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Relative modular classes of Lie algebroids. (English) Zbl 1080.22001
Let \((E,\rho_E)\) be a Lie algebroid with base \(M\) and anchor \(\rho_E\colon E\to TM\). The modular class of this Lie algebroid is a certain cohomology class denoted \(\text{Mod}\,E\) and belonging to the first Lie algebroid cohomology space \(H^1(E)\); this notion was introduced by S. Evens, J.-H. Lu and A. Weinstein [Q. J. Math., Oxf. II. Ser. 50, No. 200, 417-436 (1999; Zbl 0968.58014)] and, in the case when \(M\) reduces to a single point, the modular class is precisely the Lie algebra cohomology class of the 1-cocycle \(x\mapsto\text{Trace}\,(\text{ad}\,x)\) on the fiber. (Recall that the fibers of any Lie algebroid are Lie algebras.)
As the present authors point out, the notion of relative modular class referred to in the title was previously considered under a different name in a preprint by J. Grabowski, G. Marmo and P.W. Michor. For any Lie algebroid morphism \(\varphi\colon E\to F\) one defines \[ \text{Mod}^\varphi(E,F)=\text{Mod}\,E-\varphi^*\text{Mod}\,F \] and one calls it the relative modular class of \((E,F)\) defined by \(\varphi\). Now consider the line bundle \(L^{E,F}=\wedge^{\text{top}}E\otimes\wedge^{\text{top}}F^*\), and for \(x\in\Gamma(E)\) and \(\omega\otimes\nu\in\Gamma(L^{E,F})\) we define \(D^\varphi_x(\omega\otimes\nu)={\mathcal L}^E_x\omega\otimes\nu +\omega\otimes{\mathcal L}^F_{\varphi x}\nu\), where \({\mathcal L}^E\) and \({\mathcal L}^F\) are the Lie derivatives on the Lie algebroids \(E\) and \(F\), respectively. One of the main results of the paper under review is that the map \(x\mapsto D^\varphi_x\) is a representation of \(E\) on \(L^{E,F}\) and \(\text{Mod}^\varphi(E,F)\) is equal to the characteristic class of that representation (Theorem 3.3).
Another circle of ideas approached in the paper is the relationship between the relative modular classes and the modular classes of twisted Poisson structures in the sense of a recent preprint by Y. Kosmann-Schwarzbach and C. Laurent-Gengoux. Specifically, let \((A,\pi,\psi)\) be a twisted Lie algebroid, that is, \(\pi\in\Gamma(\wedge^2A)\) and \(\psi\in\Gamma(\wedge^3A^*)\) is a \(d_A\)-closed form satisfying \([\pi,\pi]_A=2(\wedge^3\pi^\sharp)\psi\). In particular the vector bundle \(A^*\) has a structure of Lie algebroid with the anchor \(\rho_A\circ\pi^\sharp\), where \(\pi^\sharp\colon A^*\to A\) is the bundle map associated with \(\pi\) and \(\rho_A\) is the anchor of \(A\). Then the modular class \(\theta_{\text{KL}}(A,\pi,\psi)\) of the twisted Lie algebroid \((A,\pi,\psi)\) satisfies \(2\theta_{\text{KL}}(A,\pi,\psi)= \text{Mod}^{\pi^\sharp}(A^*,A) =\text{Mod}\,(A^*)-(\pi^\sharp)^*(\text{Mod}\,A)\) (Theorem 4.1 in the paper under review).
The paper concludes with a remark on the special case of Lie algebras, that is, the case when the base of each algebroid consists of a single point. Thus, let \(H\) be a connected closed subgroup of the connected Lie group \(G\), and denote their Lie algebras by \({\mathfrak h}\) and \({\mathfrak g}\). If \(\iota\colon{\mathfrak h}\hookrightarrow{\mathfrak g}\) denotes the corresponding inclusion map, then the relative modular class \(\text{Mod}^\iota({\mathfrak h},{\mathfrak g})\) vanishes if and only if there exists a \(G\)-invariant measure on the homogeneous space \(G/H\).

MSC:
22A22 Topological groupoids (including differentiable and Lie groupoids)
53D17 Poisson manifolds; Poisson groupoids and algebroids
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