A universal Kaluzhnin--Krasner embedding theorem

TL;DR

This paper investigates the extension of the Kaluzhnin--Krasner universal embedding theorem beyond groups, Lie algebras, and Hopf algebras, identifying conditions under which it applies to other algebraic structures.

Contribution

It provides a unified analysis of the theorem's applicability to various algebraic structures and establishes that only Lie algebras admit such a universal embedding over an infinite field.

Findings

The theorem extends to crossed modules.

It cannot be adapted to associative, Jordan, or Leibniz algebras over infinite fields.

Lie algebras uniquely admit a universal Kaluzhnin--Krasner embedding in non-associative algebras.

Abstract

Given two groups $A$ and $B$, the Kaluzhnin--Krasner universal embedding theorem states that the wreath product $A\wr B$ acts as a universal receptacle for extensions from $A$ to $B$. For a split extension, this embedding is compatible with the canonical splitting of the wreath product, which is further universal in a precise sense. This result was recently extended to Lie algebras and to cocommutative Hopf algebras. The aim of the present article is to explore the feasibility of adapting the theorem to other types of algebraic structures. By explaining the underlying unity of the three known cases, our analysis gives necessary and sufficient conditions for this to happen. From those we may for instance conclude that a version for crossed modules can indeed be attained, while the theorem cannot be adapted to, say, associative algebras, Jordan algebras or Leibniz algebras, when working over an infinite field: we prove that then, amongst non-associative algebras, only Lie algebras admit a universal Kaluzhnin--Krasner embedding theorem.