On the universal $L_\infty$-algebroid of linear foliations

TL;DR

This paper constructs an $L_$-algebroid structure on resolutions of singular linear foliations, providing invariants and constant-rank replacements, with explicit computations for specific Lie algebra actions and generalizations to vector bundles.

Contribution

It explicitly constructs and computes an $L_$-algebroid for certain singular foliations, offering new invariants and methods for analyzing their structure.

Findings

Provides explicit $L_$-algebroid structures for linear actions of Lie subalgebras.

Shows how to compute fibers over singular points directly from the foliation.

Generalizes constructions to vector bundles and computes isotropy actions without resolutions.

Abstract

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 subalgebra of $\mathfrak {gl}(V)$. This $L_\infty$-algebroid provides invariants of the singular foliations, and also provides a constant-rank replacement of the singular foliation. We do this by first explicitly constructing projective resolutions of the singular foliations induced by the natural linear actions of endomorphisms of $V$ preserving a subspace $W\subset V$, the Lie algebra of traceless endomorphisms, and the symplectic Lie algebra of endomorphisms of $V$ preserving a non-degenerate skew-symmetric bilinear form $\omega$, 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, without needing the projective resolution.