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Integral basis theorem of cyclotomic Khovanov-Lauda-Rouquier algebras of type A. (English) Zbl 1387.20003

M. Khovanov and A. D. Lauda [Represent. Theory 13, 309–347 (2009; Zbl 1188.81117); Trans. Am. Math. Soc. 363, No. 5, 2685–2700 (2011; Zbl 1214.81113)] and R. Rouquier [“\(2\)-Kac-Moody algebras”, Preprint, arXiv:0812.5023] have introduced an exciting family of new algebras \(\mathcal R_n\), the quiver Hecke algebras, for each oriented quiver. They demonstrated that these algebras categorify the positive part of the enveloping algebras of the corresponding quantum groups. The algebras \(\mathcal R_n\) are naturally \(\mathbb Z\)-graded. M. Varagnolo and E. Vasserot [J. Reine Angew. Math. 659, 67–100 (2011; Zbl 1229.17019)] showed that, under this categorification, the canonical basis of the positive part of the quantum group corresponds to the image of the projective indecomposable modules in the Grothendieck rings of the quiver Hecke algebras when the Cartan matrix is symmetric.
The algebra \(\mathcal R_n\) is infinite dimensional and for every highest weight vector in the corresponding Kac-Moody algebra there is an associated finite dimensional ‘cyclotomic quotient’ \(\mathcal R_n^\Lambda\) of \(\mathcal R_n\). The cyclotomic quiver algebras \(\mathcal R_n^\Lambda\) were originally defined by Khovanov and Lauda [2009, loc. cit.; 2011, loc. cit.] and Rouquier [loc. cit.] who conjectured that these algebras should categorify the irreducible representations of the corresponding quantum group.
In this article, the author proves that the cyclotomic Khovanov-Lauda-Rouquier algebras \(\mathcal R_n^\Lambda\) in type \(A\) are \(\mathbb Z\)-free. Then, he extends the graded cellular basis of \(\mathcal R_n^\Lambda\) constructed by J. Hu and A. Mathas [Adv. Math. 225, No. 2, 598–642 (2010; Zbl 1230.20005)] to \(\mathcal R_n\) and uses this basis to give a classification of all irreducible \(\mathcal R_n\)-modules.
Reviewer: Wei Feng (Beijing)

MSC:

20C08 Hecke algebras and their representations
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16G20 Representations of quivers and partially ordered sets
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References:

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