Mathematical Research Letters

Volume 22 (2015)

Number 1

Indecomposable objects and Lusztig’s canonical basis

Pages: 245 – 278

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n1.a13

Author

Marko Stošic (Mathematical Institute SANU, Beograd, Serbia; and Centro de Análise Matemática, Geometria e Sistemas Dinâmicos (CAMGSD), Departamento de Matemática, Instituto Superior Técnico, Lisbon, Portugal)

Abstract

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

Keywords

categorification, Lusztig’s canonical basis, quantum groups, diagrammatic calculus

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