  In these notes we review some basic facts about the LLV Lie algebra. It is a
rational Lie algebra, introduced by Looijenga-Lunts and Verbitsky, acting on
the rational cohomology of a compact K\"{a}hler manifold. We study its
structure and describe one irreducible component of the rational cohomology in
the case of a compact hyperk\"{a}hler manifold.
