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