mbox{${\frak s}{\frak l}$}}
\newcommand{\fso}{\mbox{${\frak s}{\frak o}$}}
\newcommand{\fsu}{\mbox{${\frak s}{\frak u}$}}
\newcommand{\fsp}{\mbox{${\frak s}{\frak p}$}}


%Definition of blackboard Bold
\newfont{\Bbbold}{msbm10}
\newfont{\sBbbold}{msbm7}
\newcommand{\Bbb}[1]{\mbox{\Bbbold #1}}
\newcommand{\sBbb}[1]{\mbox{\sBbbold #1}}

\newcommand{\bR}{\mbox{${\Bbb R}$}}
\newcommand{\bC}{\mbox{${\Bbb C}$}}
\newcommand{\bH}{\mbox{${\Bbb H}$}}
\newcommand{\bN}{\mbox{${\Bbb N}$}}
\newcommand{\bP}{\mbox{${\Bbb P}$}}
\newcommand{\bZ}{\mbox{${\Bbb Z}$}}

\newcommand{\sbR}{\mbox{${\sBbb R}$}}
\newcommand{\sbC}{\mbox{${\sBbb C}$}}
\newcommand{\sbH}{\mbox{${\sBbb H}$}}
\newcommand{\sbN}{\mbox{${\sBbb N}$}}
\newcommand{\sbZ}{\mbox{${\sBbb Z}$}}

% Use \gr* for l.c. greek character. 
%No $ needed, but use \gr*\ for space after character
\newcommand{\gra}{\mbox{$\alpha$}}
\newcommand{\grb}{\mbox{$\beta$}}
\newcommand{\grf}{\mbox{$\phi$}}
\newcommand{\grg}{\mbox{$\gamma$}}
\newcommand{\grd}{\mbox{$\delta$}}
\newcommand{\gre}{\mbox{$\epsilon$}}
\newcommand{\grk}{\mbox{$\kappa$}}
\newcommand{\grl}{\mbox{$\lambda$}}
\newcommand{\grm}{\mbox{$\mu$}}
\newcommand{\grn}{\mbox{$\nu$}}
\newcommand{\grp}{\mbox{$\pi$}}
\newcommand{\grs}{\mbox{$\sigma$}}
\newcommand{\grt}{\mbox{$\tau$}}
\newcommand{\grth}{\mbox{$\theta$}}
\newcommand{\gro}{\mbox{$\omega$}}
\newcommand{\grr}{\mbox{$\rho$}}
\newcommand{\grx}{\mbox{$\xi$}}
\newcommand{\grz}{\mbox{$\zeta$}}

%Calligraphic letters
\newcommand{\cA}{\mbox{${\cal A}$}}
\newcommand{\cB}{\mbox{${\cal B}$}}
\newcommand{\cC}{\mbox{${\cal C}$}}
\newcommand{\cD}{\mbox{${\cal D}$}}
\newcommand{\cE}{\mbox{${\cal E}$}}
\newcommand{\cF}{\mbox{${\cal F}$}}
\newcommand{\cG}{\mbox{${\cal G}$}}
\newcommand{\cH}{\mbox{${\cal H}$}}
\newcommand{\cI}{\mbox{${\cal I}$}}
\newcommand{\cJ}{\mbox{${\cal J}$}}
\newcommand{\cK}{\mbox{${\cal K}$}}
\newcommand{\cL}{\mbox{${\cal L}$}}
\newcommand{\cM}{\mbox{${\cal M}$}}
\newcommand{\cN}{\mbox{${\cal N}$}}
\newcommand{\cO}{\mbox{${\cal O}$}}
\newcommand{\cP}{\mbox{${\cal P}$}}
\newcommand{\cQ}{\mbox{${\cal Q}$}}
\newcommand{\cR}{\mbox{${\cal R}$}}
\newcommand{\cS}{\mbox{${\cal S}$}}
\newcommand{\cT}{\mbox{${\cal T}$}}
\newcommand{\cU}{\mbox{${\cal U}$}}
\newcommand{\cV}{\mbox{${\cal V}$}}
\newcommand{\cW}{\mbox{${\cal W}$}}
\newcommand{\cX}{\mbox{${\cal X}$}}
\newcommand{\cY}{\mbox{${\cal Y}$}}
\newcommand{\cZ}{\mbox{${\cal Z}$}}

%upright letters (log-like operators) in mathmode
\newcommand{\ad}{\mbox{\rm ad}}
\newcommand{\Ad}{\mbox{\rm Ad}}
\newcommand{\Aff}{\mbox{\rm Aff}}
\newcommand{\Ann}{\mbox{\rm Ann}}
\newcommand{\Aut}{\mbox{\rm Aut}}
\newcommand{\bop}{\noindent {\em Proof. \/}}
 %beginning of proof
\newcommand{\Card}{\mbox{\rm Card}}
\newcommand{\Cent}{\mbox{\rm Cent}}
\newcommand{\ch}{\mbox{\rm ch}}
\newcommand{\Cliff}{\mbox{\rm Cliff}}
\newcommand{\Comm}{\mbox{\rm Comm}}
\newcommand{\Covar}{\mbox{\rm Covar}}
\newcommand{\depth}{\mbox{\rm depth}}
\newcommand{\sdepth}{\mbox{\scriptsize{\rm depth}}}
\newcommand{\Der}{\mbox{\rm Der}}
\newcommand{\Det}{\mbox{\rm Det}}
\newcommand{\diag}{\mbox{\rm diag}}
\newcommand{\End}{\mbox{\rm End}}

