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Derivations on the Nijenhuis-Schouten bracket algebra. (English) Zbl 0745.17009
Let $$M$$ denote a $$C^ \infty$$ $$m$$-dimensional connected paracompact manifold with tangent bundle $$TM$$ and let $$L_ i=\Gamma\Lambda^{i+1}TM$$ with $$-1\leq i\leq m-1$$ and $$\Gamma$$ the functor of sections over $$M$$. As usual $$\Lambda^ iTM$$ is the $$i$$-th exterior power of $$TM$$. If we set $$L=\sum L_ i$$ then there is a bilinear mapping $$[ ,\;]:L\times L\to L$$ called the Nijenhuis-Schouten bracket. This bracket satisfies certain properties which imply that $$L$$ is a graded Lie algebra. The author then introduces the notion of a local derivation and from this a graded Lie algebra is constructed. Properties of this algebra are described in the remainder of the paper.
##### MSC:
 17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras 17B70 Graded Lie (super)algebras 58A99 General theory of differentiable manifolds 17B65 Infinite-dimensional Lie (super)algebras
##### References:
 [1] Nijenhuis A.: Jacobi-type identities for bilinear differential concomitants of certain tensor fields I. Indagationes Math. 17 (1955), 390-403. · Zbl 0068.15001 [2] Schouten J. A.: Über Differentialkonkomitanten zweier kontravarianter Grössen. Indagationes Math. 2 (1940), 449-452. · Zbl 0023.17002 [3] Takens F.: Derivation of vector fields. Compositio Math. 26 (1973), 151-158. · Zbl 0258.58005
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