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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.
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
[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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