Division as a radial vector relationship – Alpha group

1 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-19, 2025 Division as a radial vector relationship – Alpha group A divisão como vetor de relação radial – grupo Alpha La división como vector de relación radial – grupo Alpha DOI: 10.54021/seesv6n1-037 Originals received: 4/7/2025 Acceptance for publication: 4/30/2025 Cleber Souza Corrêa PhD in Water Resources Institution: Institute of Aeronautics and Space Address: São Jose dos Campos, São Paulo, Brazil E-mail: [email protected] Thiago Braido Nogueira de Melo Master in Aeronautical and Mechanical Engineering Institution: Institute of Aeronautics and Space Address: São Jose dos Campos, São Paulo, Brazil E-mail: [email protected] ABSTRACT In this paper we present the formulation of a hypercomplex algebra called the Alpha number algebra, defined by four fundamental elements: 1, i, μ, and i.μ, with the following operational properties: The imaginary number μ is idempotent and non-commutative with the other imaginary number i. This non-commutative and partially idempotent structure provides an algebraic basis for the matrix M(θ), which parameterizes a class of operators that exhibit internal oscillatory symmetries with explicit dependence on the angle θ. The spectral analysis of M(θ) reveals the emergence of discrete natural frequencies at θ = π/2 rad and its multiples, pointing to a spontaneous geometry. The eigenvalues and associated eigenvectors characterize internal modes of vector rotation and propagation, allowing us to describe an internal vector dynamic geometry. This geometry suggests an alternative model of the internal structure of topological space and geometry, in which interactions and vector oscillations emerge directly from the algebra of the Alpha group. This formulation provides support for a new mathematical language applicable to the study of fundamental fields. Keywords: Abstract Algebra. Group Theory. Abstract Geometry. Alpha Group. RESUMO Apresentamos neste artigo a formulação de uma álgebra hipercomplexa denominada álgebra do número Alpha, definida por quatro elementos fundamentais: 1, i, μ e i.μ, com as seguintes propriedades operacionais o numero imaginário μ caracteriza idempotente, e não comutativo com o outro numero 2 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-19, 2025 imaginário i. Essa estrutura não comutativa e parcialmente idempotente fornece uma base algébrica para a matriz M(θ), a qual parametriza uma classe de operadores que exibem simetrias internas oscilatórias com dependência explícita do ângulo θ. A análise espectral de M(θ) revela a emergência de frequências naturais discretas em θ = π/2 rad e seus múltiplos, apontando para uma geometria espontânea. Os autovalores e autovetores associados caracterizam modos internos de rotação e propagação vetorial, permitindo descrever uma geometria dinâmica vetorial interna. Essa geometria sugere um modelo alternativo de estrutura interna do espaço topológico e de geometria, no qual interações e oscilações vetoriais emergem diretamente da álgebra do grupo Alpha. Tal formulação fornece subsídios para uma nova linguagem matemática aplicável à estudos dos campos fundamentais. Palavras-chave: Álgebra Abstrata. Teoria de Grupo. Geometria Abstrata. Grupo Alpha. RESUMEN En este artículo presentamos la formulación de una álgebra hipercompleja denominada álgebra de números Alpha, definida por cuatro elementos fundamentales: 1, i, μ e i.μ, con las siguientes propiedades operacionales: el número imaginario μ se caracteriza por ser idempotente y no conmutativo con el otro número imaginario i. Esta estructura no conmutativa y parcialmente idempotente proporciona una base algebraica para la matriz M(θ), que parametriza una clase de operadores que exhiben simetrías oscilatorias internas con dependencia explícita del ángulo θ. El análisis espectral de M(θ) revela la aparición de frecuencias naturales discretas en θ = π/2 rad y sus múltiplos, lo que apunta a una geometria espontánea. Los valores propios y los vectores propios asociados caracterizan los modos internos de rotación y propagación del vector, permitiendo describir una geometría dinámica vectorial interna. Esta geometría sugiere un modelo alternativo de la estructura interna del espacio topológico y la geometría, en el que las interacciones y oscilaciones vectoriales emergen directamente del álgebra del grupo Alpha. Esta formulación proporciona apoyo a un nuevo lenguaje matemático aplicable a los estudios de campos fundamentales. Palabras clave: Álgebra Abstracta. Teoría de Grupos. Geometría Abstracta. Grupo Alpha. 1 INTRODUCTION TO THE ALPHA GROUP In recent decades, significant advances in differential geometry, mathematical physics, and field theory have paved the way for the unification of structures that were previously treated separately. The Alpha group's proposal emerges in this context as a conceptual tool with geometric significance. It defines a new vector algebra that extends the classical four-dimensional vector space with 3 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-19, 2025 an unconventional basis (1, i, μ, i.