\documentclass[10pt]{amsart}
\usepackage{amsmath}
\usepackage{amssymb}


\renewcommand{\a}{\alpha}
\renewcommand{\b}{\beta}
\renewcommand{\H}{\mathcal{H}}

\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\bF}{\mathbf{F}}
\newcommand{\nm}{My Name}

\newcommand{\Hom}[3]{\mathrm{Hom}_{#1}\left(#2,#3\right)}
\newcommand{\F}[1][p]{\mathbb{F}_{#1}}
\newcommand{\mat}[4]{\begin{pmatrix} #1 & #2 \\ #3 & #4 \end{pmatrix}}
\newcommand{\sbvec}[3]{\left({#1}_{#2},\ldots, {#1}_{#3}\right)}

\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\su}{support}

\begin{document}
\centerline{\sc \nm}

$$\Q,\ \R,\ \Z,\ \C,\ \bF,\ \H$$

$$\Hom{\Q}{V}{\C}$$

$$\F,\ \F[q]$$

$$\Hom{\F}{\F[q^r]}{\F[q]}$$

$$\GL(V),\ \su(X)$$

$$\mat{1}{0}{0}{1}\mat{a}{b}{c}{d}$$

$$\sbvec{x}{1}{n}$$

$$\sbvec{\sum}{x}{y}$$

\end{document}
