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Semisimplifying Frank Lie algebras

Michiel Smet

Abstract

The Frank Lie algebras are simple Lie algebras that only occur over fields of characteristic 3. These come equipped with distinguished inner derivations that make them algebras in the category $\textbf{Rep}(α_3)$. We apply the semisimplification functor to these Frank Lie algebras and obtain modular contact Lie superalgebras. We also obtain a class of simple $J$-ternary algebras whose associated Jordan algebras are not simple.

Semisimplifying Frank Lie algebras

Abstract

The Frank Lie algebras are simple Lie algebras that only occur over fields of characteristic 3. These come equipped with distinguished inner derivations that make them algebras in the category . We apply the semisimplification functor to these Frank Lie algebras and obtain modular contact Lie superalgebras. We also obtain a class of simple -ternary algebras whose associated Jordan algebras are not simple.
Paper Structure (8 sections, 7 theorems, 37 equations)

This paper contains 8 sections, 7 theorems, 37 equations.

Table of Contents

  1. Basic definitions
  2. Building blocks
  3. Modular contact Lie superalgebras
  4. Frank algebras
  5. Link between $K(1,1; \underline{n})$ and $\mathcal{F}(n)$
  6. Rep$(\alpha_3)$ and the semisimplification functor
  7. Semisimplifying the Frank algebra
  8. On J-ternary algebras

Key Result

Lemma 1.2

The Lie superalgebra $K(1,1; \underline{n})$ is isomorphic to $W(1; \underline{n}) \oplus \mathcal{O}_{(\mathrm{div})}$. Moreover, the derived algebra $K(1,1; \underline{n})^{(1)}$ is isomorphic to $W(1; \underline{n}) \oplus \mathcal{O}'$.

Theorems & Definitions (22)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 12 more