Related papers: Trees, quaternion algebras and modular curves
We investigate the finite subgroups that occur in the Hamiltonian quaternion algebra over the real subfield of cyclotomic fields. When possible, we investigate their distribution among the maximal orders.
Let $B$ be a totally-definite quaternion algebra over a totally real field $F$, let $\mathfrak{p}$ be a prime ideal of $F$, and let $\Gamma$ be the group of reduced norm-$1$ elements of an Eichler $\mathcal{O}_F[1/\mathfrak{p}]$-order $R$…
When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in…
In this paper we construct explicit LPS-type Ramanujan graphs from each definite quaternion algebra over $\mathbb Q$ of class number 1, extending the constructions of Lubotzky, Phillips, Sarnak, and later Chiu, and answering in the…
Given a field with a set of discrete valuations $V$, we show how the genus of a division algebra over the field is related to the genus of the residue algebras at various valuations in $V$ and the ramification data. When the division…
This article is devoted to the investigation of wrap groups of connected fiber bundles over the fields of real $\bf R$, complex $\bf C$ numbers, the quaternion skew field $\bf H$ and the octonion algebra $\bf O$. Cohomologies of wrap groups…
We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and…
Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via…
In this paper, we introduce the generalized Fibonacci-Lucas quaternions and we prove that the set of these elements is an order,in the sense of ring theory, of a quaternion algebra. Moreover, we investigate some properties of these…
For a spinor field theory on the Bruhat-Tits tree, we calculate the action and the partition function of its boundary theory by integrating out the interior of the Bruhat-Tits tree. We found that the boundary theory is very similar to a…
We extend our previous computations for the relative positions of branches of quaternions to the case of local fields of even characteristic. This is a key step to understand the set of maximal orders containing a given suborder, which is…
Numerous attempts have been made to replicate the success of complex-valued algebra in engineering and science to other hypercomplex domains such as quaternions, tessarines, biquaternions, and octonions. Perhaps, none have matched the…
We extend the maximal unitarity method to amplitude contributions whose cuts define multidimensional algebraic varieties. The technique is valid to all orders and is explicitly demonstrated at three loops in gauge theories with any number…
In this paper, Grand Unified theories are discussed in terms of quaternions and octonions by using the relation between quaternion basis elements with Pauli matrices and Octonions with Gell Mann \lambda matrices. Connection between the…
Trusses, defined as sets with a suitable ternary and a binary operations, connected by the distributive laws, are studied from a ring and module theory point of view. The notions of ideals and paragons in trusses are introduced and several…
Suppose $4|n$, $n\geq 8$, $F=F_n=\mathbb{Q}(\zeta_n+\bar{\zeta}_n)$, and there is one prime $\mathfrak{p}=\mathfrak{p}_n$ above $2$ in $F_n$. We study amalgam presentations for $\operatorname{PU_{2}}(\mathbb{Z}[\zeta_n, 1/2])$ and…
In this paper we study special Fibonacci quaternions and special generalized Fibonacci-Lucas quaternions in quaternion algebras over finite fields.
We study some quadratic algebras which are appeared in the low-dimensional topology and Schubert calculus. We introduce the Jucys-Murphy elements in the braid algebra and in the pure braid group, as well as the Dunkl elements in the…
We provide an expository account of some of the Hopf algebras that can be defined using trees, labeled trees, ordered trees and heap ordered trees. We also describe some actions of these Hopf algebras on algebra of functions.
In this note we prove the classical decomposition theorems for certain subgroups of the automorphism group of the Bruhat-Tits tree associated to $\mathrm{PGL}_{2}(F)$, where $F$ is a local field. The results can be used to understand the…