\newcommand{\eop}{\mbox{${\Box}$} \vspace{1ex}}
 %end of proof

%%%%%%%%%%%%%%%%%%%%%%%%%%% empty box command %%%%%
\newcommand{\ebox}{\framebox[4mm]{\rule{0mm}{3mm}}}

%%%%%%%%%%%%  diagram box command  %%%%%%%%%
\newcommand{\dbox}[1]{\framebox[4mm]{\rule{0mm}{3mm}$#1$}}


\newcommand{\exam}{\vspace{1ex}\noindent{\em Example. \/}}
\newcommand{\exams}{\vspace{1ex}\noindent{\em Examples. \/}}
%format for examples

\newcommand{\exer}{\vspace{1ex}\noindent\nopagebreak{\bf Exercises.}}
%format for exercises

\newcommand{\gr}{\mbox{\rm gr}}
\newcommand{\Gr}{\mbox{\rm Gr}}
\newcommand{\Grass}{\mbox{\rm Grass}}
\newcommand{\GL}{\mbox{\rm GL}}
\newcommand{\sGL}{\mbox{\scriptsize{\rm GL}}}
\newcommand{\Herm}{\mbox{\rm Herm}}
\newcommand{\Hom}{\mbox{\rm Hom}}
\newcommand{\Ind}{\mbox{\rm Ind}}
\newcommand{\Inn}{\mbox{\rm Inn}}
\newcommand{\Int}{\mbox{\rm Int}}
\newcommand{\Ker}{\mbox{\rm Ker}}
\newcommand{\Lie}{\mbox{\rm Lie}}
\newcommand{\mod}{\mbox{\rm mod}}
\newcommand{\Par}{\mbox{\rm Par}}
\newcommand{\Pfaff}{\mbox{\rm Pfaff}}
\newcommand{\Pf}{\mbox{\rm Pf}}
\newcommand{\sPar}{\mbox{\scriptsize{\rm Par}}}
\newcommand{\Pin}{\mbox{\rm Pin}}
\newcommand{\sPin}{\mbox{\scriptsize{\rm Pin}}}
\newcommand{\mult}{\mbox{\rm mult}}
\newcommand{\Norm}{\mbox{\rm Norm}}
\newcommand{\Quot}{\mbox{\rm Quot}}
\newcommand{\Rad}{\mbox{\rm Rad}}
\newcommand{\Rat}{\mbox{\rm Rat}}
\newcommand{\Range}{\mbox{\rm Range}}
\newcommand{\rank}{\mbox{\rm rank}}
\newcommand{\res}{\mbox{\rm res}}
\newcommand{\Res}{\mbox{\rm Res}}
\newcommand{\Span}{\mbox{\rm Span}}
\newcommand{\Spin}{\mbox{\rm Spin}}
\newcommand{\sSpin}{\mbox{\scriptsize{\rm Spin}}}
\newcommand{\SL}{\mbox{\rm SL}}
\newcommand{\SO}{\mbox{\rm SO}}
\newcommand{\Sp}{\mbox{\rm Sp}}
\newcommand{\STab}{\mbox{\rm STab}}
\newcommand{\U}{\mbox{\rm U}}
\newcommand{\SU}{\mbox{\rm SU}}
\newcommand{\sgn}{\mbox{\rm sgn}}
\newcommand{\supp}{\mbox{\rm supp}}
\newcommand{\Tab}{\mbox{\rm Tab}}
\newcommand{\tr}{\mbox{\rm tr}}
\newcommand{\Vect}{\mbox{\rm Vect}}

%% partial derivative relative to variable #1 
%% evaluated at  point #2 (subscript)
\newcommand{\partder}[2]
{\displaystyle{\left(\frac {\partial}{\partial#1}
 \right)_{\! #2}}}

%% expression in variable #1 
%% evaluated at  point #2 (subscript)
\newcommand{\vareval}[2]
{\left(#1 \right)_{_{\scriptstyle{\! #2}}}}


%Left and right pointed brackets (Dirac terminology)
\newcommand{\bra}{\mbox{$\langle$}}
\newcommand{\ket}{\mbox{$\rangle$}}


%bilinear pairing with pointed brackets and middle bar
\newcommand{\bil}[2]{\mbox{$\langle#1\mid#2\rangle$}}

%Cases format in displayed formula.
%arg #1,#2 are first formula and condition
%arg #3,#4 are second formula and condition
%(use $...$ in conditions if math symbols occur)
\newcommand{\twocases}[4]
  {\left\{ \begin{array}{ll} 
    #1 & \mbox{#2} \\
    #3 & \mbox{#4}
  \end{array}
  \right.} 

\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{sch}[thm]{Scholium}