Distributions and quotients on degree 1 NQ-manifolds and Lie algebroids

TL;DR

This paper explores the relationship between distributions on degree 1 Q-manifolds and Lie algebroids, establishing a correspondence with IM-foliations and analyzing reduction processes and Lie 2-algebra actions.

Contribution

It provides a new characterization of distributions on degree 1 Q-manifolds in terms of Lie algebroid structures and investigates their role in reduction and Lie 2-algebra actions.

Findings

Involutive Q-invariant distributions correspond bijectively to IM-foliations.

Reduction by these distributions is characterized and analyzed.

Non-strict actions of Lie 2-algebras induce specific distributions on Q-manifolds.

Abstract

It is well-known that a Lie algebroid A is equivalently described by a degree 1 Q-manifold M. We study distributions on M, giving a characterization in terms of A. We show that involutive Q-invariant distributions on M correspond bijectively to IM-foliations on A (the infinitesimal version of Mackenzie's ideal systems). We perform reduction by such distributions, and investigate how they arise from non-strict actions of strict Lie 2-algebras on M.