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Journal of Lie Theory 32 (2022), No. 4, 997--1006
Copyright Heldermann Verlag 2022

On Extensions of Nilpotent Leibniz and Diassociative Algebras

Erik Mainellis
Dept. of Mathematics, Statistics and Computer Science, St. Olaf College, Northfield, U.S.A.


Given a pair of nilpotent Lie algebras $A$ and $B$, an extension $0\rightarrow A\rightarrow L\rightarrow B\rightarrow 0$ is not necessarily nilpotent. However, if $L_1$ and $L_2$ are extensions which correspond to lifts of homomorphism $\Phi\colon B\rightarrow \text{Out}(A)$, 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 each algebra of Loday. As an important consequence, we thereby gain its associative analogue as a special case of diassociative algebras.

Keywords: Nilpotent extensions, Leibniz algebras, diassociative, dendriform, Zinbiel.

MSC: 17A30, 17A01, 17A32.

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