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"The Edmonton Notes on Nilpotent Groups" by Philip Hall
Mark Pengitore
https://arxiv.org/abs/2507.09745 https://arxiv.org/p…

"The Edmonton Notes on Nilpotent Groups" by Philip Hall
This note is a reproduction of the well known and historically significant series of notes called "The Edmonton Notes on Nilpotent Groups" based on lectures given by Philip Hall using the copy found in the Queen Mary College Mathematics Notes series. We use the original numbering of statements, definitions, footnotes, proofs, abstract, etc. to recover as much of the original document as possible.

Companion matrices as sums of $p$-potent and nilpotent matrices
Andrada Pojar
https://arxiv.org/abs/2508.09276 https://arxiv.org/pdf/2508.09276

Companion matrices as sums of $p$-potent and nilpotent matrices
We prove that, over a field $\mathbb{F}$ of odd characteristic $p$, a companion matrix $C$ is the sum of $E$ and $N$, with $E$ $p$-potent (i.e. $E^p = E$,) and $N$ nilpotent, if and only if the trace of $C$ is an integer multiple of unity of $\mathbb{F}$.

Families of curves in Vinberg representations
Jef Laga, Beth Romano
https://arxiv.org/abs/2508.09607 https://arxiv.org/pdf/2508.09607

Families of curves in Vinberg representations
Inspired by orbit parametrizations in arithmetic statistics, we explain how to construct families of curves associated to certain nilpotent elements in $\mathbb{Z}/m\mathbb{Z}$-graded Lie algebras, generalizing work of Thorne to the $m\geq 3$ case and the non-simply laced case. We classify such families arising from subregular nilpotents in stable gradings and interpret almost all orbit parametrizations associated with algebraic curves appearing in the literature in this framework. As an extend…

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Complex structures on two-step nilpotent Lie groups
Maria Laura Barberis
https://arxiv.org/abs/2508.05885 https://arxiv.org/pdf/2508.05885

Complex structures on two-step nilpotent Lie groups
We give a characterization of the $2$-step nilpotent Lie algebras whose corresponding Lie groups admit a left invariant complex structure. This is done by considering separately the cases when the complex structure is 2-step or 3-step nilpotent in the sense of [Cordero-Fernández-Gray-Ugarte, 2000]. We also study the Hermitian geometry 2-step nilpotent Lie groups. We show that if a left invariant Hermitian metric on such Lie group is pluriclosed, then the corresponding complex structure is 2-st…

On finitely generated left nilpotent braces
H. Meng, A. Ballester-Bolinches, L. A. Kurdachenko, V. P\'erez-Calabuig
https://arxiv.org/abs/2508.07883 https://

On finitely generated left nilpotent braces
A description of finitely generated left nilpotent braces of class at most two is presented in this paper. The description heavily depends on the fact that if $B$ is left nilpotent of class at most $2$, that is $B^3 = 0$, then $B$ is right nilpotent of class at most $3$, that is $B^{(4)} = 0$. In addition, we construct a free object in the category of finitely generated left nilpotent braces of class at most $2$.

On the dimension of affine subspaces of nilpotent matrices
Simone Calamai, Elena Rubei
https://arxiv.org/abs/2508.06653 https://arxiv.org/pdf/2508.06653

On the dimension of affine subspaces of nilpotent matrices
The focus of the paper is on the maximal dimension of affine subspaces of nilpotent $n \times n $ matrices with fixed rank. In particular we obtain two results in the "border" cases rank equal to $n-1$ and rank equal to $1$.

On the Borel complexity of the space of left-orderings of nilpotent groups
Emir Molina Tauc\'an
https://arxiv.org/abs/2507.06953 https://

On the Borel complexity of the space of left-orderings of nilpotent groups
We give the first examples of nonabelian left-orderable groups such that the conjugacy orbit equivalence relation on its space of orders has infinity orbits, yet it is smooth in the Borel sense. The examples are all nilpotent groups and we provide a sufficient condition so that the space of orders of a nilpotent group has a smooth conjugacy orbit relation. We also show with different examples that nilpotence is not a sufficient condition for smoothness.

The geometric classification of nilpotent Lie-Yamaguti, Bol and compatible Lie algebras
Kobiljon Abdurasulov, Abror Khudoyberdiyev, Feruza Toshtemirova
https://arxiv.org/abs/2506.05353

The geometric classification of nilpotent Lie-Yamaguti, Bol and compatible Lie algebras
The geometric classifications of complex $4$-dimensional nilpotent Lie-Yamaguti algebras, $4$-dimensional nilpotent Bol algebras, and $4$-dimensional nilpotent compatible Lie algebras are given.

On the rigidity of k-step nilpotent graph Lie algebras
Josefina Barrionuevo, Paulo Tirao
https://arxiv.org/abs/2507.04063 https://arx…

On the rigidity of k-step nilpotent graph Lie algebras
We thoroughly explore the class of k-step nilpotent Lie algebras associated with a simple graph looking for k-step nilpotent Lie algebras which are rigid in the variety of at most k-step nilpotent Lie algebras. We find out that, besides the complete graph, the only examples arise for k=2 and graphs of at most 4 vertices. A key tool to prove non-k-rigidity in this context, is a general construction of non-trivial deformations for naturally graded nilpotent Lie algebras.