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…