Composition Algebras Research Articles

On the Leonardo quaternions sequence

In the present work, a new sequence of quaternions related to the Leonardo numbers -- named the Leonardo quaternions sequence -- is defined and studied. Binet's formula and certain sum and binomial-sum identities, some of which derived from the mentioned formula, are established. Tagiuri-Vajda's identity and, as consequences, Catalan's identity, d'Ocagne's identity and Cassini's identity are presented. Furthermore, applying Catalan's identity, and the connection between composition algebras and vector cross product algebras, Gelin-Cesàro's identity is also stated and proved. Finally, the generating function, the exponential generating function and the Poisson generating function are deduced. In addition to the results on Leonardo quaternions, known results on Leonardo numbers and on Fibonacci quaternions are extended.

Efficient Hyperdimensional Computing With Spiking Phasors.

Hyperdimensional (HD) computing (also referred to as vector symbolic architectures, VSAs) offers a method for encoding symbols into vectors, allowing for those symbols to be combined in different ways to form other vectors in the same vector space. The vectors and operators form a compositional algebra, such that composite vectors can be decomposed back to their constituent vectors. Many useful algorithms have implementations in HD computing, such as classification, spatial navigation, language modeling, and logic. In this letter, we propose a spiking implementation of Fourier holographic reduced representation (FHRR), one of the most versatile VSAs. The phase of each complex number of an FHRR vector is encoded as a spike time within a cycle. Neuron models derived from these spiking phasors can perform the requisite vector operations to implement an FHRR. We demonstrate the power and versatility of our spiking networks in a number of foundational problem domains, including symbol binding and unbinding, spatial representation, function representation, function integration, and memory (i.e., signal delay).

Tame quivers and affine bases II: Nonsimply-laced cases

In [21], we give a Ringel-Hall algebra approach to the canonical bases in the symmetric affine cases. In this paper, we extend the results to general symmetrizable affine cases by using Ringel-Hall algebras of representations of a valued quiver. We obtain a bar-invariant basis B′={C(c,tλ)|(c,tλ)∈Ga} in the generic composition algebra C⁎ and prove that B′=B′⊔(−B′) coincides with Lusztig's signed canonical basis B. Moreover, in type B˜n,C˜n, B′ is the canonical basis B.

Collineation groups of octonionic and split-octonionic planes

We present a Veronese formulation of the octonionic and split-octonionic projective and hyperbolic planes. This formulation of the incidence planes highlights the relationship between the Veronese vectors and the rank-1 elements of the Albert algebras over octonions and split-octonions, yielding to a clear formulation of the relationship with the real forms of the Lie groups arising as collineation groups of these planes. The Veronesean representation also provides a novel and minimal construction of the same octonionic and split-octonionic planes, by exploiting two symmetric composition algebras: the Okubo algebra and the paraoctonionic algebra. Besides the intrinsic mathematical relevance of this construction of the real forms of the Cayley–Moufang plane, we expect this approach to have implications in all mathematical physics related with exceptional Lie Groups of type [Formula: see text] and [Formula: see text].

On the lengths of Okubo algebras

The lengths of two particular cases of (possibly non-unital) composition algebras, namely standard composition algebras and Okubo algebras over an arbitrary field, are computed. These results finish the complete description of lengths of symmetric composition algebras and finite-dimensional flexible composition algebras.

A minimal and non-alternative realisation of the Cayley plane

The compact 16-dimensional Moufang plane, also known as the Cayley plane, has traditionally been defined through the lens of octonionic geometry. In this study, we present a novel approach, demonstrating that the Cayley plane can be defined in an equally clean, straightforward and more economic way using two different division and composition algebras: the paraoctonions and the Okubo algebra. The result is quite surprising since paraoctonions and Okubo algebra possess a weaker algebraic structure than the octonions, since they are non-alternative and do not satisfy the Moufang identities. Intriguingly, the real Okubo algebra has SU3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SU}\\left( 3\\right) $$\\end{document} as automorphism group, which is a classical Lie group, while octonions and paraoctonions have an exceptional Lie group of type G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {G}_{2}$$\\end{document}. This is remarkable, given that the projective plane defined over the real Okubo algebra is nevertheless isomorphic and isometric to the octonionic projective plane which is at the very heart of the geometric realisations of all types of exceptional Lie groups. Despite its historical ties with octonionic geometry, our research underscores the real Okubo algebra as the weakest algebraic structure allowing the definition of the compact 16-dimensional Moufang plane.

