Realizing enveloping algebras via moduli stacks

TL;DR

This paper constructs a subalgebra of constructible functions on moduli stacks that is isomorphic to the universal enveloping algebra of indecomposable functions, extending Joyce's work with new algebraic structures.

Contribution

It introduces a subalgebra isomorphic to the universal enveloping algebra and constructs a comultiplication, generalizing Joyce's results on moduli stacks and enveloping algebras.

Findings

Established an isomorphism to the universal enveloping algebra

Constructed a comultiplication on the subalgebra

Proved a degenerate form of Green's theorem

Abstract

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].