Dimension of subalgebras of a matrix algebra If $n$ is given and $A$ is a subalgebra of $M_n(\mathbb C)$, the algebra of $n \times n$ matrices with entries in the field of complex numbers, then what are the possible values of dimension of $A$ as a vector space over $\mathbb C$?
 A: Soit $E$ un $\mathbb C$-espace vectoriel de dimension $n$.
J'ai démontré entre autres les deux résultats suivants dans un article à paraître dans la revue française Quadrature :


*

*On suppose que $k$ vérifie les inégalités $k \ge 2$ et 
$k^{2}\le n$. Soit $\mathcal{A}$ une sous-algèbre de $\mathcal{L}(E)$ qui
vérifie la relation $n^{2}-kn+k^{2}-k+1 < \dim \mathcal{A} < n^{2}-kn+n.$
Alors, $\mathcal{A}$ vérifie la relation $\dim \mathcal{A}=n^{2}-kn+k^{2}.$

*Soient $n$ un entier naturel et $p$ un entier de l'intervalle $[0,n^{2}].$ On
suppose $p$ écrit sous la forme $p=n(n-k)+t,\ 0\le t \le n-1$. Alors il existe
une sous-algèbre de dimension $p$ dans $\mathcal M_n (\mathbb C )$ si
et seulement s'il existe une sous-algèbre de dimension $t$ dans $\mathcal M_k(\mathbb C)$.
A: I think that the fact that every proper subalgebra is contained in am maximal parabollic follows immediately from Jacobson's density theorem because if a subalgebra does not preserve any subspace, then $C^n$ is a simple module for it. This is of course true over any field.
In the case of Lie algebras rather than associative algebras, then a classification of maximal subalgebras of finite dimensional simple Lie algebras over the complex numbers was obtained by Dynkin. In the positive characteristic case a classiifcation can probably be obtained using arguments which were used for the classifcation of maximal subgroups of finite simple groups. This is at least what I understood talking to Liebeck and Seitz, but I am not an expert on these matters.
However, in the Lie case an elementary argument that the maximal dimension of a proper subalgebra of $sl_n(F)$ is $n^2-n$, assuming $F$ has characteristic different than 2 can be found in Y. Barnea and A. Shalev, Hausdorff dimension, pro-p groups, and Kac-Moody algebras, Trans. Amer. Math. Soc. 349 (1997), 5073-5091 (Theorem 1.7). Other related stuff (related to possible dimensions) but more on the group theoretic side can be found in the same paper. A generalization of this to other classical Lie algebras can be found in  Abért, Miklós; Nikolov, Nikolay; Szegedy, Balázs Congruence subgroup growth of arithmetic groups in positive characteristic. Duke Math. J. 117 (2003), no. 2, 367--383 (Theorem 4).
A: Rough answer : almost all small dims can appear, there are some restrictions to
large dims.
For example, considering 1 matrix all dims between 1 and n appear.
Taking centralizers of these all numbers of the form sum a_i^2 where a is a partition
of n appear.
In general, consider k-tuples of positive integers a and b such that their scalar product
a.b=n (a should be thought of as the Morita setting, b as the matrix-sizes of the semi-simple part 
of the subalgebra), then any number of the form
sum b_i^2 + subsum b_ib_j
is possible (here 'subsum' means that one takes all terms b_xb_y for all x,y in a 
substring 
1 <= i_1 < i_2 < ... < i_l <=k for any 0<=l<=k)
Edit : the subsum gives the dimension of the Jacobson radical. This answer cannot be the final one, as it only detects the subalgebras of global dimension 1. For example any n-diml algebra can be embedded in nxn matrices.
There are some obvious restriction wrt large dimensions. For example, there cannot be an 8-dml
subalgebra of 3x3 matrices as its semi-simple part can be at most C x M_2(C) and so its dimension must be
smaller or equal to 7. 
For general n there cannot be subalgebras with dimensions between the dim of the largest parabolic
subgroup of GL(n) and n^2.
Edit : a closely related question can be found here : problems concerning subspaces of mxm matrices.
