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Pre-Lie Structures for Semisimple Lie Algebras

Xerxes D. Arsiwalla, Fernando Olivie Méndez Méndez

Abstract

We address the problem of admissibility of pre-Lie structures associated with a given Lie algebra, particularly, semisimple Lie algebras over ${\mathbb C}$. Such structures are collectively referred to as Lie-admissible algebras, which are a class of nonassociative algebras such that the commutator bracket over these algebras satisfies the Jacobi identity. Among the five classes of nonassociative Lie-admissible algebras, left-symmetric algebras (LSAs) and right-symmetric algebras (RSAs), are known to be non-admissible by semisimple Lie algebras of finite dimension $n \geq 3$. Here, we examine the remaining classes starting with those corresponding to the subgroup generated by permutations of order 2: $(1 \; 3)$. These appear in the literature as anti-flexible algebras (AFAs). We discuss properties of AFAs and provide examples of finite-dimensional representations. AFAs geometrically correspond to richer structures than the flat torsion-free affine connections associated with left-symmetric algebras (LSAs) or right-symmetric algebras (RSAs). We compute Lie-admissibility criteria for AFAs and determine a few simple solution classes. Not surprisingly, solvable Lie algebras admit AFAs. Concerning semisimple ones, we report an explicit counterexample demonstrating an AFA admissible by ${\mathfrak sl(2, \, {\mathbb C})}$. We then discuss the remaining two classes of nonassociative Lie-admissible algebras, the $A_3$-associative and $S_3$-associative types. Finally, we prove that $S_3$-associative algebras are universal pre-Lie structures for any Lie algebra over ${\mathbb C}$, including semisimple ones.

Pre-Lie Structures for Semisimple Lie Algebras

Abstract

We address the problem of admissibility of pre-Lie structures associated with a given Lie algebra, particularly, semisimple Lie algebras over . Such structures are collectively referred to as Lie-admissible algebras, which are a class of nonassociative algebras such that the commutator bracket over these algebras satisfies the Jacobi identity. Among the five classes of nonassociative Lie-admissible algebras, left-symmetric algebras (LSAs) and right-symmetric algebras (RSAs), are known to be non-admissible by semisimple Lie algebras of finite dimension . Here, we examine the remaining classes starting with those corresponding to the subgroup generated by permutations of order 2: . These appear in the literature as anti-flexible algebras (AFAs). We discuss properties of AFAs and provide examples of finite-dimensional representations. AFAs geometrically correspond to richer structures than the flat torsion-free affine connections associated with left-symmetric algebras (LSAs) or right-symmetric algebras (RSAs). We compute Lie-admissibility criteria for AFAs and determine a few simple solution classes. Not surprisingly, solvable Lie algebras admit AFAs. Concerning semisimple ones, we report an explicit counterexample demonstrating an AFA admissible by . We then discuss the remaining two classes of nonassociative Lie-admissible algebras, the -associative and -associative types. Finally, we prove that -associative algebras are universal pre-Lie structures for any Lie algebra over , including semisimple ones.
Paper Structure (13 sections, 6 theorems, 85 equations)

This paper contains 13 sections, 6 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Anti-Flexible Algebras (AFAs)
  3. Properties
  4. Geometric Interpretation
  5. Examples
  6. Lie-Admissibility and Solution Classes for AFAs
  7. Determining Lie-Admissibility
  8. Computing AFAs
  9. $A_3$-Associative and $S_3$-Associative Algebras
  10. Semisimple Lie Algebras and Pre-Lie Structures
  11. Semisimple Lie Algebras and Admissibility of AFAs
  12. Semisimple Lie Algebras and $A_3$ / $S_3$ Associativity
  13. Conclusions and Discussion

Key Result

Proposition 2.1

If $({\cal A}, \, \cdot)$ is an algebra that satisfies eq. eq:associator_sigma with respect to the permutation $\sigma$, and we define the product then, the algebra $({\cal A},\, \circ)$ satisfies eq. eq:associator_sigma with respect to the permutation $\tau = (1 \space 3) \, \sigma \, (1 \space 3)$, i.e., with respect to the permutation resulting from the conjugation of $\sigma$ by $(1 \space 3)

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 20 more