Diagram categories for ${\text{U}}_q$-tilting modules at roots of unity

Abstract

We give a diagrammatic presentation of the category of ${\text{U}}_q({\mathfrak{sl}}_2)$-tilting modules $\mathfrak{T}$ for $q$ being a root of unity and introduce a grading on $\mathfrak{T}$. This grading is a "root of unity phenomenon" and might lead to new insights about link and $3$-manifold invariants deduced from $\mathfrak{T}$. We also give a diagrammatic category for the (graded) projective endofunctors on $\mathfrak{T}$, indicate how our results could generalize and collect some "well-known" facts to give a reasonably self-contained exposition.

50 pages, lots of figures, revised version (e.g. added suggestions of two referees), to appear in Transform. Groups, comments welcome

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