μ). In this structure, the elements satisfy specific algebraic relations that break with the traditional symmetry of Euclidean space, introducing a geometry rich in internal curvatures, non-trivial rotations, and radial directionality. The Alpha group is based on the operation of division as a mechanism for generating symmetries and invariances. In particular, it provides a new interpretation for the spatial orientation of radial vectors, coupled with a metric that depends on an angular parameter θ, representing the continuous transition between a flat local geometry (Euclidean) and a curved global geometry (hyperbolic and conformal). Corrêa et al. (2022) proposed a mathematical structure called the Alpha group, which is an extension of vector geometry and group theory that provides additional concepts for differential geometry. The group is based on a fourdimensional division ring. In the article by Corrêa et al. (2024), they presented a metric tensor of curvature associated with the Alpha group, which describes the hyperbolic behavior in radial regions and exponential compactification in the asymptotic limit. This generates a self-conformal geometry (Thomas (1926), Akivis and Goldberg (2011), and Schinzinger and Laura (2012)) with internal symmetries between the vector quadrants, expressing rotations between the quadrants (1↔3 and 2↔4), radial contractions and expansions, and the dynamics of the vector field along radial trajectories. However, normative structures and the algebra of the division ring of the Alpha group remained to be defined. Traditionally, the division operation is treated as a simple binary operation. Given a number a and a divisor b, a/b is calculated, which is how many times b fits into a. However, in Alpha group theory, this view is update, in which the division operation ceases to be a scalar quotient and is interpreted as a radial vector that expresses a function between two elements of space. In this aspect, what is the direction and intensity of the transformation that leads b to a? This transformation is expressed by a radial vector, a vector that starts from the origin or center of the structure and points towards the geometric reorganization generated by the division operation in the Alpha group. The radial vector has two important pieces of information: its magnitude, the division modulus (as in usual algebra), which indicates how much the vector a differs from the vector b in terms of scale. The direction, the spatial 4 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-19, 2025 orientation induced by the vector division, expresses a rotational curvature as defined by the denominator. This work seeks to standardize the norm as well as to define algebraic properties of the Alpha group by defining the radial vector division operation. In this paper, we introduce the division as a radial vector of relation, a geometric structure that captures the essence of this new vector algebra. This metric not only redefines the concept of vector norm but also paves the way for potential applications in gauge theories (Weyl, 1918; Gomis et al., 1995; and Hamilton, 2017), internal curvature, and mathematical models of compactification of infinity (Jadczyk, 2011; Gingold and Solomon, 2013; and Wieczorek, 2021). We will show the formal definition of this metric, its evolution with the parameter θ, and the geometric implications and the transition between the local (at θ = 0) and global (at θ = π/2) regimes. Finally, we will discuss how these initial results can be interpreted as the operation of radial vector division, with the potential to represent deep physical phenomena in a still unexplored geometry. 2 RESULTS The Alpha Group is an innovative mathematical structure that extends fundamental concepts of algebra and geometry, exploring division as a central operation. Unlike classical rotation and transformation groups, the Alpha Group is defined by a division matrix parameterized by an angle θ, being a conformal group that governs the dynamics of spaces of negative and positive curvature, generating a hyperbolic hyperboloid in R4, in addition to having rotations and movements between quadrants 1 and 3 and quadrants 2 and 4, which can be observed in figure 1, an aspect of the space of the Alpha group (global). The construction of the Alpha group starts from an expanded algebra, based on the elements 1, i, μ, i.μ, where: a) i2 = −1 (traditional imaginary unit); b) μ2 = μ (invariant imaginary number); c) (i.μ)2 = −μ (related to rotations and transformations in the Alpha group space); d) i.μ= - μ.i, exhibits non-commutativity. 5 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-19, 2025 Geometric Interpretation: e) Direction 1 would represent the traditional scalar axis (real number); f) The i direction would represent the planar orthogonal rotation (Euclidean space); g) The direction μ would represent the invariable radiality, or curvature; h) The i.