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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
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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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