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Abelian extensions of five-dimensional solvable Leibniz algebras
A. Kh. Khudoyberdiyev, S. A. Sheraliyeva
https://arxiv.org/abs/2506.23528 https://

Abelian extensions of five-dimensional solvable Leibniz algebras
In this work, we extend the central extension method for solvable Leibniz algebras. Using this method, a complete classification of one-dimensional abelian extensions of five-dimensional solvable Leibniz algebras with a non-trivial three-dimensional nilradical is obtained. Furthermore, we explore extensions of solvable Leibniz algebras whose nilradical is null-filiform, establishing that, in this case, there exists a unique solvable abelian extension.

Coadjoint orbits of low dimension for nilradicals of Borel subalgebras in classical types
Mikhail Ignatev, Alexey Petukhov
https://arxiv.org/abs/2507.20332 https://

Coadjoint orbits of low dimension for nilradicals of Borel subalgebras in classical types
Let $\mathfrak g$ be a classical simple Lie algebra over an algebraically closed field $\mathbb F$ of characteristic zero or large enough, and let $\mathfrak n$ be a maximal nilpotent subalgebra of $\mathfrak g$. The main tool in representation theory of $\mathfrak n$ is the orbit method, which classifies primitive ideals in the universal enveloping algebra ${\rm U}(\mathfrak n)$ and unitary representations of the unipotent group $N=\exp(\mathfrak n)$ in terms of coadjoint orbits on the dual sp…

Biderivations of some classes of solvable Leibniz algebras
Bakhtiyor Yusupov, Doston Jumaniyozov, Majidkhon Azizov
https://arxiv.org/abs/2507.18134 https://

Biderivations of some classes of solvable Leibniz algebras
In this work, we investigate anti-derivations and biderivation of Leibniz algebras. We describe general form of anti-derivations and biderivations on null-filiform and filiform Leibniz algebras. Moreover, we show how to construct Leibniz algebras, while using these biderivations. We describe general form of anti-derivations and biderivations on solvable Leibniz algebras with null-filiform and filiform nilradicals. Interesting fact that, any biderivations of solvable Leibniz algebras with null-f…

A Frattini theory for evolution algebras
Manuel Ladra, Andr\'es P\'erez-Rodr\'iguez
https://arxiv.org/abs/2507.01935 https://

A Frattini theory for evolution algebras
This paper develops a Frattini theory for evolution algebras defining the Frattini subalgebra as the intersection of all maximal subalgebras, and the Frattini ideal as the largest ideal contained in it. To this end, we revisit the notion of nilradical, whose classical definition is not directly applicable in this setting, and propose the supersolvable nilradical as a suitable alternative. This leads to necessary and sufficient conditions for the triviality of the Frattini subalgebra and ideal. …