Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-associative Frobenius algebras of type $E_7$

Jari Desmet

Abstract

Recently, Maurice Chayet and Skip Garibaldi introduced a class of commutative non-associative algebras. In previous work, we gave an explicit description of these algebras for groups of type $G_2,F_4$ and certain forms of $E_6$ in terms of octonion and Albert algebras. In this paper, we extend this further by dealing with $E_7$ in terms of generalised Freudenthal triple systems.

Non-associative Frobenius algebras of type $E_7$

Abstract

Recently, Maurice Chayet and Skip Garibaldi introduced a class of commutative non-associative algebras. In previous work, we gave an explicit description of these algebras for groups of type and certain forms of in terms of octonion and Albert algebras. In this paper, we extend this further by dealing with in terms of generalised Freudenthal triple systems.
Paper Structure (12 sections, 21 theorems, 44 equations)

This paper contains 12 sections, 21 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation theory of absolutely simple groups and Lie algebras
  4. The Chayet-Garibaldi algebras
  5. The Albert algebras
  6. Brown algebras
  7. The algebra of type $E_7$
  8. Defining multiplications on $V$
  9. Finding multiplications
  10. Calculating parameters
  11. The automorphism group
  12. Arbitrary groups of type $E_7$

Key Result

Theorem 2.2

Let $G$ be an absolutely simple algebraic group of type $E_7$ over a field $k$ with $\mathop{\mathrm{char}}\nolimits k> 19$ 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)}\math

Theorems & Definitions (52)

  • Remark 2.1
  • Theorem 2.2: chayet2020class
  • Proposition 2.3: chayet2020class
  • Definition 2.4
  • Definition 2.5: SpringerVeldkamp
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Proposition 2.8
  • proof
  • ...and 42 more