Crystal bases and two-sided cells of quantum affine algebras

By J. Beck, H. Nakajima. 2002

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Let $\g$ be an affine Kac-Moody Lie algebra. Let $\U^+$ be the positive part of the Drinfeld-Jimbo quantum enveloping algebra associated to $\g$. We construct a basis of $\U^+$ which is related to the Kashiwara-Lusztig global crystal basis (or canonical basis) by an upper triangular matrix (with respect to an explicitly defined ordering) with 1's on the diagonal and with above diagonal entries in $q_s^{-1} \Z[q_s^{-1}]$. Using this construction we study the global crystal basis $\B(\Um)$ of the modified quantum enveloping algebra defined by Lusztig. We obtain a Peter-Weyl like decomposition of the crystal $\B(\Um)$ (Theorem 4.18), as well as an explicit description of two-sided cells of $\B(\Um)$ and the limit algebra of $\Um$ at $q=0$ (Theorem 6.45).

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7883f10ca59025d51292d21adf8c737916ad596e

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Citations in the corpus

A categorical approach to Weyl modules (2009, methodology): “It was later proved in [CL] for slr+1, in [FoL] for algebras of type A, D, E. The general case can be deduced by passing to the quantum group situation and using results in [K], [ BN ].”
Affine Mirković-Vilonen polytopes (2011, background): “(iv) As mentioned at the beginning of section 1.3, Beck in [6], Beck and Nakajima in [8], and Ito in [30] construct PBW bases of Uq(n+) for g of affine type.”
Generic Extensions and Canonical Bases for Cyclic Quivers (2007, methodology): “We then prove that this basis agrees with Lusztig’s canonical basis; see [16, 30].”
Demazure Modules, Fusion Products and Q-Systems (2013, background): “Together with results established in [2], [6], it follows that the local Weyl module is the q = 1 limit of a tensor product of irreducible representations of quantum affine algebra.”
Weyl modules for the twisted loop algebras (2007, background): “Part (iii) can be deduced for the general case by using results of [2],[12],[14] for quantum affine algebras.”