DIAGRAM CATEGORIES FOR Uq-TILTING MODULES AT ROOTS OF UNITY

• DOI: 10.1007/s00031-016-9363-z
• Publication Date: 2016-01-25T07:48:05Z
• Authors: H. H. ANDERSEN; D. TUBBENHAUER
• Keywords: Geometric Topology (math.GT); Diagram categories; 01 natural sciences; Categorification; Quantum Groups; Mathematics - Geometric Topology; Mathematics - Quantum Algebra; FOS: Mathematics; Quantum Algebra (math.QA); Roots of unity; Representation Theory (math.RT); 0101 mathematics; Mathematics - Representation Theory

We give a diagrammatic presentation of the category of $\textbf{U}_q(\mathfrak{sl}_2)$-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.<br/>50 pages, lots of figures, revised version (e.g. added suggestions of two referees), to appear in Transform. Groups, comments welcome<br/>