A primer of group theory for Loop Quantum Gravity and spin-foams

Calculations in Loop Quantum Gravity (LQG) and spin-foams theory rely heavily on group theory of SU(2) and SL2(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SL_{2}({\mathbb {C}})$$\end{document}. Even though many monographs exist devoted to this theory, the different tools needed (e.g. representation theory, harmonic analysis, recoupling theory...) are often dispersed in different books, with different conventions and notations. This was the initial motivation for the compilation of the present document. Generally, these notes can serve three main purposes: a concise introduction for students to the essential mathematical tools of LQG, a convenient compendium for researchers, a translational hub between the conventions of the main references. These notes are aimed both at physicists, caring about their tools being mathematically well grounded, and at mathematicians, curious about how some of their familiar abstract structures can reveal the beauty of quantum gravity.