Roots and right factors of polynomials and left eigenvalues of matrices over Cayley-Dickson algebras

Over a composition algebra $A$, a polynomial $f(x) \in A[x]$ has a root $\alpha$ if and only $f(x)=g(x)\cdot (x-\alpha)$ for some $g(x) \in A[x]$. We examine whether this is true for general Cayley-Dickson algebras. The conclusion is that it is when $f(x)$ is linear or monic quadratic, but it is false in general. Similar questions about the connections between $f$ and its companion $C_f(x)=f(x)\cdot \overline{f(x)}$ are studied. Finally, we compute the left eigenvalues of $2\times 2$ octonion matrices.

Composition Algebras of Arbitrary Degrees

Composition Algebras of Arbitrary Degrees

Rigidity of projective symmetric manifolds of Picard number 1 associated to composition algebras

Rigidity of projective symmetric manifolds of Picard number 1 associated to composition algebras

Annihilating Entanglement Between Cones

Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The max-entanglement annihilating maps transform the k-fold maximal tensor product of a cone {textsf{C}}_1 into the k-fold minimal tensor product of a cone {textsf{C}}_2, and the pair ({textsf{C}}_1,{textsf{C}}_2) is called resilient if all max-entanglement annihilating maps are entanglement breaking. Our main result is that ({textsf{C}}_1,{textsf{C}}_2) is resilient if either {textsf{C}}_1 or {textsf{C}}_2 is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. (Phys Rev Lett 76(5):722, 1996) as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for max-entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.

Satellite Image Processing by Python and R Using Landsat 9 OLI/TIRS and SRTM DEM Data on Côte d’Ivoire, West Africa

In this paper, we propose an advanced scripting approach using Python and R for satellite image processing and modelling terrain in Côte d'Ivoire, West Africa. Data include Landsat 9 OLI/TIRS C2 L1 and the SRTM digital elevation model (DEM). The EarthPy library of Python and 'raster' and 'terra' packages of R are used as tools for data processing. The methodology includes computing vegetation indices to derive information on vegetation coverage and terrain modelling. Four vegetation indices were computed and visualised using R: the Normalized Difference Vegetation Index (NDVI), Enhanced Vegetation Index 2 (EVI2), Soil-Adjusted Vegetation Index (SAVI) and Atmospherically Resistant Vegetation Index 2 (ARVI2). The SAVI index is demonstrated to be more suitable and better adjusted to the vegetation analysis, which is beneficial for agricultural monitoring in Côte d'Ivoire. The terrain analysis is performed using Python and includes slope, aspect, hillshade and relief modelling with changed parameters for the sun azimuth and angle. The vegetation pattern in Côte d'Ivoire is heterogeneous, which reflects the complexity of the terrain structure. Therefore, the terrain and vegetation data modelling is aimed at the analysis of the relationship between the regional topography and environmental setting in the study area. The upscaled mapping is performed as regional environmental analysis of the Yamoussoukro surroundings and local topographic modelling of the Kossou Lake. The algorithms of the data processing include image resampling, band composition, statistical analysis and map algebra used for calculation of the vegetation indices in Côte d'Ivoire. This study demonstrates the effective application of the advanced programming algorithms in Python and R for satellite image processing.

The structure of symmetric tensor powers of composition algebras

Let [Formula: see text] be a composition algebra which is either the Hamilton quaternion algebra [Formula: see text] or the Cayley octonion algebra [Formula: see text] over [Formula: see text]. In a previous work, the nth symmetric power [Formula: see text] of [Formula: see text] is shown to be a direct sum of central simple algebras, corresponding to the partitions of [Formula: see text] of length [Formula: see text], such that the component corresponding to the partition [Formula: see text] is isomorphic to the component [Formula: see text] of [Formula: see text] corresponding to the partition [Formula: see text] of [Formula: see text]. In this work, we study the building blocks [Formula: see text] of these decompositions. We show that the “local” structure of [Formula: see text], i.e. the complex-like subfields of [Formula: see text], determine both the complement of [Formula: see text] in [Formula: see text] and the trace map of [Formula: see text], induced from the trace map of [Formula: see text]. We also derive a recursive trace formula on the [Formula: see text]’s. We use the “local-global” results to define positive definite symmetric bilinear forms on the vector space [Formula: see text], which has a natural structure of a commutative and associative algebra. Finally, the structure of the central simple algebra [Formula: see text] is described.

A geometrical interpretation of Okubo Spin group

In this work we define, for the first time, the affine and projective plane over the real Okubo algebra, showing a concrete geometrical interpretation of its Spin group. Okubo algebra is a flexible, composition algebra which is also a not unital division algebra. Even though, Okubo algebra has been known for more than 40 years, we believe that this is the first time the algebra was used for affine and projective geometry. After showing that all axioms of affine geometry are verified, we define a projective plane over Okubo algebra as completion of the affine plane and directly through the use of Veronese coordinates. We then present a bijection between the two constructions. Finally we show a geometric interpretation of Spin(O) as the group of collineations that preserve the axis of the plane.

