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J-ternary algebras, structurable algebras, and Lie superalgebras

Isabel Cunha, Alberto Elduque

Abstract

A Lie superalgebra is attached to any finite-dimensional J-ternary algebra over an algebraically closed field of characteristic 3, using a process of semisimplification via tensor categories. Some of the exceptional simple Lie algebras, specific of this characteristic, are obtained in this way from J-ternary algebras coming from structurable algebras and, in particular, a new magic square of Lie superalgebras is constructed, with entries depending on a pair of composition algebras.

J-ternary algebras, structurable algebras, and Lie superalgebras

Abstract

A Lie superalgebra is attached to any finite-dimensional J-ternary algebra over an algebraically closed field of characteristic 3, using a process of semisimplification via tensor categories. Some of the exceptional simple Lie algebras, specific of this characteristic, are obtained in this way from J-ternary algebras coming from structurable algebras and, in particular, a new magic square of Lie superalgebras is constructed, with entries depending on a pair of composition algebras.

Paper Structure

This paper contains 12 sections, 23 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. J-ternary algebras and short SL2-structures on Lie algebras
  3. Structurable algebras and J-ternary algebras
  4. From algebras to superalgebras via tensor categories
  5. From J-ternary algebras to Lie superalgebras in characteristic 3
  6. Lie superalgebras from prototypical J-ternary algebras
  7. Lie superalgebras from simple structurable algebras
  8. A magic square of Lie superalgebras
  9. The inner structure Lie algebra
  10. The Lie algebra S(A,A)
  11. Structure of A as a module for S(A,A)
  12. A magic square of Lie superalgebras

Key Result

Lemma 2.4

Let $U$ be a triple system satisfying equation eq:FK1, with $\epsilon\in\{\pm1\}$, and define the endomorphisms $S(x,y)$ and $T(x,y)\in\mathop{\mathrm{\mathrm{End}}}\nolimits_\mathbb{F}(U)$ by Then for any $u,v\in U$, $S(u,v)$ is a derivation of the triple system $(U,xyz)$, while $T(u,v)$ satisfies for any $x,y,z\in U$. As a consequence, the following equations hold:

Theorems & Definitions (54)

  • Definition 2.1: Hein_LieHein_degree>2
  • Definition 2.2: Allison_JTernaryAllisonBenkartGao
  • Definition 2.3: YamOno
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Theorem 2.8
  • ...and 44 more