Wreath products and Kaluzhnin-Krasner embedding for Lie algebras

The wreath product of groups $A\wr B$ is one of basic constructions in group theory. We construct its analogue, a wreath product of Lie algebras.

Consider Lie algebras $H$ and $G$ over a field $K$. Let $U(G)$ be the universal enveloping algebra. Then $\bar H=\operatorname{Hom}_K(U(G),H)$ has the natural structure of a Lie algebra, where the multiplication is defined via the comultiplication in $U(G)$. Also, $G$ acts by derivations on $\bar H$ via the (left) coregular action. The semidirect sum $\bar H \leftthreetimes G$ we call the wreath product and denote by $H\wr G$. As a main result, we prove that an arbitrary extension of Lie algebras $0\to H\to L\to G\to 0$ can be embedded into the wreath product $L\hookrightarrow H\wr G$.