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Non-associative Frobenius algebras of type $^1E_6$ with trivial Tits algebras

Jari Desmet

TL;DR

The work extends the Chayet–Garibaldi framework to inner type $E_6$ with trivial Tits algebras by modeling the $27$‑dimensional representation via the Albert algebra and constructing an explicit, commutative non‑associative algebra A(G) that realizes the group action. A key innovation is using the symmetric cube $S^3W$ and two distinct, $G$‑equivariant multiplications $igodot_1$ and $igodot_2$—one derived from cross‑product structure and the other obtained via an outer automorphism—to produce a full, explicit algebra structure on $ ext{Im}(S)$ that descends to $A(G)$. The paper derives the explicit star product $igstar$ on $A(G)$, computes its parameters, and analyzes the automorphism group, showing that the commutative case yields an extended symmetry group $G/Z(G) times C_2$ while non‑commutative cases retain the non‑outer $G$ as automorphism group. These results provide a complete E$_6$‑specific realization of Lagrangian‑like algebras associated with Albert algebras, clarifying both their internal multiplication structure and symmetry properties with potential implications for related non‑associative constructions and representation theory.

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). In a previous paper, we gave 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. In this paper, we attempt a similar approach for type $E_6$.

Non-associative Frobenius algebras of type $^1E_6$ with trivial Tits algebras

TL;DR

The work extends the Chayet–Garibaldi framework to inner type with trivial Tits algebras by modeling the ‑dimensional representation via the Albert algebra and constructing an explicit, commutative non‑associative algebra A(G) that realizes the group action. A key innovation is using the symmetric cube and two distinct, ‑equivariant multiplications and —one derived from cross‑product structure and the other obtained via an outer automorphism—to produce a full, explicit algebra structure on that descends to . The paper derives the explicit star product on , computes its parameters, and analyzes the automorphism group, showing that the commutative case yields an extended symmetry group while non‑commutative cases retain the non‑outer as automorphism group. These results provide a complete E‑specific realization of Lagrangian‑like algebras associated with Albert algebras, clarifying both their internal multiplication structure and symmetry properties with potential implications for related non‑associative constructions and representation theory.

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). In a previous paper, we gave an explicit description of these algebras for groups of type and in terms of the octonion algebras and the Albert algebras, respectively. In this paper, we attempt a similar approach for type .
Paper Structure (14 sections, 28 theorems, 51 equations, 1 table)

This paper contains 14 sections, 28 theorems, 51 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation theory of absolutely simple groups and Lie algebras
  4. Twisted forms of type $E_6$.
  5. Construction of the algebras
  6. The octonion algebra
  7. The Albert algebras
  8. Outer automorphisms on representations of type $E_6$
  9. The algebra of type $^1E_6$
  10. Defining multiplications on $V$
  11. Finding multiplication 1
  12. Finding multiplication 2
  13. Calculating parameters
  14. The automorphism group

Key Result

Theorem 2.2

Let $G$ be an absolutely simple algebraic group over a field $k$ with $\mathop{\mathrm{char}}\nolimits k\geq h+2$ or $0$, and let $\mathfrak{g} = \mathop{\mathrm{Lie}}\nolimits(G)$. Define and Then $A(G) \coloneqq (\mathop{\mathrm{Im}}\nolimits(S),\diamond)$ is a well-defined simple unital commutative non-associative algebra with counit $\varepsilon(a) \coloneqq \tfrac{1}{\dim(G)}\mathop{\mathrm

Theorems & Definitions (67)

  • Remark 2.1
  • Theorem 2.2: chayet2020class
  • Proposition 2.3: chayet2020class
  • Definition 2.4: Octonion algebra
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Definition 2.8
  • ...and 57 more