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First and second cohomology space of the Lie algebra of differential operators on a manifold with coefficients in the space of functions. (Premier et deuxième espaces de cohomologie de l’algèbre de Lie des opérateurs différentiels sur une variété, à coefficients dans les fonctions.) (French) Zbl 0971.17011

The results of this paper were announced by the author in [Cohomologie de l’algèbre de Lie des opérateurs différentiels sur une variété, à coefficients dans les fonctions, C. R. Acad. Sci., Paris, Sér. I 328, 789-794 (1999; Zbl 0968.17008)]. The author investigates the Nijenhuis-Richardson graded Lie algebra \({\mathcal E}=A (N)_{\log,n.c.}\) of multilinear antisymmetric homomorphisms \(N\times\cdots\times N\to N\), \(N= C^\infty (M)\), local and vanishing on constants. The subspace \({\mathcal E}^0= A^0 (N)_{\text{loc},n.c}= {\mathfrak gl}(N)_{\text{loc}, n.c.}\) is a Lie algebra of differential operators on a manifold \(M\) with coefficients in the space of functions. The injection \(i:{\mathcal E}^0\to {\mathfrak gl}(N)\) is a representation of \({\mathcal E}^0\) on \(N\). This yields the spaces \(\wedge_{alt} ({\mathcal E},N)_q\) (of cochains of degree \(q)\) and \(\wedge({\mathcal E}^0,N)\) with canonical derivations. For \(q=-1\) we have the restriction \[ \theta: \bigwedge^p_{alt} ({\mathcal E},N)_{-1,\text{loc}} \to\bigwedge^p ({\mathcal E}^0,N)_{\text{loc}} \] which is a homomorphism of differential spaces. The induced exact sequence \[ 0\to\ker \theta\to \bigwedge_{alt} ({\mathcal E},N)_{-1, \text{loc}} \to\bigwedge ({\mathcal E}^0, N)_{\text{loc}}\to 0 \] splits giving on cohomology \[ H_{alt}({\mathcal E})_{-1, \text{loc}}= H(\ker \theta) \oplus H({\mathcal E}^0, N)_{\text{loc}}. \] The aim of the paper is to prove that \[ H^p({\mathcal E}^0,N)_{\text{loc}}=H^p_{dR}(M),\;p=0,1,2,\;m=\dim M\geq 2. \]

MSC:

17B56 Cohomology of Lie (super)algebras
17B66 Lie algebras of vector fields and related (super) algebras
58J99 Partial differential equations on manifolds; differential operators
53D17 Poisson manifolds; Poisson groupoids and algebroids

Citations:

Zbl 0968.17008
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