There are several different ways to construct affine canonical bases, in addition to approaches by Lusztig and Kashiwara. In this paper we present a different approach to canonical bases via Hall algebras and representations of tame quivers over finite fields. The main idea is to tensor together integral bases constructed for cyclic quivers and Kronecker quivers with those from the preinjective and preprojective parts of tame quiver representations. Several different bases: a PBW type basis, a monomial basis, and a bar-invariant basis are constructed and their relations to the canonical basis are discussed. The result also answers a question by Nakajima.
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Zongzhu Lin, Jie Xiao, Guanglian Zhang, Representations of Tame Quivers and Affine Canonical Bases. Publ. Res. Inst. Math. Sci. 47 (2011), no. 4, pp. 825–885DOI 10.2977/PRIMS/53