Extensions of Nilpotent Algebras

TL;DR

This paper explores conditions under which extensions of nilpotent Lie algebras and Loday algebras remain nilpotent, establishing analogues and implications for associative and diassociative algebra structures.

Contribution

It proves new results on the nilpotency of algebra extensions, extending known Lie algebra results to Loday and diassociative algebras.

Findings

Nilpotency of extensions depends on lifts of a map to Out(A)

Analogues of Lie algebra results are established for Loday algebras

Associative algebra case is derived as a special instance

Abstract

Given a pair of nilpotent Lie algebras $A$ and $B$, an extension $0\xrightarrow{} A\xrightarrow{} L\xrightarrow{} B\xrightarrow{} 0$ is not necessarily nilpotent. However, if $L_1$ and $L_2$ are extensions which correspond to lifts of a map $\Phi:B\xrightarrow{} \text{Out}(A)$, 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.