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How to find the Cartan Subalgebra for a random algebra given by a set of generators

  1. Dec 11, 2012 #1
    Hey! I am studying group theory for particle physicists right now and I came across the following general question (Tell me if you think this rather belongs to the homework section, I am new here.)

    Say I am given a set of matrices which represent the generators of an algebra, but I don't know which algebra, i.e. I don't know the commutation relations nor anything like roots or similar about the algebra.
    Now I wonder how I could reconstruct the algebra in a smart way. I guess the first step is finding the Cartan subalgebra. Once I have that I should be ready to reconstruct all the rest.
    So it all boils down to the question: How can I find the Cartan subalgebra, in particular how can I find out what its dimension (i.e. the rank of the algebra) is, so that I know how many generators I can simultaneously diagonalise?
    I suppose this is an undergrad linear algebra question, but I would nevertheless appreciate some hint!
  2. jcsd
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