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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.

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…

The Global Glimm Property for C*-algebras of topological dimension zero
Ping Wong Ng, Hannes Thiel, Eduard Vilalta
https://arxiv.org/abs/2507.16261 https:/…

The Global Glimm Property for C*-algebras of topological dimension zero
We show that a C*-algebra with topological dimension zero has the Global Glimm Property (every hereditary subalgebra contains an almost full nilpotent element) if and only if it is nowhere scattered (no hereditary subalgebra admits a finite-dimensional representation). This solves the Global Glimm Problem in this setting. It follows that nowhere scattered C*-algebras with finite nuclear dimension and topological dimension zero are pure.