## Generalized Lie bialgebroids and Jacobi structures.(English)Zbl 1001.17025

The authors introduce and investigate a deformation of the notion of a Lie algebroid and Lie bialgebroid in the sense of K. Mackenzie and P. Xu [Duke Math. J. 73, 415-452 (1994; Zbl 0844.22005)] in the presence of 1-cocyles in such a way that any Jacobi manifold induces a canonical generalized Lie bialgebroid. More precisely, let $$[\cdot,\cdot]$$ be a Lie algebroid bracket on a vector bundle $$A$$ over $$M$$, and let $$\varphi_0\in\Gamma(A^*)$$ be a 1-cocycle, i.e. $$\text{d}\varphi_0=0$$ for the corresponding de Rham exterior differential $$\text{d}$$. Then, one can consider the deformed $$\varphi_0$$-differential $\text{d}_{\varphi_0}\omega=\text{d}\omega+\varphi_0\wedge \omega$ and the corresponding $$\varphi_0$$-Lie derivative $${\mathcal L}_{\varphi_0}$$. One can also obtain accordingly the $$\varphi_0$$-Schouten bracket $$[\cdot,\cdot]_{\varphi_0}$$. The pair $$((A,\varphi_0),(A^*,X_0))$$ of generalized Lie algebroids is called a generalized Lie bialgebroid if for all $$\mu\in\Gamma(\bigwedge^kA^*)$$, $$\nu\in\Gamma(\bigwedge^l A^*)$$ $\text{d}_{\varphi_0}[\mu,\nu]_{X_0}=[\text{d}_{\varphi_0}\mu ,\nu]_{X_0}-(-1)^k[\mu,\text{d}_{\varphi_0}\nu]_{X_0}$ (in fact, in the paper a different but equivalent condition is given). Now, let $$(\Lambda,E)$$ be a Jacobi structure on $$M$$. Then, the induced canonical Lie algebroid structure on $$A=T^*M\oplus{\mathbb R}$$ discovered by Y. Kerbrat and Z. Souici-Benhammadi [C. R. Acad. Sci., Paris, Sér. I 317, 81-86 (1993; Zbl 0804.58015)] with the 1-cocycle $$\varphi_0((\mu,f)) =-E(\mu)$$ forms a generalized Lie bialgebroid with the generalized Lie algebroid $$(TM\oplus{\mathbb R},X_0)$$, where on $$A^*=TM\oplus{\mathbb R}$$ we take the Lie algebroid structure of first-order differential operators $[(Y_1+g_1),(Y_2+g_2)]=[Y_1,Y_2]+(Y_1(g_2)-Y_2(g_1))$ and the 1-cocycle is $$X_0((Y,g))=g$$. As a kind of converse, it is proved that on the base of any generalized Lie bialgebroid a Jacobi structure is determined.
It is also shown that it is possible to construct a Lie bialgebroid from a generalized Lie bialgebroid, so a duality theorem can be deduced. Finally, some special cases are considered: triangular generalized Lie bialgebroids and generalized Lie bialgebras.

### MSC:

 17B62 Lie bialgebras; Lie coalgebras 53D17 Poisson manifolds; Poisson groupoids and algebroids 53D10 Contact manifolds (general theory)

### Keywords:

Jacobi manifolds; Lie algebroids; Schouten bracket

### Citations:

Zbl 0844.22005; Zbl 0804.58015
Full Text:

### References:

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