Stable homology of Lie algebras of derivations and homotopy invariants of wheeled operads

Abstract

We prove a theorem that computes, for any augmented operad $\mathcal{O}$, the stable homology of the Lie algebra of derivations of the free algebra $\mathcal{O}(V)$ with twisted bivariant coefficients (here stabilization occurs as $\dim(V)\to\infty$) out of the homology of the wheeled bar construction of $\mathcal{O}$; this can further be used to prove uniform mixed representation stability for the homology of the positive part of that Lie algebra with constant coefficients. This result generalizes both the Loday-Quillen-Tsygan theorem on the homology of the Lie algebra of infinite matrices and the Fuchs stability theorem for the homology of the Lie algebra of vector fields. We also prove analogous theorems for the Lie algebras of derivations with constant and zero divergence, in which case one has to consider the wheeled bar construction of the wheeled completion of $\mathcal{O}$. Similarly to how cyclic homology of an algebra $A$ may be viewed as an additive version of the algebraic $K$-theory of $A$, our results hint at the additive $K$-theoretic nature of the wheeled bar construction.