  Calculations in Loop Quantum Gravity (LQG) and spin-foams theory rely heavily
on group theory of SU(2) and SL(2,C). 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.
