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

Abstract

By using perverse sheaves on representation spaces of quivers over $k[t]/(t^n)$ 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 $k[t]/(t^n)$ and quantum generalized Kac-Moody algebra are provided.