Dynamical Non-Commutative Algebraic Geometry: Inflation, Bifurcation, and the Dynamics of Collapse across Division Algebras

TL;DR

This work introduces Dynamical Non-Commutative Algebraic Geometry (DNCAG) to study how non-commutativity in real normed division algebras inflates discrete root sets into continuous manifolds and how perturbations drive topology changes. It develops a comprehensive framework combining a Generalized Inflation Theorem, a Localization Alignment Principle, and a gradient-flow deformation retract to describe topological collapse as a symmetry-breaking process, with collapse times scaling as $T_{\rm collapse}\propto \epsilon^{-2}$ and an Entropy Scaling Law governing low-temperature behavior. The analysis spans central dynamics (breathing modes) and non-central perturbations across $\mathbb{H}$ and $\mathbb{O}$, culminating in a thermodynamic interpretation of collapse as a phase transition with a well-defined order parameter. The results provide a rigorous geometric-dynamical toolkit with potential implications for quaternionic/octonionic physics, dimensional reduction, and topological phase transitions in non-commutative settings, and lay groundwork for extending the framework to broader non-division-algebra contexts.

Abstract

We develop a framework for dynamical non-commutative algebraic geometry (DNCAG) by analyzing the evolution and stability of polynomial root manifolds in real normed division algebras ($\mathbb{H}$ and $\mathbb{O}$). We establish a Generalized Inflation Theorem, demonstrating that for central polynomials, the root set forms a homogeneous space $G/H$, where $G$ is the automorphism group of the algebra ($SO(3)$ for $\mathbb{H}$, $G_2$ for $\mathbb{O}$). This mechanism generates continuous geometry from non-commutativity. We analyze the dynamics under central modulation (breathing modes), classifying topological bifurcations ($Δ=0$). We then analyze the topological collapse induced by non-central perturbations, governed by symmetry reduction. We utilize the Localization Theorem (Gordon-Motzkin) to explain the alignment of roots with coefficient subalgebras. We formalize the dynamics of collapse using gradient flow on the potential landscape $\mathcal{V}(x) = \|P(x)\|^2$, characterizing it as a deformation retract and proving that the collapse timescale exhibits critical slowing down with quadratic scaling ($T_{\rm collapse} \propto ε^{-2}$). Finally, we introduce a thermodynamic formalism, proving an Entropy Scaling Law that rigorously characterizes the collapse as a symmetry-breaking phase transition.

Figures (5)

• Figure 1: Dynamics of Quaternionic Root Manifolds (Breathing Modes). Visualization of the solution set for $P(q,t) = q^4 + a(t)q^2 + b(t)$ in $\mathbb{H}$. The system exhibits coupled radial oscillations and changes in the radial separation ratio. An animation of these modes can be seen in http://youtu.be/WJTjyMth1FI
• Figure 2: Spectral Analysis of Breathing Modes (in $\mathbb{H}$). (1) Time evolution of radii. (2) Evolution of the Discriminant $\Delta(t)$. Markers indicate proximity to the bifurcation boundary, leading to sharp transitions (cusps). (3) Evolution of the gap distance. (4-6) Power spectra (PSD), confirming strong non-linear coupling.
• Figure 3: Visualization of Topological Collapse via Gradient Flow (in $\mathbb{H}$). The sequence illustrates the transition from $S^2$ to $S^0$ under a non-central perturbation aligned with the $i$-axis. (1) Initial state $V_0$. (2-4) Evolution under the gradient flow $-\nabla\mathcal{V}_\epsilon$. (5) Final state $V_\epsilon$: The manifold has collapsed onto the isolated roots (Alignment Principle). An animation of the collapse is shown in http://youtu.be/yaJQOuftZjE
• Figure 4: Geometric Decomposition of the Topological Collapse in $\mathbb{H}$. The plot visualizes the gradient flow on $V_0 \cong S^2$. The equator acts as an unstable equilibrium separatrix, dividing the manifold into two basins of attraction flowing to the attractors at $\pm i$.
• Figure 5: Thermodynamic Phase Transition and Statistical Symmetry Restoration (in $\mathbb{H}$). Visualization of the Gibbs probability measure $\mu_T$. (Left) Symmetric Phase ($\epsilon=0$). (Center) Ordered Phase ($\epsilon=2.5, T=0.15$). (Right) Restored Phase ($\epsilon=2.5, T=2.50$).

Theorems & Definitions (45)

• Definition 1.1: Analytic Deformation
• proof
• Theorem 2.1: Geometric Rigidity in $\mathbb{C}$
• Definition 2.2: Automorphism Group
• Theorem 2.3: Generalized Inflation Theorem
• proof
• Corollary 2.4: Dimensional Inflation Law
• proof
• Remark 2.5: Associativity and Conjugation
• Definition 3.1: Breathing Mode