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Anti-rotors and algebra invariants

Fred Greensite

TL;DR

The paper introduces a novel framework of algebraic invariants based on transformations tied to the bilinear product of real finite-dimensional unital associative algebras. Central to the approach are the Skewer map and the anti-rotor, whose uncurling metrics yield a rich family of invariants, including the anti-rotor dimension, ranks, and a dual space of unital norms that induce logarithmic and quadratic geometries on algebras. The authors develop a functorial view linking algebras to their anti-rotors, provide explicit computations for low-dimensional and structured algebras (e.g., Toeplitz, direct-product, and triangular algebras), and define Type I and Type II anti-rotors to organize an infinite landscape of potential invariants. Collectively, these results offer a new, differentiable lens on algebra structure, with tractable invariants and explicit norm geometries that extend beyond classical norms and ideals, potentially aiding classification and comparison of algebras. The work also outlines conjectures about the universality of anti-rotors across function families and orders, inviting further exploration and computational verification across broader algebra classes.

Abstract

We derive previously unrecognized geometrically-based isomorphism invariants for real finite-dimensional unital associative algebras, resulting from examination of transformations from an algebra to itself that are dependent on the algebra's bilinear product operation.

Anti-rotors and algebra invariants

TL;DR

The paper introduces a novel framework of algebraic invariants based on transformations tied to the bilinear product of real finite-dimensional unital associative algebras. Central to the approach are the Skewer map and the anti-rotor, whose uncurling metrics yield a rich family of invariants, including the anti-rotor dimension, ranks, and a dual space of unital norms that induce logarithmic and quadratic geometries on algebras. The authors develop a functorial view linking algebras to their anti-rotors, provide explicit computations for low-dimensional and structured algebras (e.g., Toeplitz, direct-product, and triangular algebras), and define Type I and Type II anti-rotors to organize an infinite landscape of potential invariants. Collectively, these results offer a new, differentiable lens on algebra structure, with tractable invariants and explicit norm geometries that extend beyond classical norms and ideals, potentially aiding classification and comparison of algebras. The work also outlines conjectures about the universality of anti-rotors across function families and orders, inviting further exploration and computational verification across broader algebra classes.

Abstract

We derive previously unrecognized geometrically-based isomorphism invariants for real finite-dimensional unital associative algebras, resulting from examination of transformations from an algebra to itself that are dependent on the algebra's bilinear product operation.
Paper Structure (29 sections, 25 theorems, 87 equations)

This paper contains 29 sections, 25 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. The Skewer and the anti-rotor
  3. The anti-rotor and resulting invariants
  4. The dual of the anti-rotor and resulting geometries
  5. The transform of an uncurling metric
  6. Special unital norms
  7. The geometries associated with an algebra
  8. Generalization of the usual algebra norm of an element
  9. Combinatorial issues
  10. Basic functorial aspects and a couple of useful results
  11. Illustrations
  12. Algebras with dimension less than 5.
  13. Two-dimensional algebras
  14. Three-dimensional algebras.
  15. The algebra with vector space of elements $\mathbb{R}^n$ and component-wise multiplication
  16. ...and 14 more sections

Key Result

Theorem 3.2

If $K:A_1\rightarrow A_2$ is an algebra isomorphism, then where $K^T \mathfrak{u}_{A_2} K \equiv \{K^T L K:L\in \mathfrak{u}_{A_2}\}$.

Theorems & Definitions (54)

  • Definition 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • Corollary 3.6
  • Definition 4.1
  • ...and 44 more