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

Ito, Ken

Hiroshima Math. J., Tome 40 (2010) no. 1, p. 133-183 / Harvested from Project Euclid

Résumé

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 $U$.

Détail

Publié le : 2010-07-15
Classification:  quantum algebra,  convex basis,  convex order,  17B37,  17B67,  20F55