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# Generic Extensions and Canonical Bases for Cyclic Quivers

Published:2007-12-01
Printed: Dec 2007
• Bangming Deng
• Jie Du
• Jie Xiao
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## Abstract

We use the monomial basis theory developed by Deng and Du to present an elementary algebraic construction of the canonical bases for both the Ringel--Hall algebra of a cyclic quiver and the positive part $\bU^+$ of the quantum affine $\frak{sl}_n$. This construction relies on analysis of quiver representations and the introduction of a new integral PBW-like basis for the Lusztig $\mathbb Z[v,v^{-1}]$-form of~$\bU^+$.
 MSC Classifications: 17B37 - Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] 16G20 - Representations of quivers and partially ordered sets