Duke Mathematical Journal

Poincaré–Birkhoff–Witt bases and Khovanov–Lauda–Rouquier algebras

Syu Kato

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We generalize Lusztig’s geometric construction of the Poincaré–Birkhoff–Witt (PBW) bases of finite quantum groups of type ADE under the framework of Varagnolo and Vasserot. In particular, every PBW basis of such quantum groups is proven to yield a semi-orthogonal collection in the module category of the Khovanov–Lauda–Rouquier (KLR) algebras. This enables us to prove Lusztig’s conjecture on the positivity of the canonical (lower global) bases in terms of the (lower) PBW bases. In addition, we verify Kashiwara’s problem on the finiteness of the global dimensions of the KLR algebras of type ADE.

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Duke Math. J., Volume 163, Number 3 (2014), 619-663.

First available in Project Euclid: 11 February 2014

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Zentralblatt MATH identifier

Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]
Secondary: 16T20: Ring-theoretic aspects of quantum groups [See also 17B37, 20G42, 81R50]


Kato, Syu. Poincaré–Birkhoff–Witt bases and Khovanov–Lauda–Rouquier algebras. Duke Math. J. 163 (2014), no. 3, 619--663. doi:10.1215/00127094-2405388. https://projecteuclid.org/euclid.dmj/1392128880

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