Generalized Cartan Matrix and Non-Semisimple Lie Algebras

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In summary, the conversation discusses the limitations of using a Cartan Weyl Basis to classify a lie algebra with a degenerate Killing form. The speaker mentions having an algebra with 5 generators and wanting to extend this method to algebras with higher dimensions, but is unsure of how to proceed due to the degeneracy of the algebra. They express frustration with the abstract nature of the literature and ask for any help or guidance.
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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.
 

FAQ: Generalized Cartan Matrix and Non-Semisimple Lie Algebras

1. What is a generalised Cartan matrix?

A generalised Cartan matrix is a mathematical structure used in the study of Lie algebras, which are a type of algebraic structure used to describe symmetries in mathematics. The generalised Cartan matrix encodes information about the root system of a Lie algebra, which is a set of vectors used to represent the algebra's symmetries.

2. How is a generalised Cartan matrix constructed?

A generalised Cartan matrix is constructed by arranging the simple roots of a Lie algebra into a matrix, and then performing certain operations on the matrix to transform it into a symmetrised and normalised form. These operations include multiplying rows and columns by certain constants, and rearranging rows and columns according to specific rules.

3. What is the significance of the numbers in a generalised Cartan matrix?

The numbers in a generalised Cartan matrix represent the structure of the root system of a Lie algebra. They indicate the number of positive and negative roots, as well as the angles between the roots. This information is important in understanding the symmetries and representations of the Lie algebra.

4. What is the relationship between a generalised Cartan matrix and a root system?

A generalised Cartan matrix is closely related to the root system of a Lie algebra. The numbers in the matrix correspond to the lengths and angles of the roots in the root system. The matrix also encodes information about the symmetries and representations of the Lie algebra.

5. How is a generalised Cartan matrix used in physics?

Generalised Cartan matrices are used in theoretical physics, specifically in the study of symmetries and representations of Lie algebras. They are used to classify and study symmetries in physical systems, such as in the study of particle physics and quantum mechanics.

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