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The Tits construction for short $\mathfrak{sl}_2$-super-structures

Gonzalo Gutierrez, Marco Farinati

TL;DR

The paper extends the Tits construction to short and very short $\mathfrak{sl}_2$-structures in the super setting by introducing $\mathfrak{J}$-ternary superalgebras $(\mathfrak{J},\mathcal{M})$, and by developing a functorial replacement for inner derivations via $\mathfrak{B}^s(\mathfrak{J},\mathcal{M})$ and the TAG construction. It proves that short $\sl_2$-Lie superalgebras correspond to such ternary data with explicit bracket formulas, and it constructs the TAG functor as a left adjoint to an extended Tits functor, thereby unifying and generalizing prior non-super and super short constructions (EBCC23, S22). The work gives detailed algebraic identities, adjunction properties, and a suite of examples (including Heisenberg, $\mathrm{osp}(1|2)$, and $\sl_3$ realizations) that illustrate the framework and its capacity to generate new short $\sl_2$-superalgebras from ternary Jordan data. Overall, the results provide a cohesive, functorial mechanism to derive short $\sl_2$-Lie superalgebras from ternary Jordan superstructures, with potential applications in constructing exceptional or otherwise intricate Lie superalgebras from Jordan-like data.

Abstract

In this paper, we generalize the Tits construction for Lie superalgebras such that $\mathfrak{sl}_2$ acts by even derivations and decompose, as $\mathfrak{sl}_2$-module, into a direct sum of copies of the adjoint, the natural and the trivial representations. This construction generalizes the one provided by Elduque et al in \cite{EBCC23}, and it is possible to described the $\mathfrak{sl}_2$-Lie superstructure in terms of $\mathcal{J}$-ternary superalgebras as a super version of the defined by Allison. We extend the Tits-Kantor-Koecher construction and the Tits-Allison-Gao functor that define a short $\mathfrak{sl}_2$-Lie superalgebra from a $\mathcal{J}$-ternary superalgebra $(\mathcal{J},\mathcal{M})$. Our setting includes and generalizes both \cite{EBCC23} and Shang's \cite{S22}.

The Tits construction for short $\mathfrak{sl}_2$-super-structures

TL;DR

The paper extends the Tits construction to short and very short -structures in the super setting by introducing -ternary superalgebras , and by developing a functorial replacement for inner derivations via and the TAG construction. It proves that short -Lie superalgebras correspond to such ternary data with explicit bracket formulas, and it constructs the TAG functor as a left adjoint to an extended Tits functor, thereby unifying and generalizing prior non-super and super short constructions (EBCC23, S22). The work gives detailed algebraic identities, adjunction properties, and a suite of examples (including Heisenberg, , and realizations) that illustrate the framework and its capacity to generate new short -superalgebras from ternary Jordan data. Overall, the results provide a cohesive, functorial mechanism to derive short -Lie superalgebras from ternary Jordan superstructures, with potential applications in constructing exceptional or otherwise intricate Lie superalgebras from Jordan-like data.

Abstract

In this paper, we generalize the Tits construction for Lie superalgebras such that acts by even derivations and decompose, as -module, into a direct sum of copies of the adjoint, the natural and the trivial representations. This construction generalizes the one provided by Elduque et al in \cite{EBCC23}, and it is possible to described the -Lie superstructure in terms of -ternary superalgebras as a super version of the defined by Allison. We extend the Tits-Kantor-Koecher construction and the Tits-Allison-Gao functor that define a short -Lie superalgebra from a -ternary superalgebra . Our setting includes and generalizes both \cite{EBCC23} and Shang's \cite{S22}.

Paper Structure

This paper contains 16 sections, 25 theorems, 254 equations.

Table of Contents

  1. Preliminaries
  2. Short $\sl_2$-Lie superalgebras
  3. Main properties of the operations $\cdot,\bullet,\star, \langle-,-\rangle_{\!\!{}_\mathfrak{g}}$ and $\partial^{{}^{\mathfrak{g}}}$.
  4. $\sl_2$-Lie theoretical formulas for the operations $\cdot,\bullet,\star,\langle-,-\rangle_{\!\!{}_\mathfrak{g}}$ and $\partial^{{}^{\mathfrak{g}}}$.
  5. $\mathfrak{J}$-ternary Superstructures
  6. The Lie algebra $\mathrm{Innder}(\mathfrak{J},\mathcal{M})$ and TKK construction for $\mathfrak{J}$-ternary algebras
  7. Bracket formulas of the inner derivations
  8. $\operatorname{TKK}(\mathfrak{J},\mathcal{M})$
  9. The functors $\operatorname{TAG}$ and $\operatorname{T}$ extended to $\sl_2{\text{-\bf supSS}}$ and ${\text{\bf J-STer}}$ respectively
  10. $\operatorname{TAG}(\mathfrak{J})$: The TAG construction for super Jordan algebras
  11. $\operatorname{TAG}(\mathfrak{J},\mathcal{M})$: The TAG construction for super $\mathfrak{J}$-ternary algebras
  12. The Lie superalgebra $\mathfrak{B} ^s(\mathfrak{J},\mathcal{M})$
  13. Definition of $\operatorname{TAG}(\mathfrak{J},\mathcal{M})$
  14. The Tits functor for ternary Jordan super algebras
  15. Adjunction property of $\operatorname{T}$ and $\operatorname{TAG}$
  16. ...and 1 more sections

Key Result

Theorem 2.1

A vector superspace $\mathfrak{g}$ with isotypic decomposition eq: sl2-corta tipo 1 is a short $\sl_2$-Lie superalgebra if and only if there exist super-symmetric maps $\cdot:\mathfrak{J}\otimes\mathfrak{J} \to \mathfrak{J}$, $\bullet:\mathfrak{J}\otimes \mathcal{M} \to \mathcal{M}$, $\partial^{{}^{

Theorems & Definitions (68)

  • Definition 1
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.3
  • Definition 1.4
  • proof
  • Remark 1
  • Definition 1.5
  • Theorem 2.1
  • ...and 58 more