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


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The category of representations of a Dynkin quiver over local ring $R=kt-(t^n)$ is not hereditary any more. The Hall algebra defined on this category doesnt 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 Lusztigs geometric setting. Moreover the canonical basis and a monomial basis of this subalgebra are constructed by using preserves sheaves. This generalizes the Lusztigs 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.



Author: Zhaobing Fan

Source: https://archive.org/







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