For a given $n$, is there any characterization for the commutative subalgebras of $M\_n(\Bbb{C})$? I would like to know how many commutative subalgebras there are for each possible dimension.


In view of Chapman's answer, I am refining my previous question:


Given $k\leq n$, is there any way of describing the commutative subalgebras of $M\_n$ which are of dimension $k$.



Commutative subalgebras of M_n
If you are only concerned about commutative subalgebras of $M\_n(\mathbb{C})$ then there is a fairly easy characterization. So any self-adjoint abelian algebra is generated by a single self adjoint element (spectral theorem). Call this element T. Then T is diagonalizable and so the algebra it form will be the algebra of polynomials over it. Since it is diagonalizable that is a unitarty $u$ with $uTu^\*$ diagonal. And the algebra has dimension $k$ exactly when T has $k$ distinct non-zero eigenvalues.


Note: This is assuming that T is invertible. If T is not invertible then the polynomial algebra since it contains the constants will have dimension $k+1$.


So then we can view the algebra generated by T as an algebra of the form $u^\*Au$ where A is an algebra of diagonal matrices.


