Hall algebra of morphism category

This paper investigates a universal PBW-basis and a minimal set of generators for the Hall algebra \(\cal{H}(C_2(\cal{P}))\) $$\mathcal{H}(C_2(\mathcal{P}))$$ , where \(C_2(\cal{P})\) $$C_2(\mathcal{P})$$ is the category of morphisms between projective objects in a finitary hereditary exact category \(\cal A\) $$\mathcal{A}$$ . When \(\cal A\) $$\mathcal{A}$$ is the representation category of a Dynkin quiver, we develop multiplication formulas for the degenerate Hall Lie algebra \(\cal{L}\) $$\mathcal{L}$$ , which is spanned by isoclasses of indecomposable objects in \(C_2(\cal{P})\) $$C_2(\mathcal{P})$$ . As applications, we demonstrate that \(\cal{L}\) $$\mathcal{L}$$ contains a Lie subalgebra isomorphic to the central extension of the Heisenberg Lie algebra and construct the Borel subalgebra of the simple Lie algebra associated with \(\cal A\) $$\mathcal{A}$$ as a Lie subquotient algebra of \(\cal{L}\) $$\mathcal{L}$$ .