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In Quantum Physics one often has to deal with commutators.

Here I want to denote by $H_0$ the set of all hermitian matrices with trace equal to zero! One can easily relate it to $\mathfrak{su}(N)=iH_0.$

Now, my question is basically if there has been a detailed study of the centralizer in $H_0$ or $\mathfrak{su}(N)$ somewhere?

I am particularly interested in questions like: Given a set of matrices $S$ spanning sum subspace of $H_0$ what is known about the centralizer

$C_{H_0}(S):=\{A \in H_0; [A,B]=0 \text{ for all B in } H_0\}$

Can we classify it in its dimension?

I am not really sure where I could find such a reference, but due to its relationship to $\mathfrak{su}(N)$ I thought that this should have been already studied somewhere.

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    $\begingroup$ If you mean by Hermitian, $A$ is its conjugate-transpose, then the trace condition is largely irrelevant for commutators: you can take any $n\times n$ Hermitian $A$ and replace it by $A':=A - cI$ where $c$ is $1/n$ times the trace of $A$, and voila, $A'$ is trace zero, Hermitian, and its centralizer is exactly the same as that of $A$. For a collection of Hermitian matrices, you can of course do exactly the same thing to each one separately. $\endgroup$
    – David Handelman
    Sep 25, 2015 at 13:11
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    $\begingroup$ yes, that is true, but I thought that the $\mathfrak{su}(N)$ reference could be valuable.Moreover, to consider the identity additionally is just pathological because as you point out: It always commutes. $\endgroup$
    – Simeon Radkov
    Sep 25, 2015 at 13:49

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As you have pointed out, one can work in the Lie algebra ${\frak{su}}(N)$. You want to classify the centralizers of various subspaces of this Lie algebra, and without loss of generality you can assume that the subspace is a Lie subalgebra, ${\frak{m}}$. Its action on $W={\Bbb{C}}^N$ preserves the inner product, hence completely reducible, so $$W=\bigoplus V_i^{k_i}, $$ where $V_i$ are mutually non-isomorphic ${\frak{m}}$-modules and $k_i$ are the multiplicities. It follows that the centralizer of ${\frak{m}}$ in ${\frak{u}}(N)$ is isomorphic to $\bigoplus {\frak{u}}(k_i)$. Taking trace zero part gives the centralizer in ${\frak{su}}(N)$.

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