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Non-associative Frobenius algebras of type $G_2$ and $F_4$

Jari Desmet

TL;DR

The paper provides explicit, algebraically rich descriptions of the Chayet–Garibaldi non-associative Frobenius algebras for the groups of type $G_2$ and $F_4$, building these descriptions from octonion and Albert algebras. It proves $A(\mathfrak g_2)\cong \mathrm{S}^2 W$ with a concrete $\star$-product and identifies two invariant multiplications on the $27$-dimensional representation $V=V(2\omega_1)$; analogously, it embeds $A(\mathfrak f_4)$ into $\mathrm{S}^2 W$ for the 26-dimensional Albert representation and constructs two invariant multiplications on the $324$-dimensional $V=V(2\omega_4)$ with explicit formulas. The automorphism-group result for $G_2$ exhibits rigidity, showing $\mathrm{Aut}(A(\mathfrak g_2)) \cong G_2$, aligning with the observed behavior for $F_4$ and higher types, and thereby completing the symmetry analysis across these cases. The work yields precise invariant products and parameter computations, enabling a full, explicit description of the algebras and their symmetry groups in characteristic $0$ or characteristic $\ge h+2$ (with the appropriate Coxeter data).

Abstract

Very recently, Maurice Chayet and Skip Garibaldi have introduced a class of commutative non-associative algebras, for each simple linear algebraic group over an arbitrary field (with some minor restriction on the characteristic). We give an explicit description of these algebras for groups of type $G_2$ and $F_4$ in terms of the octonion algebras and the Albert algebras, respectively. As a byproduct, we determine all possible invariant commutative algebra products on the representation with highest weight $2ω_1$ for $G_2$ and on the representation with highest weight $2ω_4$ for $F_4$. It had already been observed by Chayet and Garibaldi that the automorphism group for the algebras for type $F_4$ is equal to the group of type $F_4$ itself. Using our new description, we are able to show that the same result holds for type $G_2$.

Non-associative Frobenius algebras of type $G_2$ and $F_4$

TL;DR

The paper provides explicit, algebraically rich descriptions of the Chayet–Garibaldi non-associative Frobenius algebras for the groups of type and , building these descriptions from octonion and Albert algebras. It proves with a concrete -product and identifies two invariant multiplications on the -dimensional representation ; analogously, it embeds into for the 26-dimensional Albert representation and constructs two invariant multiplications on the -dimensional with explicit formulas. The automorphism-group result for exhibits rigidity, showing , aligning with the observed behavior for and higher types, and thereby completing the symmetry analysis across these cases. The work yields precise invariant products and parameter computations, enabling a full, explicit description of the algebras and their symmetry groups in characteristic or characteristic (with the appropriate Coxeter data).

Abstract

Very recently, Maurice Chayet and Skip Garibaldi have introduced a class of commutative non-associative algebras, for each simple linear algebraic group over an arbitrary field (with some minor restriction on the characteristic). We give an explicit description of these algebras for groups of type and in terms of the octonion algebras and the Albert algebras, respectively. As a byproduct, we determine all possible invariant commutative algebra products on the representation with highest weight for and on the representation with highest weight for . It had already been observed by Chayet and Garibaldi that the automorphism group for the algebras for type is equal to the group of type itself. Using our new description, we are able to show that the same result holds for type .
Paper Structure (21 sections, 47 theorems, 120 equations, 1 figure, 1 table)

This paper contains 21 sections, 47 theorems, 120 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation theory of simple algebraic groups and Lie algebras
  4. Construction of the algebras
  5. The octonion algebra
  6. The Albert algebras
  7. Standard derivations
  8. Symmetric operators and the symmetric square
  9. The algebra of type $G_2$
  10. A formula for standard derivations and their Jordan products
  11. An isomorphism to the symmetric square of the traceless octonions
  12. Defining multiplications on $V$
  13. Calculating parameters
  14. The automorphism group of type $G_2$
  15. The algebra of type $F_4$
  16. ...and 6 more sections

Key Result

Proposition 2.2

Let $\mathfrak{g}$ be a simple Lie algebra (associated to an algebraic group $G$) over $k$. Let $\pi \colon \mathfrak{g} \to \mathop{\mathrm{End}}\nolimits(V)$ be equivalent to the Weyl module $V(\lambda)$ over an algebraic closure of $k$, with $\lambda$ a dominant weight. Denote by $\delta$ half th

Figures (1)

  • Figure 1: The Fano plane mnemonic. If one follows the arrows when multiplying, then the outcome is equal to the third point on the line. Otherwise it is equal to minus the third point on the line, e.g. $e_6e_2 = e_4$.

Theorems & Definitions (105)

  • Remark 2.1
  • Proposition 2.2
  • Definition 2.3: $A(\mathfrak{g})$ as a vector space
  • Definition 2.4: The product $\diamond$
  • Definition 2.5: The bilinear form $\tau$
  • Lemma 2.6: chayet2020class
  • Proposition 2.7: chayet2020class
  • Proposition 2.8: chayet2020class
  • Definition 2.9: Octonion algebra
  • Proposition 2.10
  • ...and 95 more