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

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

2010-12-23

arxiv.org/abs/1012.5257v3

Mathematics

Representation Theory

27 pages

Scientific paper

The category of representations of a Dynkin quiver over local ring $R=k[t]/(t^n)$ is not hereditary any more. The Hall algebra defined on this category doesn't have a well defined coalgebraic structure. In the present paper, the full subcategory of this category, whose objects are the modules assigning free $R$-module to each vertex, is considered. This full subcategory is an exact category. The Ringel-Hall algebra is well defined on this exact category. There exists a coalgebraic structure on the composition subalgebra of this algebra. The geometric realization of the composition subalgebra of this Hall algebra is given under the framework of Lusztig's geometric setting. Moreover the canonical basis and a monomial basis of this subalgebra are constructed by using preserves sheaves. This generalizes the Lusztig's result about the geometric realization of quantum enveloping algebra. As a byproduct, the relation between this subalgebra and quantum generalized Kac-Moody algebra is obtained.

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