A Kaluzhnin–Krasner embedding theorem for non-associative algebras?
(2025) International Category Theory Conference CT2025 — Location: Brno (13.July.2025)
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- 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.
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Van der Linden, T. (2025). A Kaluzhnin–Krasner embedding theorem for non-associative algebras? International Category Theory Conference CT2025, Brno. https://hdl.handle.net/2078.5/267929