I have a new preprint posted on the arXiv; “Odd Jacobi manifolds and Loday-Poisson brackets”. It is a continuation of my studies of odd Jacobi structures on supermanifolds.

Odd Jacobi manifolds and Loday-Poisson brackets
Andrew James Bruce
(Submitted on 21 Jan 2013)
arXiv:1301.4799 [math-ph]

In this paper we construct a non-skewsymmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to such Poisson-like brackets as Loday-Poisson brackets. We examine the relations between the Hamiltonian vector fields with respect to both the odd Jacobi structure and the Loday-Poisson structure. Interestingly, these relations are identical to the Cartan identities.

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There are some subtle differences between even and odd structures and this preprint discusses one such difference. In particular, one can use the derived bracket formalism [3] to construct a Poisson-like bracket on the supermanifold mod the skewsymmetry.

The Loday-Poisson bracket

An odd Jacobi manifold is a supermanifold equipped with an almost Schouten structure and a homological vector field that satisfy some relations. The relations are not important for this discussion. See [1] for details.

From this data one can construct an odd Jacobi bracket, that is an odd version of a Poisson bracket with a modified Leibniz rule. The adjoint operator is a first order differential operator as opposed to a vector field.

Furthermore, by using the fact that the homological vector field is a Jacobi vector, that is it a “derivation over the odd Jacobi bracket” one can construct an even bracket using the derived bracket construction.

The resulting bracket satisfies a version of the Jacobi identity, but is not skewsymmetric. It also satisfies the Leibniz rule (from the left). Lie brackets mod the skewsymmetry were first examined by Loday, and so I call Loday brackets + Leibniz rule “Loday-Poisson brackets”.

This is in contrast to classical manifolds, where due to the work of Grabowski and Marmo [2], we know that the Jacobi identity and the Leibniz rule force the skewsymmetry. On supermanifolds we have nilpotent functions and this invalidates the assumptions of Grabowski and Marmo.

Furthermore, on an even Jacobi supermanifold there is no canonical choice of homological vector field to use, if one exists at all.

In the preprint I present several relations between the Hamiltonian vector fields with respect to the initial odd Jacobi structure and the derived Loday-Poisson structure. I note the similarity with the standard Cartan calculus.

The derived product

I have discussed the derived product on a Q-manifold here. As odd Jacobi manifolds come with a homological vector field as part of the structure, they are also Q-manifolds and have a derived product.

Interestingly, the Loday-Poisson bracket not only satisfies the Leibniz rule (from the left) for the usual product of functions on a supermanifold, but also the derived product.

That is we have a kind on non-skewsymmetric bracket that satisfies a version of the Jacobi identity and a version of the Leibniz rule over a Grassmann odd noncommutative form of multiplication. To my knowledge, these kinds of noncommutative Poisson algebras have not been studied.

References
[1] Andrew James Bruce. Odd Jacobi manifolds: general theory and applications to generalised Lie algebroids. Extracta Math. 27(1) (2012), 91-123

[2] J. Grabowski and G. Marmo. Non-antisymmetric versions of Nambu-Poisson and Lie algebroid brackets. J. Phys. A: Math. Gen. 34 (2001), 3803–3809.

[3] Yvette Kosmann–Schwarzbach. Derived brackets. Lett. Math. Phys., 69 (2004), 61-87.