On the Universal L∞-Algebroid of Linear Foliations

We compute an L -algebroid structure on a projective resolution of some classes of singular foliations on a vector space V induced by the linear action of some Lie subalgebras of gl(V ) .This L -algebroid provides invariants of the singular foliations, and also provides a constant-rank replacement of the singular foliation.The computation consists of first constructing a projective resolution of the foliation induced by the linear action of the Lie subalgebra g gl(V ) , and then computing the L -algebroid structure.We then generalize these constructions to a vector bundle E , where the role of the origin is now taken by the zero section L .We then show that the fibers over a singular point of a projective resolution of any singular foliation can be computed directly from the foliation, without needing the projective resolution.For linear foliations, we also provide a way to compute the action of the isotropy Lie algebra in the origin on these fibers directly from the foliation.