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A generalization of the first Tits construction

Thomas Moran, Susanne Pumpluen

Abstract

Let $F$ be a field of characteristic not 2 or 3. The first Tits construction is a well-known tripling process to construct separable cubic Jordan algebras, especially Albert algebras. We generalize the first Tits construction by choosing the scalar employed in the tripling process outside of the base field. This yields a new family of nonassociative unital algebras which carry a cubic map, and maps that can be viewed as generalized adjoint and generalized trace maps. These maps display properties often similar to the ones in the classical setup. In particular, the cubic norm map permits some kind of weak Jordan composition law.

A generalization of the first Tits construction

Abstract

Let be a field of characteristic not 2 or 3. The first Tits construction is a well-known tripling process to construct separable cubic Jordan algebras, especially Albert algebras. We generalize the first Tits construction by choosing the scalar employed in the tripling process outside of the base field. This yields a new family of nonassociative unital algebras which carry a cubic map, and maps that can be viewed as generalized adjoint and generalized trace maps. These maps display properties often similar to the ones in the classical setup. In particular, the cubic norm map permits some kind of weak Jordan composition law.
Paper Structure (10 sections, 16 theorems, 74 equations)

This paper contains 10 sections, 16 theorems, 74 equations.

Table of Contents

  1. Preliminaries
  2. Nonassociative algebras
  3. Cubic maps
  4. Algebras of degree three
  5. The first Tits construction
  6. The generalised first Tits construction
  7. The algebra $J(A, \mu)$
  8. The algebras $J_{lr}(A, \mu)$, $J_{rl}(A,\mu)$ and $J_{rr}(A,\mu)$
  9. Some more identities
  10. The nine-dimensional nonassociative algebras $J(K,\mu)$

Key Result

Lemma 1

Let $A$ be an associative algebra over $F$ such that $A^{+}$ is a division algebra. Then $A$ is a divison algebra.

Theorems & Definitions (33)

  • Lemma 1
  • proof
  • Remark 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Proposition 5
  • proof
  • Theorem 6
  • ...and 23 more