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Two axes in non-commutative algebras with a Frobenius form

Yoav Segev

TL;DR

The paper analyzes axial algebras with a Frobenius form and two half-type axes, showing how the axis-decomposition of products constrains the multiplication and yields derivation criteria via Miyamoto automorphisms. It introduces the polynomials $Q_c$ and $P_c$, connects their vanishing to axis structure, and derives a suite of explicit, componentwise identities that govern products across $A_0(a)$ and $A_{\\tfrac12}(a)$, including elementary proofs of key identities. The main results establish when $[L_a,L_b]$ is a derivation, and present two central, elementary proofs of a major $1/2$-weight identity for half-axes under various conditions on the mutual inner product $(a,b)$, with a separate treatment of the exceptional case $(a,b)=1$. Overall, the work provides elementary, constructive identities that reveal the algebraic structure of primitive axes of Jordan type $\\tfrac12$ and offer tools for analyzing axial algebras related to transposition groups and the Monster algebra.

Abstract

Throughout this paper $A$ is a commutative non-associative algebra over a field $\mathbb{F}$ of characteristic not $2.$ In addition $A$ posses a Frobenius form. We obtain detailed information about the multiplication in $A$ given two axes of type half in $A.$

Two axes in non-commutative algebras with a Frobenius form

TL;DR

The paper analyzes axial algebras with a Frobenius form and two half-type axes, showing how the axis-decomposition of products constrains the multiplication and yields derivation criteria via Miyamoto automorphisms. It introduces the polynomials and , connects their vanishing to axis structure, and derives a suite of explicit, componentwise identities that govern products across and , including elementary proofs of key identities. The main results establish when is a derivation, and present two central, elementary proofs of a major -weight identity for half-axes under various conditions on the mutual inner product , with a separate treatment of the exceptional case . Overall, the work provides elementary, constructive identities that reveal the algebraic structure of primitive axes of Jordan type and offer tools for analyzing axial algebras related to transposition groups and the Monster algebra.

Abstract

Throughout this paper is a commutative non-associative algebra over a field of characteristic not In addition posses a Frobenius form. We obtain detailed information about the multiplication in given two axes of type half in

Paper Structure

This paper contains 8 sections, 43 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. The polynomials $Q_c(x)$ and $P_c(x,y)$
  3. Derivations
  4. Obtaining identities in $A$
  5. The proof that $b_{\frac{1}{2}}(x_0y_0)=(b_{\frac{1}{2}}x_0)y_0+(b_{\frac{1}{2}}y_0)x_0$
  6. The proof of the identity $b_{\frac{1}{2}}(x_{\frac{1}{2}}y_{\frac{1}{2}})+(b_{\frac{1}{2}}x_{\frac{1}{2}} )y_{\frac{1}{2}} +(b_{\frac{1}{2}}y_{\frac{1}{2}} )x_{\frac{1}{2}}= \frac{1}{2}(b,y_{\frac{1}{2}} )x_{\frac{1}{2}} +\frac{1}{2}(b,x_{\frac{1}{2}} )y_{\frac{1}{2}}+(a,x_{\frac{1}{2}}y_{\frac{1}{2}})b_{\frac{1}{2}}$
  7. The proof of the identity $(b_{\frac{1}{2}}(x_{\frac{1}{2}}y_0))_0-(b_{\frac{1}{2}}x_{\frac{1}{2}})y_0+((b_{\frac{1}{2}}y_0)x_{\frac{1}{2}})_0=0.$
  8. The case $(a,b)=1$

Key Result

Theorem 1.2

Let $a, b$ be axes. Then the following are equivalent

Theorems & Definitions (84)

  • Theorem 1.2: Theorem \ref{['der']}
  • Theorem 1.3: Theorem \ref{['0012']}
  • Theorem 1.4: Theorem \ref{['SI']}
  • Theorem 1.5: Theorems \ref{['121212']} and \ref{['thm120']} respectively
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2: Serss' Lemma
  • proof
  • Lemma 2.3
  • ...and 74 more