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Derived brackets. (English) Zbl 1055.17016

The author describes several differential geometric settings that illustrate the notion of derived bracket introduced by herself [Ann. Inst. Fourier 46, No. 5, 1243–1274 (1996; Zbl 0858.17027)]. We recall that, given a graded differential Lie algebra \((V,[\cdot,\cdot],D)\) with the bracket of degree \(n\), the corresponding derived bracket is the bilinear mapping \([\cdot,\cdot]_{(D)}: V\times V\to V\) defined by \([a,b]_{(D)}=(-1)^{n+| a|+1}=[Da,b]\) for all \(a,b\in V\), where \(| a|\) stands for the degree of \(a\).
The paper is written in a clear style, and includes many instructive historical remarks, as well as a rather extensive list of references.

MSC:

17B70 Graded Lie (super)algebras
53D17 Poisson manifolds; Poisson groupoids and algebroids
17B66 Lie algebras of vector fields and related (super) algebras
17B63 Poisson algebras
17D99 Other nonassociative rings and algebras
58A50 Supermanifolds and graded manifolds

Citations:

Zbl 0858.17027
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References:

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