Realizing enveloping algebras via moduli stacks

Let $\mathrm{CF}(\mathfrak{Obj}_{\mathcal{A}})$ denote the vector space of $\mathbb{Q}$-valued constructible functions on a given stack $\mathfrak{Obj}_{\mathcal{A}}$ for an abelian category $\mathcal{A}$. In [12], Joyce proved that $\mathrm{CF}(\mathfrak{Obj}_{\mathcal{A}})$ is an associative $\mathbb{Q}$-algebra via the convolution multiplication and the subspace $\mathrm{CF}^{\mathrm{ind}} (\mathfrak{Obj}_{\mathcal{A}})$ of constructible functions supported on indecomposables is a Lie subalgebra of $\mathrm{CF}(\mathfrak{Obj}_{\mathcal{A}})$. In this paper, we extend Joyce’s result to an exact category $\mathcal{A}$ and show that there is a subalgebra $\mathrm{CF}^{\mathrm{KS}}(\mathfrak{Obj}_{\mathcal{A}})$ of $\mathrm{CF}(\mathfrak{Obj}_{\mathcal{A}})$ isomorphic to the universal enveloping algebra of $\mathrm{CF}^{\mathrm{ind}} (\mathfrak{Obj}_{\mathcal{A}})$. Moreover we construct a comultiplication on $\mathrm{CF}^{\mathrm{KS}} (\mathfrak{Obj}_{\mathcal{A}})$ and a degenerate form of Green’s theorem. This refines Joyce’s result, as well as results of [4].

Keywords

Hall algebra, stack, constructible set, universal enveloping algebra

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Published 24 August 2016