μ direction would represent the curved rotation related to the curved geometry. This set of rules defines a minimal structure of the algebra of the Alpha group, on which norms, derivatives, operators, fields, tensors, and connections can be constructed. The resulting geometric structure leads to an asymptotic compactification, where the curvature of space evolves dynamically as the angle θ varies from 0 to π/2. For θ = 0, the metric is essentially Euclidean and exhibits rapid convergence to equilibrium. For θ = π/2, a conformal compactification occurs, closing the hyperbolic hyperboloid and revealing asymptotic attraction patterns in the space of the Alpha group. This unifying property of curvature allows the Alpha group to be interpreted as a geometric model that simultaneously encompasses elliptical and hyperbolic surfaces within the same structure. This approach may have profound implications for differential geometry, theoretical physics, and the analysis of dynamical systems, opening up new possibilities for understanding the structure of space and conformal transformations. The study of the Alpha group reveals new mathematical and geometric dimensions, enabling an alternative interpretation of topology and transformations in space. Its asymptotic behavior and its relationship with dynamical systems indicate deep connections with theoretical physics and differential geometry, making it a promising subject of investigation. In this case, we seek to define its algebra and the radial vector division operation in the context of the Alpha group. 6 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-19, 2025 Figure 1. The geometric space of the Alpha Group in R4, Poincaré cut. Source: Authors 2.1 GENERAL STRUCTURE Let us consider a vector a∈R4, with components in the extended basis {1, i, μ, i.μ}, where: 1 = real number i2 = −1 (classical imagery) μ2 = μ (imaginary invariant, idempotent) (i.μ)2 = −μ (rotational, effects of curvature) i.μ = - μ.i (non-commutative) This structure has elements that are neither purely rotational (like i) nor purely projective (like μ), but that alternate sign when squared; such a situation may be typical of an oscillation with topological asymmetry. 2.2 NORM ∥a∥ The norm should capture the radial geometry and the rotation induced by the Alpha group structure (Bourbaki (2013) and Treves (2016)). It follows the following structure: 7 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-19, 2025 ∥a∥α2 = ⟨a,a⟩α where: the inner product ⟨⋅,⋅⟩α is induced by the conformal rotation matrix M(θ), so that: ⟨a,b⟩α = a⊤⋅ G(θ) ⋅ b where: G(θ)= M(θ)⊤M(θ), the conformal metric of the Alpha group. In which M(θ) is the matrix: M(θ) = (1 −cotg θ −tg θ 1 cotg θ 1 −1 −tg θ tg θ −1 1 −cotg θ 1tg θ cotg θ 1) (1) Let a∈A be a vector in the Alpha algebra, such that: a = a1 . 1 + a2 ⋅ i + a3 . μ +a4 . i.μ We define the Alpha norm as: ∥a∥α =⟨a,a⟩α := a12 + a22 +a3 +a42 μ Interpretation of terms: a12+a22: classical Euclidean components (real and imaginary); the linear term a3 in μ, reflecting the invariant nature of radiality; and the term a42 μ, which is the curved term in i.μ, producing a field structure. Note that this norm returns a value that is not purely real, since the term a42 μ is multiplied by the imaginary element μ, which is non-zero, is not a real number, and has the idempotent property, a characteristic of which shows that certain operations can be performed multiple times without altering the result. The norm is not scalar in the traditional sense: it expresses a gauge-invariant measure (Weyl, 1918) with radial curvature. The term a3 is not squared, reflecting the invariance of μ. Since the operation of μ² = μ, it is not squared like the other terms. The presence of μ 8 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-19, 2025 and iμ represents non-Euclidean geometric effects (Coxeter (1998) and Ryan (2009)) and field interactions in vector space (Lopes (2013) and Hamilton (2017)). 2.3 RADIAL VECTOR DIVISION a ⊘α b Inspired by the algebraic structure of division, especially the way the Alpha group deforms the local geometry: a⊘α b := M(θ)−1 ⋅ (a − b) But with radial vector sense. To be more in-depth, this operation can be defined as: a⊘α b := M(θ)−1 ⋅ (a ⊖α b) where: α is a conformal radial subtraction, adjusted to the local curvature induced by θ. In more specific cases: a⊘α b=M(θ)−1⋅ a ⋅ b−1 Showing a radial vector inversion in the Alpha group. The geometric interpretation associated with radial vector division: At θ = 0 rad, the norm ∥a∥α reduces to the Euclidean norm (local flat space). At θ = π/2 rad, the norm measures the rotational curvature of the projection of a in the closed space of the Alpha group. Radial vector division measures the geometric conformal difference, taking into account the rotation of the field: not just the "how much", but the "how" the direction between two vectors deforms. Considering the numerical structure of the Alpha group as a + bi + c μ + d i.