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Jacobson identities for post-Lie algebras in positive characteristic

Quentin Ehret, Nicolas Gilliers

Abstract

Let $p$ be a prime number. Given a restricted Lie algebra over a field of characteristic $p$ and a post-Lie operation over it, we prove the Jacobson identities for a $p$-structure built from the Lie bracket and the post-Lie operation, called sub-adjacent $p$-structure. Furthermore, we give sufficient conditions for the sub-adjacent Lie algebra to be restricted if equipped with this sub-adjacent $p$-structure. This construction is ''axiomatized'' by introducing the notion of restricted post-Lie algebras, and we work out several examples.

Jacobson identities for post-Lie algebras in positive characteristic

Abstract

Let be a prime number. Given a restricted Lie algebra over a field of characteristic and a post-Lie operation over it, we prove the Jacobson identities for a -structure built from the Lie bracket and the post-Lie operation, called sub-adjacent -structure. Furthermore, we give sufficient conditions for the sub-adjacent Lie algebra to be restricted if equipped with this sub-adjacent -structure. This construction is ''axiomatized'' by introducing the notion of restricted post-Lie algebras, and we work out several examples.

Paper Structure

This paper contains 27 sections, 18 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Restricted Lie algebras.
  4. Examples.
  5. Restricted morphisms and modules.
  6. Conjecture.
  7. Jacobson identities for Post-Lie algebras
  8. Post-Lie algebras.
  9. (Ordinary) $\mathcal{O}$-operators.
  10. Jacobson identities for post-Lie algebras.
  11. Pre-Lie (left-symmetric) Jacobson identities, see asqar.
  12. Post-Lie Jacobson identities.
  13. Proofs.
  14. Proof of Theorem \ref{['thm:main']}: first part.
  15. Proof of Theorem \ref{['thm:main']}: second part.
  16. ...and 12 more sections

Key Result

Theorem 1.2.1

Let $(L,[-,-])$ be a Lie algebra and let $(e_j)_{j\in J}$ be a basis of $L$ such that there are $f_j\in L$ satisfying $(\mathop{\mathrm{ad}}\nolimits_{e_j})^p=\mathop{\mathrm{ad}}\nolimits_{f_j}$. Then, there exists exactly one $p$-map $(\cdot)^{[p]}:L\rightarrow L$ such that $e_j^{[p]}=f_j \quad \t

Theorems & Definitions (48)

  • Theorem 1.2.1
  • Example 1.2.2
  • Remark 1.2.3
  • Definition 2.1.1
  • Example 2.1.2
  • Theorem 2.2.1
  • proof
  • Corollary 2.2.2
  • Theorem 2.2.3
  • proof
  • ...and 38 more