# Jacobi algebroids and quasi Q-manifolds

In “Jacobi algebroids and quasi Q-manifolds”  (arXiv:1111.4044v1 [math-ph]) I reformulate the notion of a Jacobi algebroid (aka generalised Lie algebroid or Lie algebroid in the presence of a 1-cocycle) in terms of an odd Jacobi structure of weight minus one  on the total space of the “anti-dual bundle” $$\Pi E^{*}$$. This mimics the weight minus one Schouten structure associated with a Lie algebroid. The weight is assigned as zero to the base coordinates ans one to the (anti-)fibre coordinates.

Recall that a Lie algebroid can be understood as a weight one homological vector field  on the “anti-bundle” $$\Pi E$$. What is the corresponding situation for Jacobi algebroids?

Well, this leads to a new notion, what I call a quasi Q-manifold…

A quasi Q-manifold is a supermanifold equipped with an odd vector field $$D$$ and an odd function $$q$$ that satisfy the following

$$D^{2}= \frac{1}{2}[D,D] = q \: D$$

and

$$D[q]=0$$.

The extreme examples here are

1. Q-manifolds, that is set $$q=0$$. Then $$D^{2}=0$$.
2. Supermanifolds with a distinguished (non-zero) odd function, that is set $$D=0$$.  (This includes the cotangent bundle of  Schouten and higher Schouten  manifolds)
3. The entire category of supermanifolds if we set $$D=0$$ and $$q =0$$.

The theorem here is that a Jacobi algebroid,  understood as a weight minus one Jacobi structure on $$\Pi E^{*}$$ is equivalent to  $$\Pi E$$ being a weight one  quasi Q-manifold.  I direct the interested reader to the preprint for details.

A nice example is $$M:= \Pi T^{*}N \otimes \mathbb{R}^{0|1}$$, where $$N$$ is a pure even classical manifold.  The supermanifold $$M$$ is in fact an odd contact manifold or equivalently an odd Jacobi manifold of weight minus one, see arXiv:1101.1844v3 [math-ph]. Then  it turns out that $$M^{*} := \Pi TN\otimes \mathbb{R}^{0|1}$$  is a weight one quasi Q-manifold. It is worth recalling that $$\Pi T^{*}N$$ has a canonical Schouten structure (in fact odd symplectic) and that $$\Pi TN$$ is a Q-manifold where the homological vector field is identified with the de Rham differential on $$N$$.  Including the “extra odd direction” deforms these structures.

As far as I can tell quasi Q-manifolds are a new class of supermanifold that generalises Q-manifolds and Schouten manifolds.  It is not know if other examples of such structures outside the theory of Lie and Jacobi algebroids are interesting. Only time will tell…