Geometric approach to Hall algebra of representations of Quivers over local ring

2010/12/23 by Zhaobing Fan, Fan, Zhaobing
Mathematics · Physics and Astronomy · #20G99 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT) #math.RT #msc:20G99

paper · pdf · doi:10.48550/arxiv.1012.5257

Simplify some proofs and some other revisions on the earlier version

openalex publication_date 2010/12/23 · arxiv created 2014/10/22 · arxiv updated 2014/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

By using perverse sheaves on representation spaces of quivers over k[t]/(tⁿ) and jet schemes over flag varieties, we construct a geometric composition algebra \mathbf K under Lusztig's framework on geometric realizations of the negative part of quantum algebras. Simple perverse sheaves in \mathbf K form the canonical basis of \mathbf K. The relationships among the algebra \mathbf K, the composition algebra of locally projective representations of quivers over k[t]/(tⁿ) and quantum generalized Kac-Moody algebra are provided.

Citations

• Notes on affine canonical and monomial bases
• A geometric version of BGP reflection functors
• Notes on Perverse Sheaves and Vanishing Cycles
• Hall algebras and quantum groups

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