Semi-derived Ringel-Hall algebras and Hall algebras of odd-periodic relative derived categories (Q6595562)

Ringel-Hall algebras were first introduced in [\textit{C. M. Ringel}, Banach Cent. Publ. 433--447 (1990; Zbl 0778.16004)] and generalized to the Kac-Moody type in [\textit{J. A. Green}, Invent. Math. 120, No. 2, 361--377 (1995; Zbl 0836.16021)], which provide a nice framework to realize Lie algebras and quantum enveloping algebras. Inspired by the work in [\textit{T. Bridgeland}, Ann. Math. (2) 177, No. 2, 739--759 (2013; Zbl 1268.16017)], semi-derived Ringel-Hall algebras were introduced in [\textit{M. Gorsky}, Int. Math. Res. Not. 2018, No. 1, 138--159 (2018; Zbl 1411.18020)] and further developed in [\textit{M. Lu} and \textit{L. Peng}, Adv. Math. 383, Article ID 107668, 72 p. (2021; Zbl 1460.18007)] and [\textit{J. Lin} and \textit{L. Peng}, J. Algebra 526, 81--103 (2019; Zbl 1410.18013)].\N\NLet \(t\) be a positive integer and \(\mathcal{A}\) be a hereditary abelian category satisfying some finiteness conditions. In this paper under review, the authors define the notion of semi-derived Ringel-Hall algebra of \(\mathcal{A}\) from the category \(\mathcal{C}_{\mathbb{Z}/t}(\mathcal{A})\) of \(\mathbb{Z}/t\)-graded complexes in a slightly different way. They obtain a natural basis, as well as the generators and defining relations of the semi-derived Ringel-Hall algebra. The aim of this paper is to establish a relationship between the semi-derived Ringel-Hall algebras and the derived Hall algebras of the odd-periodic relative derived categories. Via the generators, the authors directly construct an embedding of the derived Hall algebra in the ``larger'' semi-derived Ringel-Hall algebra, which is by definition the tensor algebra of the semi-derived Ringel-Hall algebra and the extended quantum torus by adding ``the half elements'' to the quantum torus.