Pre-symplectic left-symmetric algebras
Abstract
A pre-symplectic left-symmetric algebra 𝒜 is a left-symmetric algebra endowed with a nondegenerate skew-symmetric bilinear form ω such that all left multiplication operators are symmetric with respect to ω. In this setting, the underlying subadjacent Lie algebra (𝒜−,ω) forms a flat T-symplectic Lie algebra. This paper provides a systematic investigation into the structural properties of pre-symplectic left-symmetric algebras. In particular, we introduce a distinguished subclass termed \emph{Milnor pre-symplectic algebras}, and prove that any pre-symplectic left-symmetric algebra whose commutator ideal is nondegenerate necessarily belongs to this subclass. Next, we investigate the Levi-Civita product associated with symplectic Lie algebras. We show that this product always yields a right-symmetric algebra, and we prove that it forms a left-symmetric algebra if and only if it is associative. Furthermore, we provide a characterization of symplectic Lie algebras in terms of representations of left-symmetric algebras, and conclude by establishing a construction method for these structures known as the T*-extension. Furthermore, we develop a double extension procedure for pre-symplectic left-symmetric algebras by means of commutative associative algebras. We show that every such algebra with a degenerate commutator ideal can be reconstructed via this extension process. More generally, we show that any pre-symplectic left-symmetric algebra is either a Milnor pre-symplectic algebra or can be obtained through a finite sequence of successive double extensions starting from a Milnor pre-symplectic algebra. As a concrete application of these structural results, we provide a complete classification of pre-symplectic left-symmetric algebras of dimension less than or equal to $4$.
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BibTeX
@misc{benayadi2026presymplectic,
title = {{Pre-symplectic left-symmetric algebras}},
author = {Saïd Benayadi and Hamza El Ouali},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.08517},
eprint = {2608.08517},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.08517},
}
RIS
TY - UNPB
TI - Pre-symplectic left-symmetric algebras
AU - Saïd Benayadi
AU - Hamza El Ouali
PY - 2026
DA - 2026/10/31
PB - arXiv
DO - 10.48550/arxiv.2608.08517
UR - https://arxiv.org/abs/2608.08517
AB - A pre-symplectic left-symmetric algebra 𝒜 is a left-symmetric algebra endowed with a nondegenerate skew-symmetric bilinear form ω such that all left multiplication operators are symmetric with respect to ω. In this setting, the underlying subadjacent Lie algebra (𝒜^−,ω) forms a flat T-symplectic Lie algebra. This paper provides a systematic investigation into the structural properties of pre-symplectic left-symmetric algebras. In particular, we introduce a distinguished subclass termed \emph{Milnor pre-symplectic algebras}, and prove that any pre-symplectic left-symmetric algebra whose commutator ideal is nondegenerate necessarily belongs to this subclass. Next, we investigate the Levi-Civita product associated with symplectic Lie algebras. We show that this product always yields a right-symmetric algebra, and we prove that it forms a left-symmetric algebra if and only if it is associative. Furthermore, we provide a characterization of symplectic Lie algebras in terms of representations of left-symmetric algebras, and conclude by establishing a construction method for these structures known as the T^*-extension. Furthermore, we develop a double extension procedure for pre-symplectic left-symmetric algebras by means of commutative associative algebras. We show that every such algebra with a degenerate commutator ideal can be reconstructed via this extension process. More generally, we show that any pre-symplectic left-symmetric algebra is either a Milnor pre-symplectic algebra or can be obtained through a finite sequence of successive double extensions starting from a Milnor pre-symplectic algebra. As a concrete application of these structural results, we provide a complete classification of pre-symplectic left-symmetric algebras of dimension less than or equal to $4$.
ER -