Prof. Grabowski has placed a review of the various brackets found in geometry and physics [1]. He also covers some of the ideas of superalgebra and graded differential geometry as many of the brackets really have their roots there. The review is based on a mini-course held at XXI Fall Workshops on Geometry and Physics, Burgos (Spain), 2012.

I have posted here about the review here as it contains a lot of the background material needed to understand my own research. In particular I am interested in brackets found in supergeometry, including super versions of Poisson, Jacobi and Loday brackets.

Brackets?

Rather generally, a bracket is understood as a non-associative operation on a vector space or a module. The principle example here is a Lie bracket. The review focuses on Lie brackets, such as Poisson and Jacobi brackets as well as Loday brackets, which are a non-skewsymmetric generalisation of a Lie bracket.

Interestingly, various forms of brackets arise in a wide context in contemporary mathematics. For example, Poisson brackets are found in classical and quantum mechanics as well as the theory of cluster algebras and geometric representation theory.

Prof. Janusz Grabowski

Prof Grabowski is Head of the Department of Mathematical Physics and Differential Geometry at the Institute of Mathematics within Polish Academy of Sciences. His personal homepage can be found here.

Reference

[1] Janusz Grabowski, Brackets, arXiv:1301.0227 [math.DG], 2012.