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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.

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    $\begingroup$ A nonzero algebra can't be both nilpotent and unital. $\endgroup$ – YCor Mar 9 at 15:44
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    $\begingroup$ This is surely a hopeless task. There are billions of nilpotent semigroups of order 8 up to isomorphism and although I suppose some of them might have isomorphic semigroup algebras I suspect most don't. $\endgroup$ – Benjamin Steinberg Mar 9 at 18:38
  • $\begingroup$ Thanks, fixed the assumptions, no unit $\endgroup$ – Eugene Starling Mar 9 at 21:41
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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 $\text{rad}(A)$.

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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 $\leq 5$.

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  • $\begingroup$ An English language version of Suprenenko and Tyshkevich has been available since at least 1971 (although it may be out of print now). I remember that I did a summer research project on nilpotent algebras in third year (undergraduate), using it as the basic reference. $\endgroup$ – David Handelman Mar 9 at 22:55
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    $\begingroup$ Infinitely many classes doesn't mean that the classification is hopeless. For instance, for (complex) 7-dimensional nilpotent Lie algebras there's a complete (and very useful) classification, among more than 100 items it includes five 1-parameter families , in which there is a simple description of isomorphism. $\endgroup$ – YCor Mar 9 at 23:12
  • $\begingroup$ could you please give a ref to this classification? $\endgroup$ – Eugene Starling Mar 10 at 7:26

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