Duke Mathematical Journal

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

Syu Kato

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 11 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1392128880

Digital Object Identifier
doi:10.1215/00127094-2405388

Mathematical Reviews number (MathSciNet)
MR3165425

Zentralblatt MATH identifier
1292.17012

Subjects
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]

Citation

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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