Non-associative algebras - Lie, Jordan, alternative, Leibniz, Novikov, Poisson-type and n-ary among them - are classified in two complementary ways. Algebraically, algebras of a fixed small dimension are listed up to isomorphism by computing central extensions of nilpotent algebras of one dimension lower; geometrically, the same family is studied as an affine variety of structure constants acted on by GL(V), where irreducible components, rigid algebras and degenerations between orbit closures give the global picture. Students learn how polynomial identities cut out a variety of algebras, how the central-extension method enumerates nilpotent algebras dimension by dimension, and how orbit closures and degeneration levels organise the whole variety.
This arxiv paper by Ivan Kaygorodov provides lecture notes on the classification and structure of nonassociative algebraic structures. Presented at the University of Beira Interior (Covilha, Portugal…