  In the context of varieties of representations of arbitrary quivers, possibly
carrying loops, we define a generalization of Lusztig Lagrangian subvarieties.
From the combinatorial study of their irreducible components arises a structure
richer than the usual Kashiwara crystals. Along with the geometric study of
Nakajima quiver varieties, in the same context, this yields a notion of
generalized crystals, coming with a tensor product. As an application, we
define the semicanonical basis of the Hopf algebra generalizing quantum groups,
which was already equipped with a canonical basis. The irreducible components
of the Nakajima varieties provide the family of highest weight crystals
associated to dominant weights, as in the classical case.
