Canonical bases and KLR-algebras. (English) Zbl 1229.17019

Varagnolo, M.; Vasserot, E.

J. Reine Angew. Math. 659, 67-100 (2011).

Let \(_{\mathcal{A}}\mathbf{f}\) be Lusztig’s integral form of the negative half of the quantum universal enveloping algebra associated with a quiver \(\Gamma\). There is an isomorphism between \(_{\mathcal{A}}\mathbf{f}\) and the direct sum of Grothendieck groups of the categories of graded finitely generated modules over certain Khovanov-Lauda-Rouquier algebras. It the present paper the authors show that this isomorphism maps the canonical basis of \(_{\mathcal{A}}\mathbf{f}\) to classes of indecomposable projective modules.

Reviewer: Volodymyr Mazorchuk (Uppsala)

MSC:

17B37	Quantum groups (quantized enveloping algebras) and related deformations
16E30	Homological functors on modules (Tor, Ext, etc.) in associative algebras

Keywords:

Khovanov-Lauda-Rouquier algebra; canonical basis; categorification; quantum group; Cartan datum; Yoneda extension algebra; quantum universal enveloping algebra

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