# Cartan/Weyl basis. Please check I understand it properly

• bartadam
In summary, the conversation discusses the process of verifying that a six dimensional lie algebra is semi-simple by checking the killing form, finding maximally commuting elements, and calculating their matrix representation. The conversation also touches on calculating eigenvalues and commutators, and concludes by questioning the method used for calculating generators.
bartadam
I have a six dimensional lie algebra. I checked it was semi-simple by checking if the killing form was not invertible. I found a set of two (I think) maximally commuting elements which I called H1 and H2 and found their matrix representation by calculating the structure constants.

I have put these matrices into maple and calculated
$$H=aH_1+bH_1$$

and calculated the eigenvalues of H and got four (as expected) eigenvalues as follows,...

$$\sqrt{2}i(a+b),\ -\sqrt{2}(a+b),\ \sqrt{2}(a-b)\ and\ -\sqrt{2}i(a-b)$$

So I think the commutators are as follows
$$[\stackrel{\rightarrow}{H},E^{\pm \alpha}]=\pm\sqrt{2}i(1,1)E^{\pm\alpha}$$
$$[\stackrel{\rightarrow}{H},E^{\pm \beta}]=\pm\sqrt{2}i(1,-1)E^{\pm\beta}$$

plus all the others which follow from cartan weyl procedure. Where $$\stackrel{\rightarrow}{H}=(H_1,H_2)$$

This makes the algebra $$A1+A1=D2$$ as it has two orthogonal root spaces of A1.

Am I doing this properly?

Unfortunately I have not calculated the generators $$E^{\alpha}$$ to check the above because the algebra in the basis which I derived it is is such that calculating these is horrifically tedious if you understand me. Are the generators just going to be the dot product of the eigenvectors with the original basis?

## 1. What is the Cartan/Weyl basis?

The Cartan/Weyl basis is a set of basis vectors used in the theory of Lie algebras. It consists of a set of simple roots, which are linearly independent vectors that span the Lie algebra, and their associated coroots, which are dual to the simple roots.

## 2. How is the Cartan/Weyl basis used in Lie algebra representation theory?

The Cartan/Weyl basis is used to decompose the Lie algebra into a direct sum of simple subalgebras. This decomposition is then used to construct irreducible representations of the Lie algebra.

## 3. What is the difference between a simple root and a coroot in the Cartan/Weyl basis?

A simple root is a vector in the Cartan/Weyl basis that is used to label the simple subalgebras in the Lie algebra decomposition. A coroot is the dual of a simple root and is used to label the weights of an irreducible representation of the Lie algebra.

## 4. Are the Cartan/Weyl basis and the standard basis the same?

No, the Cartan/Weyl basis and the standard basis are different. The standard basis is a set of basis vectors that span the entire vector space, while the Cartan/Weyl basis is specific to Lie algebras and is used for decomposition and representation theory.

## 5. Can the Cartan/Weyl basis be used for any Lie algebra?

Yes, the Cartan/Weyl basis can be used for any finite-dimensional complex Lie algebra. However, for different types of Lie algebras, the specific form of the simple roots and coroots may differ.

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