Modified Ringel-Hall Algebras, Green's formula and Derived Hall Algebras

2017/07/26 by Lin, Ji, Peng, Liangang
#16T10 #18E10 #18E30 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1707.08292

Abstract

In this paper we define the modified Ringel-Hall algebra \cm\ch(\ca) of a hereditary abelian category \ca from the category C^b(A) of bounded ℤ-graded complexes. Two main results have been obtained. One is to give a new proof of Green's formula on Ringel-Hall numbers by using the associative multiplication of the modified Ringel-Hall algebra. The other is to show that in certain twisted cases the derived Hall algebra can be embedded in the modified Ringel-Hall algebra. As a consequence of the second result, we get that in certain twisted cases the modified Ringel-Hall algebra is isomorphic to the tensor algebra of the derived Hall algebra and the torus of acyclic complexes and so the modified Ringel-Hall algebra is invariant under derived equivalences.

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• Semi-derived Ringel-Hall algebras and Hall algebras of odd-periodic relative derived categories
• Semi-derived Ringel-Hall algebras and Drinfeld double
• From Green's formula to Derived Hall algebras
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• Derived Hall Algebras