Loday-Quillen-Tsygan theorem on Quivers

TL;DR

This paper extends the Loday-Quillen-Tsygan theorem to Lie algebras associated with quivers, relating their homology to factorization homology over loops and paths in the quiver structure.

Contribution

It generalizes the classical theorem to quiver-based Lie algebras, connecting homology to stratified factorization algebras and factorization homology.

Findings

Homology expressed as sum over loops and paths in the quiver

Extension of classical theorem to quiver-structured Lie algebras

Connection between Lie algebra homology and factorization algebra

Abstract

The well-known Loday-Quillen-Tsygan theorem calculates the Lie algebra homology of the infinite general linear Lie algebra $\mathfrak{gl}(A)$ over an unital associative algebra $A$. We generalize the Loday-Quillen-Tsygan theorem to an infinite Lie algebra associated with a (framed) quiver, where we assign to each vertex $v$ an infinite general linear Lie algebra $\mathfrak{gl}(A_v)$, to each edge $e$ an infinite matrix module and to each framed vertex a (anti)-fundamental representation. Given this data, each loop or path ending on framed vertices of the quiver defined a stratified factorization algebra over $S^1$ or $[0,1]$ respectively. We show that the corresponding Lie algebra homology can be expressed as summing the factorization homology over all loops and framed paths of the quiver.