[Paper Review] Mackenzie theory and Q-manifolds

Theodore Voronov|ArXiv.org|Aug 3, 2006

Homotopy and Cohomology in Algebraic Topology12 references14 citations

TL;DR

This paper provides a simplified characterization of Mackenzie's double Lie algebroids using homological vector fields on supermanifolds, showing that the complex compatibility conditions are equivalent to the commutativity of two homological vector fields. The key contribution is a coordinate-free, supergeometric reformulation that extends naturally to $n$-fold Lie algebroids and clarifies the duality structure underlying double and multiple vector bundles.

ABSTRACT

We give a simple characterization of Mackenzie's double Lie algebroids in terms of homological vector fields. Application to the `Drinfeld double' of Lie bialgebroids is given and an extension to the multiple case is suggested.

Motivation & Objective

• To provide a simpler, more conceptual characterization of Mackenzie's double Lie algebroids, which were previously defined by a complex system of compatibility conditions.
• To establish an equivalence between Mackenzie's double Lie algebroids and a new notion of double Lie antialgebroids defined via homological vector fields.
• To extend the theory to $n$-fold Lie algebroids by generalizing the supergeometric framework.
• To clarify the role of duality and symmetry in the structure of double and multiple vector bundles through the use of supermanifold techniques.
• To provide a foundation for the theory of $n$-fold Lie bialgebroids and their cotangent doubles as $(n+1)$-fold Lie algebroids.

Proposed method

• Use of supermanifolds and the parity reversion functor $\Pi$ to encode the structure of double and multiple vector bundles.
• Definition of homological vector fields $Q$ of total weight 1 on the total space of an $n$-fold vector bundle as the central object.
• Introduction of partial parity reversion $\Pi_r$ and dualization $\mathrm{D}_r$ in each direction to generate the 12 neighbors of a double vector bundle.
• Proof that Mackenzie’s Condition III subsumes all other compatibility conditions, reducing the entire structure to a single commutativity condition $[Q_1, Q_2] = 0$ for two homological vector fields.
• Use of weight grading and multilinearity of transition functions to define total and partial weights on multiple vector bundles.
• Application of the supergeometric framework to show that the cotangent double of a Lie bialgebroid is an $(n+1)$-fold Lie algebroid, supporting the generalization to higher $n$.

Experimental results

Research questions

• RQ1 Can Mackenzie’s complex system of compatibility conditions for double Lie algebroids be reduced to a simpler, more intrinsic condition?
• RQ2 Is there a supergeometric formulation of double Lie algebroids using homological vector fields that captures the entire structure?
• RQ3 How do the various dual and parity-reversed versions (neighbors) of a double vector bundle reflect the underlying algebraic structure?
• RQ4 Can the theory of double Lie algebroids be naturally extended to $n$-fold Lie algebroids?
• RQ5 What is the role of the cotangent double in the context of $n$-fold Lie algebroids and $n$-fold Lie bialgebroids?

Key findings

• Mackenzie’s double Lie algebroids are equivalent to double Lie antialgebroids defined by a pair of homological vector fields $Q_1$ and $Q_2$ of weights $(1,0)$ and $(0,1)$ that commute: $[Q_1, Q_2] = 0$.
• The entire system of Mackenzie’s compatibility conditions is subsumed by his Condition III, which is equivalent to the commutativity of the two homological vector fields.
• Out of the 12 neighbors of a double vector bundle, five admit structures with simple compatibility conditions, four of which are reformulations of Condition III, and one is the commutativity condition.
• The theory generalizes naturally to $n$-fold Lie algebroids via the definition that an $n$-fold vector bundle is an $n$-fold Lie algebroid if its complete parity reversal $\Pi^n$ carries a homological vector field $Q$ of total weight 1.
• The cotangent double of a Lie bialgebroid is shown to be an $(n+1)$-fold Lie algebroid, supporting the conjecture that $n$-fold Lie bialgebroids are characterized by a $QS$- or $QP$-structure of appropriate weight.
• The framework provides a natural path to defining $n$-fold Lie bialgebroids as $n$-fold Lie algebroids whose duals are also $n$-fold Lie algebroids, with a compatible bracket structure.