Classification of finite-dimensional (nilpotent) associative algebras What is known about the classification of finite-dimensional (nilpotent) associative algebras? I am assuming that algebras are over a field of characteristic zero. If it is simple, then it has to be an algebra of matrices, so only the nilpotent ones are of interest. Quick search shows some results up to dimension four, but nothing beyond. 
 A: In
Belitskii, Genrich; Lipyanski, Ruvim; Sergeichuk, Vladimir V., Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild, Linear Algebra Appl. 407, 249-262 (2005). ZBL1159.17304.
it is proved that the classification of local commutative associative algebras $A$ over an algebraically closed field of characteristic different from two, where the cube of the radical is zero, is wild (contains the problem of classifying pairs of square matrices up to simultaneous conjugacy).
This means that the classification of nilpotent algebras must be wild, since giving a local algebra $A$ is the same as giving the nilpotent algebra $\text{rad}(A)$.
A: There are already infinitely many isomorphism classes of finite-dimensional commutative associative (nilpotent) $\mathbb{C}$-algebras of rank $\geq 6$. See for example Suprunenko and Tueshkevich's book on commutative matrices (it's in Russian, I'm not sure if there is an English translation of it) for the construction. So, the classification question is rather hopeless already in the commutative case. On the other hand, all such commutative algebras can be fully listed for rank $\leq 5$.
