[Math] Cartan Lie Algebra of the Unitary Group $U(N)$

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I am having trouble understanding the Lie Algebra terminology. What is the Cartain Lie algebra of the unitary group $U(n)$? It must be in many textbooks, but they explain it very generally in terms of root lattices and it's usually for general lie groups $G$ and lie algebras $\mathfrak{g}$.

Is it safe to say "Cartain" just means "diagonal"? The diagonal unitary matrices are those whose eigenvalues are $|\lambda| = |e^{2\pi i t} |= 1$.

$$ \left(\begin{array}{cccc}e^{2\pi i t} & 0 & \dots & 0 \\
0 & e^{2\pi i t} & \dots & 0 \\
\vdots & \vdots & & \vdots \\
0 & 0 & \dots & e^{2\pi i t}\end{array} \right) $$

This is why we can talk about the maximal torus $\vec{t} \in [0, 2\pi]^n$. The tangent space of this torus would be $T(S^1) = \mathbb{R}^n$ itself.

Is that the Cartan Lie Algebra for $U(n)$ ?

In the special unitary group there a restriction $\sum t_i = 0$ which defines a hyperplane in $\mathbb{R}^n$ so that's my guess for the special unitary group.

I am surprised I can't get a more straightfoward answer. I need to know since I need to a compute an integral defined over a the Cartan algebra but I don't know what that is.

Best Answer

You seem to be on the right track. Here is a succinct piece of information from Hans Samelson's notes:

For $\mathfrak{gl}(n, \mathbb{C})$ a CSA is the set of all diagonal matrices—clearly an object of interest.

Since $\mathfrak{u}_n(\mathbb{C})$ is a sub-algebra of $\mathfrak{gl}_n(\mathbb{C})$, we can take the sub-algebra of $\mathfrak{u}_n(\mathbb{C})$ as being of a more general form:

$\left(\begin{array}{cccc}e^{2\pi i t_1} & 0 & \dots & 0 \\ 0 & e^{2\pi i t_2} & \dots & 0 \\ \vdots & \vdots & & \vdots \\ 0 & 0 & \dots & e^{2\pi i t_m}\end{array} \right)$


Another way to put this is that the Cartan sub-algebra is the maximal toral sub-algebra.

We shall call a toral subalgebra $\mathfrak{h} ⊂ \mathfrak{g}$ a Cartan subalgebra if it has maximal dimension among all toral subalgebras of $\mathfrak{g}$.

As for what is a toral sub-algebra, as per Wikipedia, a Toral Lie Algrebra is a diagonalisable sub-algebra of $\mathfrak{gl}_n(\mathbb{C})$. To bring this round full-circle, if we take from Wikipedia again, the second property given for a Maximal Torus:

The maximal tori in G are exactly the Lie subgroups corresponding to the maximal abelian, diagonally acting subalgebras of $\mathfrak g$ (cf. Cartan subalgebra)

where a maximal torus for $U(n)$ has the form

$T = \{diag(e^{i\theta_1}, e^{i\theta_2},\ldots, e^{i\theta_n}) : \forall j, \theta_j\in \mathbb{R}\}$


Hopefully this provides some clarity on your query.

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