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In this paper, we study 2-representations of 2-quantum groups (in the senseof Rouquier and Khovanov-Lauda) categorifying tensor products of irreduciblerepresentations. Our aim is to construct knot homologies categorifyingReshetikhin-Turaev invariants of knots for arbitrary representations, whichwill be done in a follow-up paper.  We consider an algebraic construction of these categories, via an explicitdiagrammatic presentation, generalizing the cyclotomic quotient of the quiverHecke algebra. One of our primary results is that these categories coincidewhen both are defined.  We also investigate finer structure of these categories. Like many similarrepresentation-theoretic categories, they are standardly stratified and satisfya double centralizer property with respect to their self-dual modules. Thestandard modules of the stratification play an important role, as Vermas do inmore classical representation theory, as test objects for functors.  The existence of these representations has consequences for the structure ofpreviously studied categorifications; it allows us to prove the non-degeneracyof Khovanov and Lauda's 2-category (that its Hom spaces have the expecteddimension) in all symmetrizable types, and that the cyclotomic quiver Heckealgebras are symmetric Frobenius.

Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products