Homology of the Lie Algebra of Locally Generated Derivations of a Discrete and Proper Metric Space

Abstract

We associate to each proper discrete metric space $X$ a Lie algebra that acts by locally generated derivations on an infinite tensor product of matrix algebras indexed by the points of $X$. We compute the homology of this Lie algebra with trivial scalar coefficients when $X$ is the integer lattice in $n$-dimensional Euclidean space.