Notation/Site for Representations of an Algebra

[bob2]

I'm currently reading the paper "Higher Spin extension of cosmological spacetimes in 3d: asymptotically flat behaviour with chemical potentials in thermodynamics"
I'm looking at equation (3) on page 4. I know that symmetrization brackets work like this
A_(a b) = (A_ab + A_ba)/2. However I have never seen the expression given for [J_ab, J_cd] with the two brackets. How should one interpret this?
Furthermore: Does anyone know a site where one can look up representations of Lie algebras? I'm specifically looking for a representation of this algebra (isl(3,R))
Thanks so much in advance!

 

[jvicens]

Hi there,

I can provide some insight into your questions regarding the paper and the expression in equation (3).

Firstly, the expression [J_ab, J_cd] with the two brackets is known as the commutator bracket, denoted by [ , ]. It is commonly used in the study of Lie algebras, which are mathematical structures that describe the symmetry of a system. In this context, J_ab represents a generator of the Lie algebra, and the commutator [J_ab, J_cd] is used to determine the commutation relation between these generators.

To interpret this expression, you can think of it as a mathematical operation that calculates the difference between the products of J_ab and J_cd in two different orders. In other words, it measures the extent to which these two generators "commute" with each other.

Regarding your second question, there are several resources available online where you can find representations of Lie algebras. One such site is the Encyclopedia of Mathematics, which has a section specifically dedicated to Lie algebras and their representations. Additionally, you can also refer to books and articles on the subject for more detailed information.

I hope this helps clarify your doubts. Happy reading!