# Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields

@inproceedings{Andersen2010KacMoodyFS, title={Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields}, author={Kasper K. S. Andersen and Lisa Carbone and Diego Penta}, year={2010} }

Let A be the generalized Cartan matrix of rank 2 Kac-Moody algebra g. We write g = g(a, b) when A has non-diagonal entries −a and −b. To each such A, its Weyl group and corresponding root lattice, we associate a ‘Fibonacci type ’ integer sequence. These sequences are derived from the coordinates of the real root vectors in the root space. Each element of each sequence can be expressed as a polynomial in the non-diagonal entries of the generalized Cartan matrix, whose coefficients are shallow…

## 7 Citations

### Weyl group orbits on Kac–Moody root systems

- Mathematics
- 2014

Let D ?> be a Dynkin diagram and let &Pgr; = { &agr; 1 ,..., &agr; ℓ } ?> be the simple roots of the corresponding Kac–Moody root system. Let h ?> denote the Cartan subalgebra, let W denote the Weyl…

### A Lightcone Embedding of the Twin Building of a Hyperbolic Kac-Moody Group

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2020

Let A be a symmetrizable hyperbolic generalized Cartan matrix with Kac-Moody algebra g = g(A) and (adjoint) Kac-Moody group G = G(A)=$\langle\exp(ad(t e_i)), \exp(ad(t f_i)) \,|\, t\in C\rangle$…

### The geometry of rank 2 hyperbolic root systems

- Mathematics
- 2015

Let $\Delta$ be a rank 2 hyperbolic root system. Then $\Delta$ has generalized Cartan matrix $H(a,b)= \left(\begin{smallmatrix} ~2 & -b\\ -a & ~2 \end{smallmatrix}\right)$ indexed by…

### Root subsystems of rank 2 hyperbolic root systems

- Mathematics
- 2015

Let $\Delta$ be a rank 2 hyperbolic root system. Then $\Delta$ has generalized Cartan matrix $H(a,b)= \left(\begin{smallmatrix} ~2 & -b\\ -a & ~2 \end{smallmatrix}\right)$ indexed by…

### K−Fibonacci sequences and minimal winning quota in Parsimonious games ∗

- Mathematics
- 2014

Parsimonious games are a subset of constant sum homogeneous weighted majority games unequivocally described by their free type representation vector. We show that the minimal winning quota of…

### On the Topology of Kac–Moody groups

- Materials ScienceMathematische Zeitschrift
- 2013

We study the topology of spaces related to Kac–Moody groups. Given a Kac–Moody group over C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}…

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