## 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

### Keywords:

derived bracket; Courant bracket; Loday-Leibniz algebra

Zbl 0858.17027
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