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.

27 pages, TikZ figures

References in corpus (1)

• Graded decomposition numbers for cyclotomic Hecke algebras

Cited by in corpus (1)

• The ${\mathfrak{sl}}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality