Higher Koszul Brackets on the Cotangent Complex

Let A = k[x(1), x(2), ..., x(n)]/I be a commutative algebra where k is a field, char(k) = 0, and I subset of S := k[x(1), x(2), ..., x(n)] a Poisson ideal. It is well known that [dx(i), dx(j)] := d{x(i),x(j)} defines a Lie bracket on the A-module Omega(A vertical bar k) of Kahler differentials, making (A, Omega(A vertical bar k)) a Lie-Rinehart pair. If A is not regular, that is, Omega(A vertical bar k) is not projective, the cotangent complex L-A vertical bar k serves as a replacement for Omega(A vertical bar k). We prove that L-A vertical bar k is an L-infinity-algebroid compatible with the Lie-Rinehart pair (A, Omega(A vertical bar k)). The L-infinity-algebroid structure comes from a P-infinity-algebra structure on the resolvent of the morphism S -> A. We identify examples when this L-infinity-algebroid simplifies to a dg Lie algebroid, concentrating on cases where S is Z(>= 0)-graded and I and { , } are homogeneous.