# generalized Gell-Mann matrices

Does anyone know of an implementation of the generalized Gell-Mann matrices? They are defined here: http://en.wikipedia.org/wiki/Generali...

There is a nice Mathematica demonstration about the dimension 3 case: http://demonstrations.wolfram.com/Eve...

Failing that, I would like to figure out how to put them in from scratch. Specifically, given a dimension, I would like a list of the Gell-Mann matrices and to be able to multiply and to take the commutator of two of them. The problem is that I do not have any ideas for turning these rules into sage code and creating a collection of matrices in this way.

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The problem with the generalization is that the diagonal ones are not unique. So you will have to decide how you are going to do that on your own. As far as the other matrices are concerned the code should not be very difficult. The most efficient way of doing it is to start with the raising and lowering operators i.e. symmetric matrices which have a single one in the off-diagonal element. The transpose of that would be the lowering operator. The just the combinations \sigma_{+} \pm i \sigma_{-} gives you all the off-diagonal elements.

Just out of curiosity are you working on GUTs?

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Thank you for this. I am pretty new to SAGE so I don't have experience with these operators, but I am sure they are well documented and now I have a lead. I am not a physicist. I am interested in these matrices as a nice basis for $\mathfrak su(n)$ to calculate some representations explicitly. More generally, I am working in the realm of symmetric spaces.

( 2011-11-16 06:10:58 -0500 )edit

Thank you for this. I am pretty new to SAGE so I don't have experience with these operators, but I am sure they are well documented and now I have a lead.

( 2011-11-16 06:10:59 -0500 )edit