"Brackets" by Janusz Grabowski

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.


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.


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

Fallen space heros of 2012

Last year we lost three very important people in the field of space exploration and astronomy.

Sally Ride
May 26, 1951 – July 23, 2012 (aged 61)

Ride in 1984

She was the first American woman to enter into low Earth orbit in 1983 and remains the youngest American astronaut to be launched into space.

Niel Armstrong
August 5, 1930 – August 25, 2012 (aged 82)

Armstrong in 1969

He was the first man to step foot on the Moon way back in 1969. Armstrong’s first space flight, as command pilot of Gemini 8, in 1966, made him NASA’s first civilian astronaut to fly in space.

Patrick Moore
4 March 1923- 9 December 2012 (aged 89)

Patrick Moore

Known as the presenter of the world’s longest-running television series with the same original presenter, the BBC’s The Sky at Night. Moore was a former president of the British Astronomical Association. He was a co-founder and former president of the Society for Popular Astronomy (SPA). As an author he wrote over 70 books on astronomy. He was the reason many astronomers here in the UK became astronomers.