algebraic-groups.sty

% !TEX root = 6490.tex

% special characters
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cT}{\mathcal{T}}
\newcommand{\cU}{\mathcal{U}}

\newcommand{\dA}{\mathbf{A}}
\newcommand{\dalpha}{\bm{\alpha}}
\newcommand{\dC}{\mathbf{C}}
\newcommand{\dF}{\mathbf{F}}
\newcommand{\dH}{\mathbf{H}}
\newcommand{\dmu}{\bm{\mu}}
\newcommand{\dN}{\mathbf{N}}
\newcommand{\dP}{\mathbf{P}}
\newcommand{\dR}{\mathbf{R}}
\newcommand{\dQ}{\mathbf{Q}}
\newcommand{\dS}{\mathbf{S}}
\newcommand{\dV}{\mathbf{V}}
\newcommand{\dZ}{\mathbf{Z}}

\newcommand{\eR}{\mathrm{R}}

\newcommand{\fa}{\mathfrak{a}}
\newcommand{\fb}{\mathfrak{b}}
\newcommand{\fe}{\mathfrak{e}}
\newcommand{\ff}{\mathfrak{f}}
\newcommand{\fg}{\mathfrak{g}}
\newcommand{\fh}{\mathfrak{h}}
\newcommand{\fH}{\mathfrak{H}}
\newcommand{\fm}{\mathfrak{m}}
\newcommand{\fp}{\mathfrak{p}}
\newcommand{\ft}{\mathfrak{t}}

\newcommand{\sA}{\mathscr{A}}
\newcommand{\sE}{\mathscr{E}}
\newcommand{\sF}{\mathscr{F}}
\newcommand{\sH}{\mathscr{H}}
\newcommand{\sL}{\mathscr{L}}
\newcommand{\sM}{\mathscr{M}}
\newcommand{\sN}{\mathscr{N}}
\newcommand{\sO}{\mathscr{O}}
\newcommand{\sR}{\mathscr{R}}
\newcommand{\sV}{\mathscr{V}}

% algebraic groups 
\newcommand{\typeA}{\mathrm{A}}
\newcommand{\typeB}{\mathrm{B}}
\newcommand{\typeC}{\mathrm{C}}
\newcommand{\typeD}{\mathrm{D}}
\newcommand{\typeE}{\mathrm{E}}
\newcommand{\typeF}{\mathrm{F}}
\newcommand{\typeG}{\mathrm{G}}

\newcommand{\Ga}{\mathbf{G}_\mathrm{a}}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\GSp}{GSp}
\newcommand{\Gm}{\mathbf{G}_\mathrm{m}}
\DeclareMathOperator{\HSpin}{HSpin}
\DeclareMathOperator{\Iso}{Iso}
\DeclareMathOperator{\Or}{O}
\DeclareMathOperator{\PGL}{PGL}
\DeclareMathOperator{\Pin}{Pin}
\DeclareMathOperator{\PSL}{PSL}
\DeclareMathOperator{\Sim}{Sim}
\DeclareMathOperator{\SL}{SL}
\DeclareMathOperator{\SO}{SO}
\DeclareMathOperator{\Sp}{Sp}
\DeclareMathOperator{\Spin}{Spin}
\DeclareMathOperator{\SU}{SU}

\newcommand{\gl}{\mathfrak{gl}}
\newcommand{\Sl}{\mathfrak{sl}}
\newcommand{\so}{\mathfrak{so}}
\newcommand{\fsp}{\mathfrak{sp}}

