Differences between a Cartan subalgebra and a Levi subalgebra?

Question

Let $\mathfrak{h}$ be a Cartan subalgebra and $\mathfrak{l}$ be a Levi subalgebra of $\mathfrak{gl_n}$, where $\mathfrak{h}$ and $\mathfrak{l}$ are both semisimple subalgebras.

This is a simple question but I am not sure how to answer this for myself: how are they different?

Cartan algebras of $\mathfrak{gl}_n$ are not semisimple. – Mariano Suárez-Álvarez Feb 06 '13 at 08:00 — Mariano Suárez-Álvarez, Feb 06 '13 at 08:00

score 0 · Accepted Answer · answered Feb 06 '13 at 07:56

You mean, what are they in the case of $\mathfrak{gl}_n$? (I assume we are working over the complex numbers.)

A Cartan subalgebra would be given by the set of diagonal matrices.

A Levi subalgebra would be $\mathfrak{sl}_n$, the set of matrices with trace zero.

Differences between a Cartan subalgebra and a Levi subalgebra?

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