Jacobi structure revisited (Q2773137): Difference between revisions

The authors introduce notions of Jacobi algebroids and Jacobi bialgebroids, which turn out to be equivalent to objects already studied by \textit{D. Iglesias} and \textit{J. C. Marrero} in [J. Geom. Phys. 40, No. 2, 176-199 (2001; Zbl 1001.17025)]. NEWLINENEWLINENEWLINEThe paper results from an attempt to understand the Lie algebroid structure on \(T^{\ast}M\oplus_{M} \mathbb R\) usually associated with a Jacobi structure on a manifold \(M\). Since a Jacobi bracket is just a Lie bracket on \(V=C^{\infty}(M)\) given by a bilinear first order differential operator, the authors begin with studying the structure of the algebra \({\mathcal A}\text{Diff}_{1}(V)\) of skew-linear first order differential operators on \(V\), endowed with the Richardson-Nijenhuis bracket. The resulting algebra is a Gerstenhaber algebra, but with a graded bracket satisfying NEWLINE\[NEWLINE[X,Y\wedge Z]=[X,Y]\wedge Z+(-1)^{|X|(|Y|+1)}Y\wedge [X,Z]- \widetilde{D}(X)\wedge Y\wedge ZNEWLINE\]NEWLINE where \(\widetilde{D}\) is a graded linear map of degree -1, instead of the Leibniz rule. Such algebras, are called Gerstenhaber-Jacobi algebras. NEWLINENEWLINENEWLINEIt is already known that Schouten-Nijenhuis algebras i.e. Gerstenhaber structures on the Grassmann algebra \({\mathcal A}(L)\) associated with a vector bundle \(L\) are in one to one correspondence with Lie algebroids (see the paper of \textit{Y. Kosmann-Schwarzbach} [Acta Appl. Math. 41, 153-165 (1995; Zbl 0837.17014)]). In view of this identification, the authors define the Jacobi-algebroids as Schouten-Jacobi algebras, i.e Gerstenhaber-Jacobi algebras on \({\mathcal A}(L)\). They then prove that this notion is equivalent to the notion of a Lie algebroid with the presence of a 1-cocycle as defined by Iglesias and Marrero.NEWLINENEWLINENEWLINEIn the last sections, the authors develop a lifting procedure which transports the Schouten-Jacobi bracket on \({\mathcal A}(L)\) into the Schouten bracket of multivector fields on the total space \(L\). The main result here is a natural construction associating a Lie algebroid with any local Lie algebra in the sense of Kirillov.NEWLINENEWLINENEWLINEFinally, it is shown that a consistent notion of a Jacobi bialgebroid can be obtained just by imitating the classical definition of a Lie bialgebroid. This notion reduces exactly to the notion of a generalized Lie bialgebroid given by Iglesias and Marrero, it has the advantage to be introduced in a more natural way.