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Binary operations in classical and quantum mechanics. (English) Zbl 1068.17009

Grabowski, Janusz (ed.) et al., Classical and quantum integrability. Dedicated to Włodzimierz Tulczyjew. Papers from the workshop, Warsaw, August 27 – September 1, 2001. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Cent. Publ. 59, 163-172 (2003).
The authors give a study of binary operations in order to find a minimal set of requirements for the structures in classical and quantum mechanics.
As observed by P. A. M. Dirac, when considering the quantum Poisson bracket of observables, the Leibniz rule implies that fixing an argument in the bracket, we get a derivation. Similarly, the authors show that natural compatibility conditions imply some properties, which become superfluous to require.
More precisely, it is shown that the differentiability of the bracket and the Jacobi identity imply the skew-symmetry.
As a consequence, this result tells us for instance that Loday algebras (as defined in [J.-L. Loday, Enseign. Math., II. Sér. 39, 269–293 (1993; Zbl 0806.55009)]) do not exist, except for the skew-symmetric ones). In other words, it is proved that breaking the skew-symmetry in the context is impossible.
For the entire collection see [Zbl 1011.00047].

MSC:

17B65 Infinite-dimensional Lie (super)algebras
17B63 Poisson algebras
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
53D17 Poisson manifolds; Poisson groupoids and algebroids
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations

Citations:

Zbl 0806.55009
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