% special operators
\DeclareMathOperator{\adjoint}{ad}
\DeclareMathOperator{\automorphisms}{Aut}
\DeclareMathOperator{\centralizer}{C}
\DeclareMathOperator{\characters}{X}
\DeclareMathOperator{\clifford}{Cl}
\DeclareMathOperator{\derivations}{Der}
\DeclareMathOperator{\diag}{D} % diagonalizable group
\DeclareMathOperator{\diagonal}{diag} % diagonal matrix
\DeclareMathOperator{\dgalois}{DGal} % differential Galois group
\DeclareMathOperator{\dynkin}{Dyn}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\filtration}{fil}
\DeclareMathOperator{\flag}{Fl}
\DeclareMathOperator{\galois}{Gal}
\DeclareMathOperator{\graded}{gr}
\DeclareMathOperator{\grassmannian}{Gr}
\DeclareMathOperator{\h}{H}
\DeclareMathOperator{\image}{im}
\DeclareMathOperator{\induce}{ind}
\DeclareMathOperator{\inner}{Inn}
\DeclareMathOperator{\lie}{Lie}
\DeclareMathOperator{\lierad}{rad}
\DeclareMathOperator{\matrices}{M}
\DeclareMathOperator{\norm}{N}
\DeclareMathOperator{\normalizer}{N}
\DeclareMathOperator{\permutations}{Perm}
\DeclareMathOperator{\picard}{Pic}
\DeclareMathOperator{\proj}{Proj}
\DeclareMathOperator{\rank}{rk}
\DeclareMathOperator{\restrict}{res}
\DeclareMathOperator{\roots}{R} % root datum
\DeclareMathOperator{\spectrum}{Spec}
\DeclareMathOperator{\stabilizer}{Stab}
\DeclareMathOperator{\stricttransporter}{stTransp}
\DeclareMathOperator{\trace}{tr}
\DeclareMathOperator{\transporter}{Transp}
\DeclareMathOperator{\volume}{vol}
\DeclareMathOperator{\weil}{R} % Weil restriction
\DeclareMathOperator{\weyl}{W}
\DeclareMathOperator{\zentrum}{Z} % center

% categories
\newcommand{\affine}[1]{\mathsf{Aff}_{#1}}
\newcommand{\algebras}[1]{\mathsf{Alg}_{#1}}
\newcommand{\groups}{\mathsf{Grp}}
\newcommand{\modules}{\mathsf{Mod}}
\newcommand{\reductivegroups}{\mathsf{RdGp}}
\newcommand{\rep}{\mathsf{Rep}}
\newcommand{\rootdata}{\mathsf{RtDt}}
\newcommand{\schemes}[1]{\mathsf{Sch}_{#1}}
\newcommand{\sets}{\mathsf{Set}}
\DeclareMathOperator{\sheaves}{Sh}

% miscellaneous commands
\newcommand{\derived}{\mathcal{D}}
\newcommand{\epic}{\twoheadrightarrow}
\newcommand{\etale}{\textnormal{\'et}}
\newcommand{\fppf}{\mathrm{fppf}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\iso}{\xrightarrow\sim}
\newcommand{\jet}{\mathrm{J}}
\newcommand{\mat}[4]{\begin{pmatrix} #1 & #2 \\ #3 & #4 \end{pmatrix}}
\newcommand{\monic}{\hookrightarrow}
\newcommand{\nilpotent}{\mathrm{n}}
\newcommand{\rad}{\mathcal{R}}
\newcommand{\semisimple}{\mathrm{ss}}
\newcommand{\separable}{\mathrm{sep}}
\newcommand{\tangent}{\mathrm{T}}
\newcommand{\torsion}[2]{\prescript{}{#1}{#2}}
\newcommand{\transpose}[1]{\prescript{\mathrm{t}}{}{#1}}
\newcommand{\unipotent}{\mathrm{u}}
\newcommand{\urad}{\mathcal{R}_\mathrm{u}}
\newcommand{\zariski}{\mathrm{zar}}

% theorem environments
\newtheorem{conjecture}[subsubsection]{Conjecture}
\newtheorem{corollary}[subsubsection]{Corollary}
\newtheorem{lemma}[subsubsection]{Lemma}
\newtheorem{theorem}[subsubsection]{Theorem}
\newtheorem{definition}[subsubsection]{Definition}
\theoremstyle{definition} % contents are not italic
\declaretheorem[sibling=subsubsection, qed=$\triangleright$]{example}

\newenvironment{hard}{$\star$}{$\star$}