  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 $\H(C_1(\mathscr{P}))$ 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 in
the generators in $\H(C_1(\mathscr{P}))$, and obtain quantum Serre relations in
a quotient of certain twisted version of $\H(C_1(\mathscr{P}))$. Moreover, we
establish relations between the degenerate Hall algebra, twisted Hall algebra
of $A$ and those of $C_1(\mathscr{P})$, respectively.