μ, we have figure 2, which presents the relation of the radial vector division close to zero radians, and in figures 3, 4, and 5, approaching π/2 radians. They show the rotational dynamics in which close to π/2 radians in coordinate conforms to the asymptotic compactification in the formation 9 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-19, 2025 of the closed hyperboloid (figure 1). The projection of the relation of the radial vector division was generated in a Python script using Google Colab. The angle between the vectors is preserved, but the length of the vectors can scale (increase) proportionally and directionally. Rotational curvature means that this conformity changes with direction, particularly along radial trajectories. And as θ varies from 0 to π/2, the vector a describes a trajectory with exponential growth on certain axes (hyperbolic) and conformal rotation on others (elliptical). Hence, the resulting curvature is directionally dependent and conformal: the local structure is scaled, but the angles and rotations are preserved in the overall orientation, with a geometric twist. Figure 2. Radial vector division near zero radians, representing a (local) Euclidean space. Source: Authors Figure 3. Radial vector division, angle θ at 1.43 radians. Source: Authors 16 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-19, 2025 negative curvature within the same geometric structure, as well as presenting dynamics between its quadrants. This approach can have profound implications in differential geometry, theoretical physics, and the analysis of dynamic systems, opening up new possibilities for understanding the structure of space and conformal transformations. The matrix M(θ) has two highly relevant properties, one acting on the curvature of the vector space and at the same time generating rotations, creating an internal dynamic. Therefore, as θ traverses the interval [0, π/2], the Alpha group scans a space of possible geometries, where each point represents a different topology and, therefore, a different physical state of the field. These results associated with the matrix M(θ) can be associated by similarity in a coupled rotational structure when the angle is θ = π/2; they would reach their maximum expression of rotation and dynamics, and it is at this point that it manifests itself as a geometric. They have transverse propagation; the generated vectors move in a trajectory that is not linear but rather helical or oscillatory, very similar to the propagation of an electromagnetic wave in a vacuum. They present conformal dynamics; the geometry of the Alpha group would allow this propagation to be understood as a manifestation of curvature and conformal deformation of space. They also show aspects of duality, with the fact that the Alpha group uses imaginary numbers such as i, μ, and iμ revealing an internal duality, with imaginary, real, conformal, and rotational parts, just as light carries energy, momentum, and spin in an inseparable duality. The matrix M(θ), at this point, behaves like a dynamic operator that rotates vectors around intertwined axes, propagates these rotations in a synchronized manner, and stabilizes an oscillatory structure in 4D space. When θ = π/2, in fact, the behavior of the system reveals itself with its full symmetry, and the complete rotations between the axes and the vector structure converge to a dynamic, almost wave-like pattern. This analogy can even open space for deeper connections, in which the Alpha group has a structure that carries symmetry and propagation. Its radial lines can be like conformal lines of force. Such results and geometric, rotational, and vectorial aspects would open the possibility of this mathematical model representing a primary geometric e algebraic structure. The Alpha group can be seen as a geometrized manifestation of a field; that is, space itself would be organized dynamically and rotationally in a vectorial field generated by conformal deformations. The Alpha Group's proposal 17 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-19, 2025 represents a significant expansion of the foundations of applied mathematics by introducing a new algebraic and geometric structure based on elements such as the imaginary number μ, which is idempotent and non-commutative, with possible applications in physics. Its results, which involve an internal oscillating vector geometry, have the potential to unify areas of mathematics. From an academic perspective, this approach opens new fronts of research in algebra, differential geometry, and field theory, encouraging interdisciplinary dialogue between mathematics, physics, and the philosophy of science. Socially, by proposing a new symbolic and visual language of nature, the Alpha Group can renew the teaching of fundamental sciences, in addition to inspiring future technologies based on dynamic symmetries of topological space. Despite the conceptual advances proposed by the Alpha Group, this research is still in its early stages and has limitations that deserve consideration. The complete formalization of its algebra, as well as its compatibility with already established structures, requires rigorous indepth study. Furthermore, the absence of direct experimental or observational models imposes restrictions on the empirical validation of the theory. For future studies, it is recommended to develop more robust symbolic computational applications, the analysis of the dynamic behavior of systems based on the matrix M(θ). These advances may consolidate the Alpha Group as a promising tool for reformulating the foundations of modern mathematics. 18 Studies in Engineering and Exact Sciences, Curitiba, v.6, n.1, p.01-19, 2025 REFERENCES AKIVIS, MAKS A.; GOLDBERG, VLADISLAV V. CONFORMAL DIFFERENTIAL GEOMETRY AND ITS GENERALIZATIONS. HOBOKEN: JOHN WILEY & SONS, 2011. BOURBAKI, Nicolas. 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