Related papers: Planar networks and simple Lie groups beyond type …

We perform a detailed analysis of the statistical properties of Poisson networks and show that the metric and topological properties of random cellular structures, can not be derived from simple models of random networks based on a poisson…

Double Bruhat cells in a semisimple group are intersections of cells in two Bruhat decompositions corresponding to two opposite Borel subgroups. They form a geometric framework for the study of total positivity in semisimple groups; they…

We present here two detailed examples of additive categorifications of the cluster algebra structure of a coordinate ring of a maximal unipotent subgroup of a simple Lie group. The first one is of simply-laced type ($A_3$) and relies on an…

For $n \geq 2$ let $\Delta$ be a Dynkin diagram of rank $n$ and let $I = {1, >..., n}$ be the set of labels of $\Delta$. A group $G$ admits a weak Phan system of type $\Delta$ over $\mathbb{C}$ if $G$ is generated by subgroups $U_i$, $i \in…

We study the dual ${\rm G}^\ast$ of a standard semisimple Poisson-Lie group ${\rm G}$ from a perspective of cluster theory. We show that the coordinate ring $\mathcal{O}({\rm G}^\ast)$ can be naturally embedded into a cluster Poisson…

Let $G$ be a connected complex semi-simple Lie group, and let $Z_{{\bf u}}$ be an $n$-dimensional Bott-Samelson variety of $G$, where ${\bf u}$ is any sequence of simple reflections in the Weyl group of $G$. We study the Poisson structure…

We study the relationship between two sets of coordinates on a double Bruhat cell, the cluster variables introduced by Berenstein, Fomin, and Zelevinsky and the $\CX$-coordinates defined by the coweight parametrization of Fock and…

We describe transposed Poisson structures on generalized Witt algebras $W(A,V, \langle \cdot,\cdot \rangle )$ and Block Lie algebras $L(A,g,f)$ over a field $F$ of characteristic zero, where $\langle \cdot,\cdot \rangle$ and $f$ are…

Curtis-Ingerman-Morrow characterize response matrices for circular planar electrical networks as symmetric square matrices with row sums zero and non-negative circular minors. In this paper, we study this positivity phenomenon more closely,…

The family of planar graphs is a particularly important family and models many real-world networks. In this paper, we propose a principled framework based on the widely-known Apollonian packing process to generate new planar network, i.e.,…

We consider the symplectic groupoid of pairs $(B,\mathbb{A})$ with $\mathbb A$ unipotent upper-triangular matrices and $B\in GL_n$ being such that $\widetilde {\mathbb A}=B{\mathbb A} B^{\text{T}}$ are also unipotent upper-triangular…

We derive properties of Latent Variable Models for networks, a broad class of models that includes the widely-used Latent Position Models. These include the average degree distribution, clustering coefficient, average path length and degree…

Let G be a connected, simply connected Poisson-Lie group with quasitriangular Lie bialgebra g. An explicit description of the double D(g) is given, together with the embeddings of g and g^*. This description is then used to provide a…

Let $\Sigma $ be a compact connected and oriented surface with nonempty boundary and let $G$ be a Lie group equipped with a bi-invariant pseudo-Riemannian metric. The moduli space of flat principal $G$-bundles over $\Sigma$ which are…

Let $\cA$ be a commutative unital Banach algebra, $\g$ be a semisimple complex Lie algebra and $G(\cA)$ be the 1-connected Banach--Lie group with Lie algebra $\g \otimes \cA$. Then there is a natural concept of a parabolic subgroup $P(\cA)$…

Reductive Lie Groups, such as the orthogonal groups, the Lorentz group, or the unitary groups, play essential roles across scientific fields as diverse as high energy physics, quantum mechanics, quantum chromodynamics, molecular dynamics,…

We consider network models of quantum localisation in which a particle with a two-component wave function propagates through the nodes and along the edges of an arbitrary directed graph, subject to a random SU(2) rotation on each edge it…

A generalized Euler parameterization of a compact Lie group is a way for parameterizing the group starting from a maximal Lie subgroup, which allows a simple characterization of the range of parameters. In the present paper we consider the…

The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative…

Network models with latent geometry have been used successfully in many applications in network science and other disciplines, yet it is usually impossible to tell if a given real network is geometric, meaning if it is a typical element in…