Minimal Generators of Hall Algebras of 1-cyclic Perfect Complexes

Abstract

Let A be the path algebra of a Dynkin quiver over a finite field, and let C 1 (P) be the category of 1-cyclic complexes of projective A-modules. In the present paper, we give a PBW-basis and a minimal set of generators for the Hall algebra H (C 1 (P)) of C 1 (P). Using this PBW-basis, we firstly prove the degenerate Hall algebra of C 1 (P) is the universal enveloping algebra of the Lie algebra spanned by all indecomposable objects. Secondly, we calculate the relations in the generators in H (C 1 (P)), and obtain quantum Serre relations in a quotient of certain twisted version of H (C 1 (P)). Moreover, we establish relations between the degenerate Hall algebra, twisted Hall algebra of A and those of C 1 (P), respectively.2010 Mathematics Subject Classification. 16G20, 17B20, 17B37.

Hall algebras and quantum symmetric pairs I: Foundations

“…Remark A variant of Bridgeland's Hall algebra via the module category $$\operatorname{mod}\nolimits ({k}Q\otimes R_1)$$ is studied in an interesting paper by H. Zhang [59] independent of our work, who established a connection to $${\mathbf {U}}^+$$.…”

Modified Ringel–Hall algebras, Green's formula and derived Hall algebras

“…Zhang [31] found the generators and relations of Bridgeland's Hall algebra of Z/t-graded complexes over projectives for t > 2 or t = 0, here Z/0-graded complex means the bounded complex. Zhang also realized the Lie algebra spanned by all indecomposable objects via Hall algebras of Z/1-graded perfect complexes in [32].…”