Realizing enveloping algebras via moduli stacks

Abstract

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

33 pages

References in corpus (2)

• Motivic Donaldson-Thomas invariants and McKay correspondence
• Degenerations for derived categories