  In this paper, we study 2-representations of 2-quantum groups (in the sense
of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible
representations. Our aim is to construct knot homologies categorifying
Reshetikhin-Turaev invariants of knots for arbitrary representations, which
will be done in a follow-up paper.
  We consider an algebraic construction of these categories, via an explicit
diagrammatic presentation, generalizing the cyclotomic quotient of the quiver
Hecke algebra. One of our primary results is that these categories coincide
when both are defined.
  We also investigate finer structure of these categories. Like many similar
representation-theoretic categories, they are standardly stratified and satisfy
a double centralizer property with respect to their self-dual modules. The
standard modules of the stratification play an important role, as Vermas do in
more classical representation theory, as test objects for functors.
  The existence of these representations has consequences for the structure of
previously studied categorifications; it allows us to prove the non-degeneracy
of Khovanov and Lauda's 2-category (that its Hom spaces have the expected
dimension) in all symmetrizable types, and that the cyclotomic quiver Hecke
algebras are symmetric Frobenius.
