Minimal Generators of Hall Algebras of 1-cyclic Perfect Complexes

Abstract

Abstract Let $A$ be the path algebra of a Dynkin quiver over a finite field, and let $C_1(\mathscr{P})$ 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 ${\mathcal{H}}\,(C_1(\mathscr{P}))$ of $C_1(\mathscr{P})$. Using this PBW-basis, we firstly prove the degenerate Hall algebra of $C_1(\mathscr{P})$ is the universal enveloping algebra of the Lie algebra spanned by all indecomposable objects. Secondly, we calculate the relations among the generators in ${\mathcal{H}}\,(C_1(\mathscr{P}))$, and obtain quantum Serre relations in a quotient of certain twisted version of ${\mathcal{H}}\,(C_1(\mathscr{P}))$. Moreover, we establish relations between the degenerate Hall algebra, twisted Hall algebra of $A$ and those of $C_1(\mathscr{P})$, respectively.