Indecomposable 1-morphisms of \dot{U}^+_3 and the canonical basis of U_q^+(sl_3)

Abstract

We compute the indecomposable objects of \dot{U}^+_3 - the category that categorifies the positive half of the quantum sl_3, and we decompose an arbitrary object into indecomposable ones. On decategorified level we obtain the Lusztig's canonical basis of the positive half U^+_q(sl_3) of the quantum sl_3. We also categorify the higher quantum Serre relations in U_q^+(sl_3), by defining a certain complex in the homotopy category of ${\dot{U}}_3^{+}$ that is homotopic to zero. We work with the category ${\dot{U}}_3^{+}$ that is defined over the ring of integers. This paper is based on the (extended) diagrammatic calculus introduced to categorify quantum groups.

arXiv comment27 pages, TikZ figures

References in corpus (3)

• Graded decomposition numbers for cyclotomic Hecke algebras
• Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products
• Canonical bases and Khovanov-Lauda algebras

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