Related papers: Root subsystems of rank 2 hyperbolic root systems

In analogy to the theory of nilpotent orbit in finite-dimensional semisimple Lie algebras, it is known that the principal $\mathfrak{sl}_2$ subalgebras can be constructed in hyperbolic Kac-Moody Lie algebras. We obtained a series of…

A skew polynomial ring $R=K[x;\sigma,\delta]$ is a ring of polynomials with non-commutative multiplication. This creates a difference between left and right divisibility, and thus a concept of left and right evaluations and roots. A…

The analog of the principal SO(3) subalgebra of a finite dimensional simple Lie algebra can be defined for any hyperbolic Kac Moody algebra g(A) associated with a symmetrizable Cartan matrix A, and coincides with the non-compact group…

Cipriani and Sauvageot have shown that for any $L^2$-generator $L^{(2)}$ of a tracially symmetric quantum Markov semigroup on a C*-algebra $\mathcal{A}$ there exists a densely defined derivation $\delta$ from $\mathcal{A}$ to a Hilbert…

We show that the set $SA(G)$ of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic group $G$ is dense in the product of the sets $SA(P)$ over all maximal parabolic subgroups $P$. The set $BSA(G)$…

An $n \times n$ matrix $H$ is Butson-Hadamard if its entries are $k^{\text{th}}$ roots of unity and it satisfies $HH^* = nI_n$. Write $BH(n, k)$ for the set of such matrices. Suppose that $k = p^{\alpha}q^{\beta}$ where $p$ and $q$ are…

Let $\Delta$ be a hyperbolic triangle with a fixed area $\varphi$. We prove that for all but countably many $\varphi$, generic choices of $\Delta$ have the property that the group generated by the $\pi$--rotations about the midpoints of the…

A subclass of coded systems containing synchronized systems is the family of half-synchronized systems. In this note, we will consider them and show that the property half-synchronized lifts under hyperbolic maps. This enables us to define…

Let $f \in { \mathbb R} ( t) [x]$ be given by $ f(t, x) = x^n + t \cdot g(x) $ and $\beta_1 < \dots < \beta_m$ the distinct real roots of the discriminant $\Delta_{(f, x)} (t)$ of $f(t, x)$ with respect to $x$. Let $\gamma$ be the number of…

This paper investigates the effective categoricity of ultrahomogeneous structures. It is shown that any computable ultrahomogeneous structure is $\Delta^0_2$ categorical. A structure A is said to be weakly ultrahomogeneous if there is a…

This paper continues a geometric study of Harvey's Complex of Curves, whose ultimate goal is to apply the theory of hyperbolic spaces and groups to algorithmic questions for the Mapping Class Group and geometric properties of Kleinian…

We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the later one…

We state the relation between the variety of binary forms of given rank and the dual of the multiple root loci. This is a new result for the suprageneric rank, as a continuation of the work by Buczy\'nski, Han, Mella and Teitler. We…

The nonnegative integer rank of a matrix is a variant of the classical nonnegative rank, introduced in the 1980s, where factorizations are required to have integer entries. While computing nonnegative integer rank is generally very hard, we…

The $\mathcal{N}{=}\,2$ supersymmetric hyperbolic Calogero-Sutherland model obtained in arXiv:1902.08023 by gauging the $\mathcal{N}{=}\,2$ superfield matrix system is studied. Classical and quantum $\mathcal{N}{=}\,2$ supersymmetry…

There are various results connecting ranks of incidence matrices of graphs and hypergraphs with their combinatorial structure. Here, we consider the generalized incidence matrix $N_2$ (defined by inclusion of pairs in edges) for one natural…

The 727-dimensional root space associated with the level-2 root $\bLambda_1$ of the hyperbolic Kac--Moody algebra $E_{10}$ is determined using a recently developed string theoretic approach to hyperbolic algebras. The explicit form of the…

We describe a straightforward construction of the pseudo-split absolutely pseudo-simple groups of minimal type with irreducible root systems of type $BC_n$; these exist only in characteristic $2$. We also give a formula for the dimensions…

The origin of the bosonic and fermionic solutions, constructed in [1,2,3], to the symmetry equations corresponding to the two-dimensional bosonic and N=(2|2) supersymmetric Toda lattices is established, and algebras of the corresponding…

We discuss several topics related to the notion of strong hyperbolicity which are of interest in general relativity. After introducing the concept and showing its relevance we provide some covariant definitions of strong hyperbolicity. We…