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The variety of nilpotent pairs $(A,B)$ with $[A,B] = \lambda I$
Vlad Roman, Robert M. Guralnick
https://arxiv.org/abs/2508.20199 https://arxiv.org/pdf/2508…

The variety of nilpotent pairs $(A,B)$ with $[A,B] = λI$
Let $k$ be an algebraically closed field of characteristic $p >0$. We consider the variety of nilpotent pairs $(A,B)$ with $[A,B]=λI$, namely the set of pairs $ X = \{ (A,B) \in M_n(k) \times M_n(k) \mid A,B \text{ nilpotent}, [A,B]=λI, λ\in k \}$. We prove that if $n=pr$, then $X$ is irreducible of dimension $n^2$.

On Zel'manov's global nilpotence theorem for Engel Lie algebras
Michael Vaughan-Lee
https://arxiv.org/abs/2507.21677 https://arxiv.org/pdf/2507.216…

On Zel'manov's global nilpotence theorem for Engel Lie algebras
I give a proof of Zel'manov's theorem that if $L$ is an $n$-Engel Lie algebra over a field $F$ of characteristic zero then $L$ is (globally) nilpotent. This is a very important result which extends Kostrikin's theorem that $L$ is locally nilpotent if the characteristic of $F$ is zero or some prime $p>n$. Zel'manov's proof contains some striking original ideas, and I wrote this note in an effort to understand his arguments. I hope that my efforts will be of use to other mathematicians in underst…

Sums of two nilpotent quaternionic matrices
Cl\'ement de Seguins Pazzis
https://arxiv.org/abs/2508.19627 https://arxiv.org/pdf/2508.19627

Sums of two nilpotent quaternionic matrices
Let $\mathcal{Q}$ be a quaternion division algebra over a field, and $n \geq 2$ be an integer. In a recent article, de La Cruz et al have proved that every $n$-by-$n$ matrix with entries in $\mathcal{Q}$ and pure quaternionic trace is the sum of three nilpotent matrices, and they have shown that some are not the sum of two nilpotent matrices. Here, we give a simple characterization of the square matrices with entries in $\mathcal{Q}$ that are the sum of two nilpotent ones. When $n \geq 3$, th…

Eight-Dimensional Symplectic Nilpotent Lie Groups with Lagrangian Normal Subgroups: A Complete Classification
T. A\"it Aissa, M. W. Mansouri
https://arxiv.org/abs/2508.16978

Eight-Dimensional Symplectic Nilpotent Lie Groups with Lagrangian Normal Subgroups: A Complete Classification
We investigate symplectic nilpotent Lie groups with Lagrangian normal subgroups. We show that there exists a bijection between the isomorphism classes of nilpotent Lie groups with Lagrangian normal subgroups and the isomorphism classes of geodesically complete, flat, nilpotent Lie groups with Lagrangian extension cohomology class. Finally, we provide a complete classification of eight-dimensional symplectic nilpotent Lie groups with Lagrangian normal subgroups, identifying exactly ninety-five s…

Relay Selection in Wireless Networks as Restless Bandits
Mandar R. Nalavade, Ravindra S. Tomar, Gaurav S. Kasbekar
https://arxiv.org/abs/2508.20625 https://

Relay Selection in Wireless Networks as Restless Bandits
We consider a wireless network in which a source node needs to transmit a large file to a destination node. The direct wireless link between the source and the destination is assumed to be blocked. Multiple candidate relays are available to forward packets from the source to the destination. A holding cost is incurred for each packet stored at every relay in each time slot. The objective is to design a policy for selecting a relay in each time slot to which the source attempts to send a packet,…

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Translates in Step-Two SI/Z Nilpotent Groups
Vignon Oussa
https://arxiv.org/abs/2508.18529 https://arxiv.org/pdf/2508.18529

Translates in Step-Two SI/Z Nilpotent Groups
We prove a linear--independence principle for finite families of translates in irreducible, square--integrable modulo the center (SI/Z) representations of step--two nilpotent Lie groups with one--dimensional center. Fix a one--parameter subgroup $\exp(\mathbb R X)$ and a noncentral element $Y$ (not assumed to commute with $X$). For every nonzero smooth vector $f$, any finite collection $\{π(\exp(x_k X))f\}_{k=1}^m$ of translates along $\exp(\mathbb R X)$, together with the single translate $π…

An extension of the Floquet-Bloch theory to nilpotent groups and its applications
Atsushi Katsuda
https://arxiv.org/abs/2509.16848 https://arxiv.org/pdf/25…

An extension of the Floquet-Bloch theory to nilpotent groups and its applications
We develop a generalized Floquet-Bloch theory for discrete torsion-free nilpotent groups by exploiting their Malcev completions. Our main result is a branching formula that relates finite-dimensional representations of a discrete nilpotent lattice to infinite-dimensional unitary representations of its simply connected nilpotent Lie group. This generalization enables to extend the following two classical asymptotic problems (i) a Chebotarev density analogue for prime closed geodesics on compact …

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On self-similarity of finitely generated torsion-free nilpotent groups
Adilson Berlatto, Tulio Santos
https://arxiv.org/abs/2509.16947 https://arxiv.org/pd…

On self-similarity of finitely generated torsion-free nilpotent groups
This paper aims to investigate the self-similarity property in finitely-generated torsion-free nilpotent groups. We establish connections between geometric equivalence and self-similarity in these groups. Moreover, we show that any 2-generated torsion-free nilpotent group of class 3 has a faithful representation as a transitive self-similar group. Additionally, we provide an example of a 4-generated torsion-free nilpotent group of class 3 that does not admit such a representation. This example,…