Multiplication Formulas and Canonical Bases for Quantum Affine gln

Du, Jie ; Zhao, Zhonghua

Canadian journal of mathematics, Tome 70 (2018) no. 4, pp. 773-803

Résumé

We will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra $\mathfrak{H}\vartriangle \,(n)$ of a cyclic quiver $\Delta \,(n)$ . As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, a fact established by J. Y. Guo and C. M. Ringel, and derive a recursive formula to compute them. We will further use the formula and the construction of a monomial basis for $\mathfrak{H}\vartriangle \,(n)$ given by Deng, Du, and Xiao together with the double Ringel-Hall algebra realisation of the quantum loop algebra ${{U}_{v}}({{\widehat{\mathfrak{g}\mathfrak{l}}}_{n}})$ given by Deng, Du, and Fu to develop some algorithms and to compute the canonical basis for $U_{v}^{+}({{\widehat{\mathfrak{g}\mathfrak{l}}}_{n}})$ . As examples, we will show explicitly the part of the canonical basis associated with modules of Lowey length at most 2 for the quantum group ${{U}_{v}}({{\widehat{\mathfrak{g}\mathfrak{l}}}_{2}})$ .

Détail

DOI : 10.4153/CJM-2017-009-4

Mots-clés : 16G20, 20G42, Ringel-Hall algebra, quantum group, cyclic quiver, monomial basis, canonical basis

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