Extensions of Nilpotent Algebras

Abstract

Given a pair of nilpotent Lie algebras $A$ and $B$, an extension $0\to A\to L\to B\to 0$ is not necessarily nilpotent. However, if $L_1$ and $L_2$ are extensions which correspond to lifts of a map $\Phi :B\to \text{Out}(A)$, it has been shown that $L_1$ is nilpotent if and only if $L_2$ is nilpotent. In the present paper, we prove analogues of this result for the algebras of Loday. As an important consequence, we thereby gain its associative analogue as a special case of diassociative algebras.

arXiv comment9 pages