Generalized Cartan Matrix and Non-Semisimple Lie Algebras

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SUMMARY

The discussion focuses on the challenges of classifying a non-semisimple Lie algebra with a degenerate Killing form, which prevents the application of the Cartan-Weyl basis. The user is currently working with an algebra consisting of 5 generators and anticipates extending this to 11 generators. They aim to explore higher-dimensional algebras through the study of root spaces and Dynkin diagrams, despite the abstract nature of existing literature on the topic.

PREREQUISITES
  • Understanding of Lie algebras and their properties
  • Familiarity with the Killing form and its implications
  • Knowledge of Cartan-Weyl basis and its application
  • Basic concepts of root spaces and Dynkin diagrams
NEXT STEPS
  • Research the classification of non-semisimple Lie algebras
  • Study the implications of degenerate Killing forms in Lie algebra theory
  • Explore advanced techniques for analyzing root spaces
  • Learn about the construction and interpretation of Dynkin diagrams
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Mathematicians, theoretical physicists, and researchers in algebraic structures, particularly those focusing on Lie algebras and their classifications.

bartadam
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I have a lie algebra whose killing form is degenerate, hence not semi simple by cartan's second criterion.

So I cannot apply a Cartan Weyl Basis to classify the algebra. I currently have an algebra with 5 generators. Later I will have one with 11 generators and I am hoping I can spot how i can continue this to algebras with higher dimension with the method I am using by studying root spaces and dynkin diagrams and so on.

Because the algebra isn't semi-simple, I simply do not have any idea where to start and the literature is all very abstract.

Help :cry:
 
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Anyone? Any help would be good.
 

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