Geometric realisations of the unipotent enveloping algebra of a quiver

TL;DR

This paper compares and generalizes geometric constructions of unipotent enveloping algebras associated with quivers, establishing their integral forms and linking them to cohomological Hall algebras, advancing the understanding of BPS algebras in 2-Calabi-Yau categories.

Contribution

It provides a unified framework for geometric realizations of unipotent enveloping algebras of quivers, including integral forms and coproduct compatibility, and connects these to cohomological Hall algebras.

Findings

Geometric realizations hold for the integral form of the Lie algebra.

The top cohomological Hall algebra is isomorphic to the positive part of the enveloping algebra.

Compatibility with coproducts is established through modified generators.

Abstract

We compare and generalise the various geometric constructions (due to Ringel, Lusztig, Schofield, Bozec, Davison...) of the unipotent generalised Kac-Moody algebra associated with an arbitrary quiver. These constructions are interconnected through several geometric operations, including the stalk Euler characteristic of constructible complexes, the characteristic cycle, the Euler obstruction map, and the intersection multiplicities of Lagrangian subvarieties. We provide a proof that these geometric realisations hold for the integral form of the Lie algebra. Furthermore, by modifying the generators of the enveloping algebra, we ensure compatibility with the natural coproducts that can be defined in terms of restriction diagrams. As a result, we establish that the top cohomological Hall algebra of the strictly seminilpotent stack is isomorphic to the positive part of the enveloping algebra of the generalised Kac-Moody algebra associated with the quiver. This appears to be one of the cornerstones needed to describe the BPS algebra of very general $2$-Calabi-Yau categories, which is the subject of the author's work on BPS Lie algebras for $2$-Calabi-Yau categories with Davison and Schlegel Mejia.