$Q$-Manifolds and Mackenzie Theory

TL;DR

This paper introduces a simplified supermanifold-based characterization of double Lie algebroids, making their structure more accessible and paving the way for extensions to multiple Lie algebroids.

Contribution

It provides an alternative, simpler description of double Lie algebroids using supermanifold language and commuting homological vector fields, clarifying and extending Mackenzie's original framework.

Findings

Equivalent description of double Lie algebroids via supermanifolds

Simplification of Mackenzie's original definition

Extension of the theory to multiple Lie algebroids

Abstract

Double Lie algebroids were discovered by Kirill Mackenzie from the study of double Lie groupoids and were defined in terms of rather complicated conditions making use of duality theory for Lie algebroids and double vector bundles. In this paper we establish a simple alternative characterization of double Lie algebroids in a supermanifold language. Namely, we show that a double Lie algebroid in Mackenzie's sense is equivalent to a double vector bundle endowed with a pair of commuting homological vector fields of appropriate weights. Our approach helps to simplify and elucidate Mackenzie's original definition; we show how it fits into a bigger picture of equivalent structures on `neighbor' double vector bundles. It also opens ways for extending the theory to multiple Lie algebroids, which we introduce here.