  This is an expanded version of the lectures given by the author on the 3rd
school "Lie algebras, algebraic groups and invariant theory" in Togliatti,
Russia. In these notes we explain the concept of a categorical Kac-Moody action
by studying an example of the category of rational representations of a general
linear group in positive characteristic. We also deal with some more advanced
topics: a categorical action on the polynomial representations and crystals of
categorical actions.
