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.


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


Zbl 0858.17027
Full Text: DOI arXiv


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