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arXiv · 2107.03095

Tame quivers and affine bases I: a Hall algebra approach to the canonical bases

Jie Xiao·Han Xu·Minghui Zhao
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Abstract

For quantum group of affine type, Lusztig gave an explicit construction of the affine canonical basis by simple perverse sheaves. In this paper, we construct a bar-invariant basis by using a PBW basis arising from representations of the corresponding tame quiver. We prove that this bar-invariant basis coincides with Lusztig's canonical basis and obtain a concrete bijection between the elements in theses two bases. The index set of these bases is listed orderly by modules in preprojective, regular non-homogeneous, preinjective components and irreducible characters of symmetric groups. Our results are based on the work of Lin-Xiao-Zhang and closely related with the work of Beck-Nakajima. A crucial method in our construction is a generalization of that by Deng-Du-Xiao.

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Jie Xiao, Han Xu, Minghui Zhao. 2023-08-28. Tame quivers and affine bases I: a Hall algebra approach to the canonical bases. https://doi.org/10.1016/j.jalgebra.2023.06.023

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