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

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



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…

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