Minimal generators of Hall algebras of 1-cyclic perfect complexes

Let $A$ be the path algebra of a Dynkin quiver over a finite field, and let $C_1(\mathcal{P})$ 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 $\stackrel{˝}{(}{C}_1(\mathcal{P}))$ of $C_1(\mathcal{P})$. Using this PBW-basis, we firstly prove the degenerate Hall algebra of $C_1(\mathcal{P})$ is the universal enveloping algebra of the Lie algebra spanned by all indecomposable objects. Secondly, we calculate the relations in the generators in $\stackrel{˝}{(}{C}_1(\mathcal{P}))$, and obtain quantum Serre relations in a quotient of certain twisted version of $\stackrel{˝}{(}{C}_1(\mathcal{P}))$. Moreover, we establish relations between the degenerate Hall algebra, twisted Hall algebra of $A$ and those of $C_1(\mathcal{P})$, respectively.