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, 16G20 show english descriptions Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]
Representations of quivers and partially ordered sets 17B37 - Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23]
16G20 - Representations of quivers and partially ordered sets

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