Pre-Lie Structures for Semisimple Lie Algebras
Xerxes D. Arsiwalla, Fernando Olivie M\'endez M\'endez

TL;DR
This paper investigates the existence of pre-Lie structures in semisimple Lie algebras, focusing on classes like anti-flexible and $S_3$-associative algebras, and establishes that $S_3$-associative algebras universally admit pre-Lie structures.
Contribution
It introduces new classes of Lie-admissible algebras, especially $S_3$-associative algebras, and proves their universal pre-Lie structure property for all Lie algebras over ${f C}$.
Findings
Counterexample of an anti-flexible algebra for ${\mathfrak sl(2, \mathbb C)}$
$S_3$-associative algebras are universal pre-Lie structures
Solvable Lie algebras admit anti-flexible 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…
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Taxonomy
TopicsAdvanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models
