We give a diagrammatic presentation of the category of U _q (\( \mathfrak{s}\mathfrak{l} \) ₂)-tilting modules for q being a root of unity and introduce a grading on . This grading is a “root of unity phenomenon” and might lead to new insights about link and 3-manifold invariants deduced from . We also give a diagrammatic category for the (graded) projective endofunctors on , indicate how our results could generalize, and collect some “well-known” facts to give a reasonably self-contained exposition.

We show how to use Jantzen’s sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of $\mathbf{U}_{q}$ -tilting modules (for any field $\mathbb{K}$ and any parameter $q\in \mathbb{K}-\{0,-1\}$ ). As an application, we recover the semisimplicity criteria for the Hecke algebras of types $\mathbf{A}$ and $\mathbf{B}$ , the walled Brauer algebras and the Brauer algebras from our more general approach.

We show how to use Jantzen's sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of $\textbf{U}_q$-tilting modules (for any field $\mathbb{K}$ and any parameter $q\in\mathbb{K}-\{0,-1\}$). As an application, we recover the semisimplicity criteria for the Hecke algebras of types $\textbf{A}$ and $\textbf{B}$, the walled Brauer algebras and the Brauer algebras from our more general approach.

In this paper we study of the BGG-categories $\mathcal O_q$ associated to quantum groups. We prove that many properties of the ordinary BGG-category $\mathcal O$ for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when $q$ is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for $\mathcal O$ and for finite dimensional $U_q$-modules we are able to determine all irreducible characters as well as the characters of all indecomposable tilting modules in $\mathcal O_q$. As a consequence of our study of the root of unity case we deduce that the non-root of unity case (including the generic case) behaves like $\mathcal O$.