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Realizing enveloping algebras via moduli stacks

Let $\mathop{\rm CF}\nolimits(\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$ denote the vector space of $\mathbb{Q}$-valued constructible functions on a given stack $\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A}$ for an exact category $\mathcal{A}$. By using the Ringel--Hall algebra approach, Joyce proved that $\mathop{\rm CF}\nolimits(\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$ is an associative $\mathbb{Q}$-algebra via the convolution multiplication and the subspace $\mathop{\rm CF}\nolimits^{\rm ind}\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$ of constructible functions supported on indecomposables is a Lie subalgebra of $\mathop{\rm CF}\nolimits(\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$ in [10]. In this paper, we show that there is a subalgebra $\mathop{\rm CF}\nolimits^{\text{KS}}(\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$ of $\mathop{\rm CF}\nolimits(\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$ isomorphic to the universal enveloping algebra of $\mathop{\rm CF}\nolimits^{\rm ind}(\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$. Moreover we construct a comultiplication on $\mathop{\rm CF}\nolimits^{\text{KS}}(\mathop{\mathfrak{Obj}\kern .05em}\nolimits_\mathcal{A})$ and a degenerate form of Green's theorem. This generalizes Joyce's work, as well as results of [3].