pres_definitions.tex

\newcommand{\inserttikzfigure}[2]{
											\begin{center}
												\begin{figure}[h]
													\centering 
													\input{#1}
													\caption{#2}
												\end{figure} 
											\end{center}
										}										
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% THEOREMS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%% CMUC %%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SYMBOLS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\et}{\,\&\,}
\renewcommand{\k}{\Bbbk}
\renewcommand{\C}{\mathbb{C}}
\newcommand{\octo}{\mathbb{O}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\lie}[1]{\mathfrak{#1}}
\newcommand{\rep}[1]{\mathbb{#1}}
\newcommand{\bun}[1]{\mathcal{#1}}
\newcommand{\oppar}{\overline{\lie{p}}}
\newcommand{\Hom}{\mathrm{Hom}}
\newcommand{\pproj}[1]{\Pi_{#1}^\lie{g}}
\newcommand{\floor}[1]{\left[ #1 \right]}

\newcommand{\roots}{\Phi} % the set of roots
\newcommand{\sroots}{\Delta} % the set of positive simple roots
\newcommand{\weights}{\Lambda} % weight lattice

\renewcommand{\a}{\alpha}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% OPERATORS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newcommand{\im}{\mathrm{im}\,}
%\newcommand{\lap}{\square}
%\newcommand{\td}{\mathrm{d}^\lie{g}}
%\newcommand{\id}{\mathrm{Id}\,}
%\newcommand{\plap}{\lap_\lie{g}}
%\newcommand{\lcd}{\partial\,}
% \renewcommand{\lhd}{\mathop{\partial^*}}
% \renewcommand{\lhd}{\mathop{\delta}}
 \renewcommand{\lhd}{\operatorname{\delta}}

\newcommand{\bigo}{\mathcal{O}}

\newcommand{\conj}[1]{\overline{#1}}
\newcommand{\jscal}[2]{\langle #1\,|\, #2 \rangle}
\newcommand{\oscal}[2]{\left( #1 \,|\, #2 \right)}
 

% \DeclareMathOperator{\hom}
\DeclareMathOperator{\proj}{\mathrm{proj}}
\DeclareMathOperator{\im}{\mathrm{im}}
\DeclareMathOperator{\lap}{\square}
\DeclareMathOperator{\id}{\mathrm{Id}}
\DeclareMathOperator{\td}{\mathrm{d}^\lie{g}}
\DeclareMathOperator{\plap}{\lap_\lie{g}}
\DeclareMathOperator{\lcd}{\partial}
% \DeclareMathOperator{\lhd}{\partial^*}
\DeclareMathOperator{\tr}{\mathrm{Tr}}
\DeclareMathOperator{\ad}{\mathrm{ad}}
\DeclareMathOperator{\Ad}{\mathrm{Ad}}
\DeclareMathOperator{\Id}{\mathrm{Id}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ENVIRONMENTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newenvironment{smatrix}{\left(\begin{smallmatrix}}{\end{smallmatrix}\right)}

%%%%%%%%%%%%%%%%%%%%%%%%%% CD %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\dsum}{\oplus}               % small direct sum
\newcommand{\Dsum}{\bigoplus}            % big direct sum
\newcommand{\tens}{\mathbin{\otimes}}    % small tensor product
\newcommand{\Tens}{\bigotimes}           % big tensor product
\newcommand{\cartan}{\mathbin{\odot}}    % small Cartan product
\newcommand{\Cartan}{\bigodot}           % big Cartan product
\newcommand{\subideal}{\ltimes}          % semidirect product for algebras
\newcommand{\idealsub}{\rtimes}          %  with ideal on right or left
\newcommand{\subnormal}{\ltimes}         % semidirect product for groups with
\newcommand{\normalsub}{\rtimes}         %  normal subgroup on right or left
\newcommand{\intersect}{\mathinner{\cap}}% intersection
\newcommand{\setdif}{\mathbin{\mathrm -}}% or {\smallsetminus}
\newcommand{\skwend}{\mathinner{\vartriangle}} % skew endomorphism
%
\newcommand{\act}{\mathinner{\cdot}}
\newcommand{\dual}{^{*\!}}
\newcommand{\Cinf}{\mathrm{C}^\infty}
%
\newcommand{\oper}[3][n]{\newcommand{#2}{\mathop{\mathrm{#3}}\ifx
  n#1\nolimits\else\limits\fi}}
\newcommand{\rsoper}[3][n]{\newcommand{#2}{\mathop{\mathrmsl{#3\mkern1mu}}\ifx
  n#1\nolimits\else\limits\fi}}
\newcommand{\symb}[2]{\newcommand{#1}{{\mathit{#2}}}}
\newcommand{\rssymb}[2]{\newcommand{#1}{{\mathrmsl{#2\mkern1mu}}}}
\newcommand{\calsymb}[2]{\newcommand{#1}{{\mathcal{#2}}}}
\newcommand{\bbsymb}[2]{\newcommand{#1}{{\mathbb{#2}}}}

