Odd Jacobi Manifolds and Loday-Poisson Brackets

We construct a nonskew symmetric 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. Furthermore, we show that the Loday-Poisson bracket satisfies the Leibniz rule over the noncommutative product derived from the homological vector field. In memory of Jean-Louis Loday (1946–2012) 1. Introduction In this paper, we construct a Loday-Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold ; see [1]. The key observation here is that the homological vector field is a Jacobi vector field with respect to the odd Jacobi structure. Due to this, one can directly employ the notion of derived brackets as defined by Kosmann-Schwarzbach [2, 3]. We will draw heavily from these works (the notion of a derived bracket can be traced back to J. L. Koszul (1990) in unpublished notes. The notion was also put forward by Th. Th. Voronov (1993), but remains unpublished). On any differential super Lie algebra, , with bracket of even or odd parity, one can construct the “derived bracket” This bracket is not in general a Lie bracket (or a Lie antibracket) due to the lack of skew symmetry. However, the bracket does satisfy a form of the Jacobi identity. Such Lie algebra mod, the skew symmetry was first introduced by Loday under the name “Leibniz algebras” [4, 5]; we will shortly be a little more precise here. On an odd Jacobi manifold, we have a differential Lie antialgebra, that is, an odd Lie bracket provided by the odd Jacobi bracket and a differential provided by the homological vector field. In essence on an odd Jacobi manifold, one also has “a Poisson bracket mod the skew symmetry.” We make the previous statement more precise and then investigate the relations between the Hamiltonian vector fields given by the odd Jacobi and the Loday-Poisson structure. Furthermore, we study the notion of an odd form of noncommutative multiplication on the smooth functions on an odd Jacobi manifold generated by the homological vector field. It is shown that the Loday-Poisson bracket satisfies the right Leibniz rule over both the standard product of smooth functions and the odd derived product. Remark 1. Grabowski and Marmo in [6, 7] prove that on a classical (purely even) manifold the Jacobi-Loday identity together with the (generalized) Leibniz rule for a binary bracket on implies the skew symmetry: there can only be