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 $\mathbb C$?

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 ktuples 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 matrixsizes of the semisimple 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 ndiml algebra can be embedded in nxn matrices. There are some obvious restriction wrt large dimensions. For example, there cannot be an 8dml subalgebra of 3x3 matrices as its semisimple 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. 


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 :



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 $n^2n$, assuming $F$ has characteristic different than 2 can be found in Y. Barnea and A. Shalev, Hausdorff dimension, prop groups, and KacMoody algebras, Trans. Amer. Math. Soc. 349 (1997), 50735091 (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, 367383 (Theorem 4). 

