We prove that the hierarchy of higher antibrackets (aka higher Koszul brackets, aka Koszul braces) of a linear operator ∆ on a commutative superalgebra can be defined by some universal formulas involving iterated Nijenhuis–Richardson brackets having as arguments ∆ and the multiplication operators. As a byproduct, we can immediately extend higher antibrackets to noncommutative algebras in a way ...

Citation: Dalmaijer ES, Nijenhuis BG and Van der Stigchel S (2016) Commentary: Life is unfair, and so are racing sports: some athletes can randomly benefit from alerting effects due to inconsistent starting procedures. Front. Psychol. 7:119. doi: 10.3389/fpsyg.2016.00119 Commentary: Life is unfair, and so are racing sports: some athletes can randomly benefit from alerting effects due to inconsi...

Abstract We give several explicit examples of compact manifolds with a 1-parameter family almost complex structures having arbitrarily small Nijenhuis tensor in the C 0 -norm. The 4-dimensional possess no structure, whereas 6-dimensional example does not left invariant and whether it possesses structure appears to be unknown.

We define quantum bi-Hamiltonian systems, by analogy with the classical case, as derivations in operator algebras which are inner derivations with respect to two compatible associative structures. We find such structures by means of the associative version of Nijenhuis tensors. Explicit examples, e.g. for the harmonic oscillator, are given.

After a brief summary of the main properties of Poisson manifolds and Lie algebroids in general, we survey recent work on the modular classes of Poisson and twisted Poisson manifolds, of Lie algebroids with a Poisson or twisted Poisson structure, and of Poisson–Nijenhuis manifolds. A review of the spinor approach to the modular class concludes the paper.