A Kaluzhnin–Krasner embedding theorem for non-associative algebras?

For (non-abelian) groups A and B, the Universal Embedding Theorem of Kaluzhnin and Krasner 6 says that the (unrestricted) wreath product A ≀ B acts as a universal receptacle for any group G viewed as an extension from A to B. Recently, versions of the result were obtained for Lie algebras 7 and cocommutative Hopf algebras 1. The aim of this talk is to report on joint work with Bo Shan Deval and Xabier García-Martínez 2, where we attempt to prove a version of the theorem for general varieties of non-associative algebras over a field. In such a variety, the objects are vector space equipped with a bilinear multiplication, eventually subject to a set of polynomial identities. By establishing a connection with the concept of local algebraic cartesian closedness 4, we find a universal Kaluzhnin–Krasner embedding theorem, valid in semi-abelian categories 5. Via the results of 3, this allows us to fully characterise those varieties of non-associative algebras over an infinite field where the embedding theorem holds.