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On the computation of the Lichnerowicz–Jacobi cohomology. (English) Zbl 1092.53060

Let \((\Lambda,E)\) be a Jacobi structure on a manifold \(M\). This defines a canonical Lie algebroid structure on the first jet bundle \(J^1(M,\mathbb{R})\). The corresponding Lie algebroid cohomology (with trivial coefficients) is called by the authors the Lichnerowicz-Jacobi cohomology of \((M,\Lambda,E)\) (LJ-cohomology, in short). First, the authors give an easy proof that the LJ-cohomology is invariant under conformal tranformations of the form: \[ \Lambda_f=f\Lambda,\quad E_f=\Lambda(df)+fE. \] where \(f\in C^\infty(M)\) is a positive function. Second, the authors consider the following classes:
(i) Poisson manifolds. They recall the results of A. Lichnerowicz [J. Math. Pures Appl., IX. Sér. 57, 453–488 (1978; Zbl 0407.53025] for a Poisson manifold \((M,\Lambda)\), which relate the Lichnerowicz-Poisson cohomology (LP-cohomology, in short) and the LJ-cohomology. For \(\mathfrak{g}\) a Lie algebra of compact type, these together with the computation of the LP-cohomology of \(\mathfrak{g}^*\), due to V. L. Ginzburg and A. Weinstein [J. Am. Math. Soc. 5, No. 2, 445–453 (1992; Zbl 0766.58018)], show that \[ H^\bullet_{LJ}(\mathfrak{g}^*)=(H^\bullet(\mathfrak{g})\otimes\text{Inv})\oplus (H^{\bullet-1}(\mathfrak{g})\otimes\text{Inv}), \] where Inv denotes the algebra of Casimirs.
(ii) Contact manifolds. For a contact manifold \(M\), they show that its LJ-cohomology is given by: \[ H^\bullet_{LJ}(M)=H^\bullet_{dR}(M)\oplus H^{\bullet-1}_{dR}(M). \]
(iii) Locally conformal symplectic manifolds. Let \((M,\Omega)\) be a locally conformal symplectic manifold with Lee 1-form \(\omega\). Under some finiteness assumptions, they show that the relation between the LJ-cohomology and the de Rham cohomology, is the same as in the symplectic case.
(iv) Unit sphere in a Lie algebra. Let \(\mathfrak{g}\) be a Lie algebra with a scalar product \(\langle\cdot,\cdot\rangle\). They show that the unit sphere in \(\mathfrak{g}\) carries a Jacobi structure. For a Lie algebra of compact type, they show that: \[ H^\bullet_{LJ}(S^{n-1}(\mathfrak{g}^*))=H^\bullet(\mathfrak{g})\otimes\text{Inv}, \] where here Inv is the algebra of smooth functions on the sphere, invariant under all Hamiltonian functions (the “Casimirs”).
The title of this paper can be a bit misleading: the LJ-cohomology, like the Poisson cohomology is usually quite hard (to say the least) to compute. What is done in the paper, is not any specific computation, but rather explaining the relationship with other known cohomologies.

MSC:

53D17 Poisson manifolds; Poisson groupoids and algebroids
53D10 Contact manifolds (general theory)
17B56 Cohomology of Lie (super)algebras
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