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.



Centralizer of hermitian matrices with zero trace
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 ${\frak{su}}(N)$. 


