Strict lifts and cohomological obstructions in symplectic Lie--Rinehart--Jacobi prequantization
Abstract
Let (G,ρ,ω) be a symplectic Lie--Rinehart--Jacobi algebra over a commutative algebra A, equivalently a Lie--Rinehart algebra (G,ρ̅) endowed with a $1$-cocycle χ and a nondegenerate $2$-form closed for the twisted differential dρ=dρ̅+χ∧. Starting from the standard extension by the $2$-cocycle ω, we study the strict lifting problem for its canonical $1$-form η in the LRJ setting. We prove that the strict symmetries of η are precisely the elements Xf+fe and that f↦ Xf+fe identifies the Jacobi Lie algebra (A,{ ,}) with this Lie algebra of strict symmetries. For locally Hamiltonian elements, the obstruction to a strict lift is the class [ixω]∈ H1ρ(G,A), yielding the exact sequence 0→ham→hamloc→ H1ρ(G,A)→0. We also relate Okassa's symplectic curl to the variation of dη and establish a conformal triviality criterion. In the smooth case the framework reduces to locally conformally symplectic geometry, and on a closed surface of genus g≥2 we obtain structures that are not conformally trivial and whose obstruction space has dimension 2g−2.
Related
- Obstructions in Symplectic Embedding Problems for Four-Dimensional Convex Toric Domains
- Multiplicity of brake Reeb orbits on real prequantization bundles
- Contractible Closed Reeb Orbits on Prequantization Bundles under Weak Index Conditions
- Anchor Radicals and Metric Rigidity in n-Lie--Rinehart Algebras
- Categorical Lie-Rinehart modules and Shen-Larsson functors
BibTeX
@misc{bossoto2026strict,
title = {{Strict lifts and cohomological obstructions in symplectic Lie--Rinehart--Jacobi prequantization}},
author = {Basile Guy Richard Bossoto and Servais Cyr Gatsé and Olivier Mabiala Mikanou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32370},
eprint = {2609.32370},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32370},
}
RIS
TY - UNPB
TI - Strict lifts and cohomological obstructions in symplectic Lie--Rinehart--Jacobi prequantization
AU - Basile Guy Richard Bossoto
AU - Servais Cyr Gatsé
AU - Olivier Mabiala Mikanou
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32370
UR - https://arxiv.org/abs/2609.32370
AB - Let (G,ρ,ω) be a symplectic Lie--Rinehart--Jacobi algebra over a commutative algebra A, equivalently a Lie--Rinehart algebra (G,ρ̅) endowed with a $1$-cocycle χ and a nondegenerate $2$-form closed for the twisted differential d_ρ=d_ρ̅+χ∧. Starting from the standard extension by the $2$-cocycle ω, we study the strict lifting problem for its canonical $1$-form η in the LRJ setting. We prove that the strict symmetries of η are precisely the elements X_f+fe and that f↦ X_f+fe identifies the Jacobi Lie algebra (A,{ ,}) with this Lie algebra of strict symmetries. For locally Hamiltonian elements, the obstruction to a strict lift is the class [i_xω]∈ H^1_ρ(G,A), yielding the exact sequence 0→ham→hamloc→ H^1_ρ(G,A)→0. We also relate Okassa's symplectic curl to the variation of dη and establish a conformal triviality criterion. In the smooth case the framework reduces to locally conformally symplectic geometry, and on a closed surface of genus g≥2 we obtain structures that are not conformally trivial and whose obstruction space has dimension 2g−2.
ER -