Affine Grassmannians for triality groups

We study affine Grassmannians for ramified triality groups. These groups are of type D43, so they are forms of the orthogonal or the spin groups in 8 variables. They can be given as automorphisms of certain twisted composition algebras obtained from the octonion algebras. Using these composition algebras, we give descriptions of the affine Grassmannians for these triality groups as functors classifying suitable lattices in a fixed space.

Geometric characterisation of subvarieties of 𝓔6(𝕂) related to the ternions and sextonions

Abstract The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety 𝓔6(𝕂) over an arbitrary field 𝕂. The characterised varieties arise as Veronese representations of certain ring projective planes over quadratic subalgebras of the split octonions 𝕆’ over 𝕂 (among which the sextonions, a 6-dimensional non-associative algebra). We describe how these varieties are linked to the Freudenthal–Tits magic square, and discuss how they would even fit in, when also allowing the sextonions and other “degenerate composition algebras” as the algebras used to construct the square.

Misconfiguration-Free Compositional SDN for Cloud Networks

Cloud computing provides a new paradigm to offer flexible IT infrastructures. In IaaS clouds, tenants deploy software-defined networking (SDN) policies to simplify network management and customize network behaviors. However, programming SDN networks is error-prone no matter using low-level APIs or high-level programming languages. Specifically, SDN policies may contain misconfigurations that do not break the pre-defined network invariants (e.g., black holes), but either degrade the deployment efficiency or mistakenly translate tenants intents. Prior studies for checking either traditional access control policies or network-wide invariants, are thus fail to detect these misconfigurations. To address this gap, this paper presents PMM, a misconfiguration checking tool for compositional SDN that works at the data plane of cloud networks. We first propose a new data structure, minimal interval set, to represent the match patterns of rulesets. This representation serves the basis for composition algebra construction and misconfiguration checking. We then propose the principles, algorithms and also optimisations for fast and accurate checking. We finally implement PMM in Covisor. Experiments with both real-world rulesets and synthetic rulesets show that PMM can detect misconfigurations of SDN policies in cloud networks within hundreds of milliseconds.

Twisted composition algebras and Arthur packets for triality $\operatorname{Spin}_8$

Twisted composition algebras and Arthur packets for triality $\operatorname{Spin}_8$

View Composition Algebra for Ad Hoc Comparison.

Comparison is a core task in visual analysis. Although there are numerous guidelines to help users design effective visualizations to aid known comparison tasks, there are few techniques available when users want to make ad hoc comparisons between marks, trends, or charts during data exploration and visual analysis. For instance, to compare voting count maps from different years, two stock trends in a line chart, or a scatterplot of country GDPs with a textual summary of the average GDP. Ideally, users can directly select the comparison targets and compare them, however what elements of a visualization should be candidate targets, which combinations of targets are safe to compare, and what comparison operations make sense? This article proposes a conceptual model that lets users compose combinations of values, marks, legend elements, and charts using a set of composition operators that summarize, compute differences, merge, and model their operands. We further define a View Composition Algebra (VCA) that is compatible with datacube-based visualizations, derive an interaction design based on this algebra that supports ad hoc visual comparisons, and illustrate its utility through several use cases.

On the lengths of standard composition algebras

We suggest a new method which allows us to compute the lengths of (possibly non-unital) standard composition algebras over an arbitrary field with

A compositional mediation model for a binary outcome: Application to microbiome studies.

The delicate balance of the microbiome is implicated in our health and is shaped by external factors, such as diet and xenobiotics. Therefore, understanding the role of the microbiome in linking external factors and our health conditions is crucial to translate microbiome research into therapeutic and preventative applications. We introduced a sparse compositional mediation model for binary outcomes to estimate and test the mediation effects of the microbiome utilizing the compositional algebra defined in the simplex space and a linear zero-sum constraint on probit regression coefficients. For this model with the standard causal assumptions, we showed that both the causal direct and indirect effects are identifiable. We further developed a method for sensitivity analysis for the assumption of the no unmeasured confounding effects between the mediator and the outcome. We conducted extensive simulation studies to assess the performance of the proposed method and applied it to real microbiome data to study mediation effects of the microbiome on linking fat intake to overweight/obesity. An R package can be downloaded from https://github.com/mbsohn/cmmb. Supplementary files are available at Bioinformatics online.