
Journal of Lie Theory 33 (2023), No. 3, 925952 Copyright Heldermann Verlag 2023 On the Universal L_{∞}Algebroid of Linear Foliations Karandeep J. Singh Dept. of Mathematics, Katholieke Universiteit Leuven, Belgium karandeep.singh@kuleuven.be [Abstractpdf] We compute an $L_\infty$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 $\mathfrak{gl}(V)$. This $L_\infty$algebroid provides invariants of the singular foliations, and also provides a constantrank 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 $\mathfrak{g}\subset \mathfrak{gl}(V)$, and then computing the $L_\infty$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. Keywords: Singular foliations, Linfinityalgebroids, projective resolutions. MSC: 22E45, 13D02, 17B55. [ Fulltextpdf (217 KB)] for subscribers only. 