Non-degenerate bilinear forms and left-symmetric structures
Abstract
Chu proved that a symplectic structure ω on an even-dimensional Lie algebra induces a left-symmetric structure which is defined by ω(xscriptstyleΔω y,z)=−ω(y,[x,z]). El Bourkadi and Mansouri proved that a cosymplectic structure on an odd-dimensional Lie algebra g induces the left-symmetric structure, which is defined by using a linear isomorphism from g to g* associated with the cosymplectic structure. In this paper, for a non-degenerate bilinear form ϕ on a Lie algebra g, we give a necessary and sufficient condition for the product scriptstyleΔϕ on g defined by ϕ(xscriptstyleΔϕ y,z)=−ϕ(y,[x,z]) to be a left-symmetric structure. We also prove that the left-symmetric structure scriptstyleΔϕ is complete if and only if the Lie algebra g is unimodular. Moreover, we formulate the notion of double extension for non-degenerate bilinear forms and prove that a certain class of non-degenerate bilinear forms is obtained by double extension.
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BibTeX
@misc{kato2026nondegenerate,
title = {{Non-degenerate bilinear forms and left-symmetric structures}},
author = {Naoki Kato},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33063},
eprint = {2609.33063},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33063},
}
RIS
TY - UNPB
TI - Non-degenerate bilinear forms and left-symmetric structures
AU - Naoki Kato
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33063
UR - https://arxiv.org/abs/2609.33063
AB - Chu proved that a symplectic structure ω on an even-dimensional Lie algebra induces a left-symmetric structure which is defined by ω(xscriptstyleΔ_ω y,z)=−ω(y,[x,z]). El Bourkadi and Mansouri proved that a cosymplectic structure on an odd-dimensional Lie algebra g induces the left-symmetric structure, which is defined by using a linear isomorphism from g to g^* associated with the cosymplectic structure. In this paper, for a non-degenerate bilinear form ϕ on a Lie algebra g, we give a necessary and sufficient condition for the product scriptstyleΔ_ϕ on g defined by ϕ(xscriptstyleΔ_ϕ y,z)=−ϕ(y,[x,z]) to be a left-symmetric structure. We also prove that the left-symmetric structure scriptstyleΔ_ϕ is complete if and only if the Lie algebra g is unimodular. Moreover, we formulate the notion of double extension for non-degenerate bilinear forms and prove that a certain class of non-degenerate bilinear forms is obtained by double extension.
ER -