\rsoper\kernel{ker}
\rsoper\image{im}
% \rsoper\alt{alt}           % alternating part
\rsoper\sym{sym}           % symmetric part
\rsoper\trace{tr}          % trace of an endomorphism
\rsoper\detm{det}          % determinant
\rsoper\divg{div}
\rsoper\Sdivg{Sdiv}
\rsoper\Twist{Twist}
\rsoper\project{proj}
\rsoper\represent{repr}
\rssymb\ev{ev}
\rssymb\iden{id}
%
\newcommand{\cross}{\mathbin{{\times}\!}\low}
\newcommand{\from}{\colon}
\newcommand{\isom}{\cong}
\newcommand{\hQuabla}{\hat{\pmb\square}}
\newcommand{\Quabla}{\pmb{\square}}
\newcommand{\Lie}{{\mathcal L}}
%
\newcommand{\conf}{\mathsf{c}}
\newcommand{\cip}{\ip}
\newcommand{\cform}{\eta}
\newcommand{\g}{\lie{g}}
\newcommand{\p}{\lie{p}}
\newcommand{\Lieb}[1]{[#1]}
\bbsymb\E{E}\bbsymb\W{W}
\newcommand{\T}{\lie{m}}
\newcommand{\inner}{\mathbin{\lrcorner}}
\newcommand{\capinner}{\mathbin{\lower3pt\hbox{$\urcorner$}}}
\newcommand{\low}{^{\vphantom x}}
\newcommand{\dT}{d\low_{\T}}
\newcommand{\dTd}{d\low_{\T^*}}
\newcommand{\delT}{\delta\low_{\T}}
\newcommand{\delTd}{\delta\low_{\T^*}}
\newcommand{\delTdM}{\delta\low_{T\dual M}}
\newcommand{\gM}{\g\low_M}
\newcommand{\InvDer}{\nabla^\cform}
\newcommand{\TwConn}{\nabla^\g}
\newcommand{\deltw}{\delta^\g}
\newcommand{\dtw}{d^\g}
\newcommand{\Rtw}{R^\g}
\newcommand{\QuaTw}{\Quabla_\g}
\newcommand{\Qtw}{Q}
\newcommand{\PiTw}{\Pi}
\calsymb\cD{D}\calsymb\cG{G}\calsymb\cH{H}\calsymb\cK{K}\calsymb\cL{L}

\newcommand{\OpTw}{\cD}
\newcommand{\cptw}{\mathbin{\scriptstyle\sqcup}}
\newcommand{\delinv}{\delta^\cform}
\newcommand{\dinv}{d^\cform}
\newcommand{\Rinv}{R^\cform}
\newcommand{\hQuaInv}{\hQuabla_\cform}
\newcommand{\QuaInv}{\Quabla_\cform}
\newcommand{\Qinv}{Q_\cform}
\newcommand{\PiInv}{\Pi^\cform}
\newcommand{\OpInv}{\cD^\cform}
\newcommand{\Opinv}{\cD_\cform}
\newcommand{\cpinv}{\cptw\low_\cform}
\newcommand{\TwIso}{\Psi_\W}
\newcommand{\captw}{\mathbin{\scriptstyle\sqcap}}
\newcommand{\capinv}{\captw\low_\cform}
\newcommand{\Dirac}{{\smash{\raise1pt\hbox{$\not$}}\mkern-1.5mu D}}

%%%%%%%%%%%%% F4/Spin9
\newcommand{\field}{\Bbbk}
\newcommand{\reals}{\mathbb{R}}
\newcommand{\complex}{\mathbb{C}}
\newcommandtwoopt{\projplane}[2][2][P]{\mathbb{O}{#2}^{#1}}
\newcommandtwoopt{\jalg}[2][G][\octo]{J_{#1}(#2)}

\newcommand{\set}[1]{\left\lbrace #1\right\rbrace}
\renewcommand{\mid}{\middle \mid}
\newcommand{\sform}[1]{\mathrm{II}(#1)}

\newenvironment{psmatrix}
  {\left(\begin{smallmatrix}}
  {\end{smallmatrix}\right)}
	
	%%%%%%%%%%%%%%%%%% G2/P1
	
	\newcommand{\uenv}[1]{\mathfrak{U(#1)}}


\newcommand{\pd}[1]{\partial_{#1}}
\newcommand{\euler}{\mathbb{E}}
\newcommand{\wdeg}{\mathrm{wdeg}\,}
\newcommand{\actpol}{\widehat{\pi}_{\lambda}}

\newcommand{\cas}{\mathrm{Cas}_\mathbb{V}\,}
\newcommand{\vsingtop}{v_\mathrm{sing}^\mathrm{top}}

%%%% 1-graded 

\newcommand{\SL}{\mathrm{SL}}
\def\diag{\mathop {\rm diag} \nolimits}
\let\veps=\varepsilon
%\newcommand*\widebar[1]{\@ifnextchar^{{\wide@bar{#1}{0}}}{\wide@bar{#1}{1}}}
\newcommand{\widebar}[1]{\overline{#1}}
\def\htt{\mathop {\rm ht} \nolimits}
\newcommand{\eus}{\EuScript}
%\let\eus=\EuScript
\let\rarr=\rightarrow
\let\mfrak=\mathfrak
\newcommand{\mcal}{\mathcal}
\def\End{\mathop {\rm End} \nolimits}
\newcommand*\riso{%
  \xrightarrow[]{\raisebox{-0.25em}{\smash{\ensuremath{\sim}}}}%
}
\def\Sol{\mathop {\rm Sol} \nolimits}