Find Cartan Subalgebra for Algebra Given Generators

In summary, the conversation discusses the problem of reconstructing an algebra when given a set of matrices representing its generators. The first step is finding the Cartan subalgebra, which can be used to reconstruct the rest of the algebra. The question then becomes how to find the dimension of the Cartan subalgebra, which determines how many generators can be simultaneously diagonalized. There are specific algorithms for finding the Cartan subalgebra, but they can be more complicated than those used in linear algebra. In certain cases, the Cartan subalgebra is the maximal toral subalgebra, which is Abelian and can be simultaneously diagonalized in the adjoint representation. However, in general, it is only nilpotent and an
  • #1
CuriosusNN
3
0
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!
 
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  • #2
Algorithms for Lie algebras given the multiplication table are notoriously a bit more complicated than what we have in the rest of linear algebra, especially if the scalar field isn't real or complex. For algebraic closed scalar fields of characteristic not two, the CAS is the maximal toral subalgebra, i.e. the largest subalgebra which can be simultaneously diagonalized in the adjoint representation. It is also Abelian in this case, which is a strong requirement. However, in general we have only nilpotency. Hence an algorithm should look for the semisimple parts of the Jordan-Chevalley decomposition of the adjoint representation.
 

What is a Cartan subalgebra?

A Cartan subalgebra is a maximal abelian subalgebra of a given Lie algebra. It is named after the mathematician Élie Cartan and is an important concept in the theory of Lie algebras.

Why is finding a Cartan subalgebra important?

Finding a Cartan subalgebra is important because it allows us to decompose a given Lie algebra into simpler components, making it easier to study and understand. It also helps in the classification of Lie algebras.

How do you find a Cartan subalgebra for a given Lie algebra?

The most common method for finding a Cartan subalgebra is by using the root space decomposition of the Lie algebra. This involves finding a set of linearly independent elements called roots, which form a basis for the Cartan subalgebra.

What are generators in the context of finding a Cartan subalgebra?

Generators are elements of a Lie algebra that generate the entire algebra through a linear combination. In the context of finding a Cartan subalgebra, the generators are used to determine the root space decomposition and ultimately, the Cartan subalgebra.

Are there any applications of finding a Cartan subalgebra?

Yes, there are many applications of finding a Cartan subalgebra in mathematics and physics. For example, it is used in the study of Lie groups, representation theory, and quantum mechanics. It also has applications in geometry and topology.

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