2016/08/10 by Lu, Ming, Peng, Liangang
#16W30 #18E10 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1608.03106
Let A be an arbitrary hereditary abelian category that may not have enough projective objects. For example, A can be the category of finite-dimensional representations of a quiver or the category of coherent sheaves on a smooth projective curve or on a weighted projective line. Inspired by the works of Bridgeland and Gorsky, we define the semi-derived Ringel-Hall algebra of A, denoted by SDHℤ/2(A), to be the localization of a quotient algebra of the Ringel-Hall algebra of the category of ℤ/2-graded complexes over A. We obtain the following three main results. The semi-derived Ringel-Hall algebra has a natural basis. A twisted version of the semi-derived Ringel-Hall algebra of A is isomorphic to the Drinfeld double of the twisted extended Ringel-Hall algebra Htwe(A) of A. If A has a tilting object T, then its semi-derived Ringel-Hall algebra is isomorphic to the ℤ/2-graded semi-derived Hall algebra SDHℤ/2(add T) of the exact category add T defined by Gorsky, and so is isomorphic to Bridgeland's Hall algebra of \mod (End(T)op).