Hiroshima Mathematical Journal

A new description of convex bases of PBW type for untwisted quantum affine algebras

Ken Ito

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In [8] we classified all ``convex orders'' on the positive root system $\Delta_+$ of an arbitrary untwisted affine Lie algebra ${\mathfrak g}$ and gave a concrete method of constructing all convex orders on $\Delta_+$. The aim of this paper is to give a new description of ``convex bases'' of PBW type of the positive subalgebra $U^+$ of the quantum affine algebra $U=U_q({\mathfrak g})$ by using the concrete method of constructing all convex orders on $\Delta_+$. Applying convexity properties of the convex bases of $U^+$, for each convex order on $\Delta_+$, we construct a pair of dual bases of $U^+$ and the negative subalgebra $U^-$ with respect to a $q$-analogue of the Killing form, and then present the multiplicative formula for the universal $R$-matrix of $U$.

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Hiroshima Math. J., Volume 40, Number 2 (2010), 133-183.

First available in Project Euclid: 2 August 2010

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Zentralblatt MATH identifier

Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]
Secondary: 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]

quantum algebra convex basis convex order


Ito, Ken. A new description of convex bases of PBW type for untwisted quantum affine algebras. Hiroshima Math. J. 40 (2010), no. 2, 133--183. doi:10.32917/hmj/1280754419. https://projecteuclid.org/euclid.hmj/1280754419

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