Loday-Quillen-Tsygan theorem on Quivers

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.

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