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Chirality and Racemization on Isotopy Classes of Quasigroups

Takao Inoué

Abstract

We develop a structural and dynamical theory of chirality for quasigroups formulated at the level of isotopy classes. Interpreting isotopy as a gauge symmetry of re-coordinatization and mirror parastrophy as handedness reversal, we introduce a gauge-invariant continuous-time two-state Markov model in which transitions occur only between a quasigroup and its mirror. We prove that this dynamics descends to the isotopy quotient, yielding a reduced generator governed by a single class-dependent rate $k([Q])$. Symmetric mirror transitions lead to convergence toward a racemic equilibrium, whereas the vanishing condition $k([Q])=0$ characterizes dynamical chiral stability. By restricting admissible transitions to those generated by intrinsic symmetries, we show that $k([Q])=0$ is equivalent to the absence of mirror-isotopisms. A concrete example of order $7$ demonstrates the existence of structurally chiral quasigroup classes.

Chirality and Racemization on Isotopy Classes of Quasigroups

Abstract

We develop a structural and dynamical theory of chirality for quasigroups formulated at the level of isotopy classes. Interpreting isotopy as a gauge symmetry of re-coordinatization and mirror parastrophy as handedness reversal, we introduce a gauge-invariant continuous-time two-state Markov model in which transitions occur only between a quasigroup and its mirror. We prove that this dynamics descends to the isotopy quotient, yielding a reduced generator governed by a single class-dependent rate . Symmetric mirror transitions lead to convergence toward a racemic equilibrium, whereas the vanishing condition characterizes dynamical chiral stability. By restricting admissible transitions to those generated by intrinsic symmetries, we show that is equivalent to the absence of mirror-isotopisms. A concrete example of order demonstrates the existence of structurally chiral quasigroup classes.
Paper Structure (14 sections, 10 theorems, 39 equations, 1 figure)

This paper contains 14 sections, 10 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction (Motivation and Aim)
  2. A Two-State Dynamical Model on Isotopy Classes
  3. Quasigroups and Isotopy as Gauge Symmetry
  4. Mirror Operation as Parastrophy
  5. Gauge-Invariant Markov Dynamics
  6. Two-State Racemization Model
  7. Symmetric Racemization and Equilibrium
  8. Structural Vanishing of the Mirror Rate: When Does $k([Q])=0$?
  9. Autotopisms and Mirror-Isotopisms
  10. Intrinsic Mirror Dynamics Generated by Symmetry
  11. Structural Criterion for $k([Q])=0$
  12. Restricting Admissible Mirror Transitions to Intrinsic Symmetry
  13. Structural Characterization of $k([Q])=0$
  14. Conclusion

Key Result

Lemma 2.3

For any $g=(\alpha,\beta,\gamma)\in G$ and $Q\in\mathcal{Q}(X)$, where $g^{\#}:=(\beta,\alpha,\gamma)$.

Figures (1)

  • Figure 1: Schematic representation of chiral interconversion as a two-state dynamical system. The two nodes represent a pair of mirror-related states, conventionally denoted $(R)$ and $(S)$ in chemistry. In the case of thalidomide, these correspond to enantiomers which are known to interconvert rapidly under physiological conditions, so that even the administration of a single enantiomer may lead to the presence of its mirror form in vivo. When the surrounding environment does not distinguish left from right, the transition rates are effectively symmetric, and the resulting dynamics drives the system toward a racemic equilibrium with equal probabilities $\tfrac{1}{2}$ for each state. This diagram serves as a conceptual motivation for the algebraic model developed in this paper, where a quasigroup and its mirror parastrophe play the role of the two chiral states, and racemization is modeled as a gauge-invariant two-state Markov process on isotopy classes.

Theorems & Definitions (34)

  • Definition 2.1: Isotopy action
  • Definition 2.2: Mirror parastrophe
  • Lemma 2.3: Compatibility of mirror and isotopy
  • proof
  • Corollary 2.4
  • Definition 2.5: Isotopy-invariant observables
  • Theorem 2.6: Descent of two-state dynamics to isotopy classes
  • proof
  • Corollary 2.7: Racemization equilibrium
  • Remark 2.8
  • ...and 24 more