On two Hall algebra approaches to odd periodic triangulated categories

Given an odd-periodic algebraic triangulated category, we compare Bridgeland’s Hall algebra in the sense of Bridgeland (2013) and Gorsky (2014), and the derived Hall algebra in the sense of Toen (2006), Xiao and Xu (2008) and Xu and Chen (2013), and show that the former one is the twisted form of the tensor product of the latter one and a suitable group algebra.