A proof of the Zariski closure conjecture and quasi-Frobenius exponential algebras
Abstract
Let G=exp(g) be an exponential solvable Lie group, π∈Ĝ, and Ωπ the coadjoint orbit attached to π by the orbit method. The Zariski closure conjecture asserts that the Poisson characteristic variety VA(πν) of the deformed representation coincides with the Zariski closure of Ωπ in g*. Known until now only in special situations, including the nilpotent case, it is proved here in full generality. Since the Hilbert space of π is a quantization of the symplectic manifold Ωπ, the statement says that the passage back from the representation to the orbit, a dequantization, loses nothing: the orbit is recovered, up to Zariski closure, from the algebra alone. We also characterize the quasi-Frobenius exponential Lie algebras, those whose dual vector space is a Zariski-dense coadjoint orbit, through the triviality of the annihilator of an irreducible representation.
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BibTeX
@misc{baklouti2026proof,
title = {{A proof of the Zariski closure conjecture and quasi-Frobenius exponential algebras}},
author = {Ali Baklouti},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00757},
eprint = {2611.00757},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00757},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A proof of the Zariski closure conjecture and quasi-Frobenius exponential algebras
AU - Ali Baklouti
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00757
UR - https://arxiv.org/abs/2611.00757
AB - Let G=exp(g) be an exponential solvable Lie group, π∈Ĝ, and Ω_π the coadjoint orbit attached to π by the orbit method. The Zariski closure conjecture asserts that the Poisson characteristic variety VA(π_ν) of the deformed representation coincides with the Zariski closure of Ω_π in g^*. Known until now only in special situations, including the nilpotent case, it is proved here in full generality. Since the Hilbert space of π is a quantization of the symplectic manifold Ω_π, the statement says that the passage back from the representation to the orbit, a dequantization, loses nothing: the orbit is recovered, up to Zariski closure, from the algebra alone. We also characterize the quasi-Frobenius exponential Lie algebras, those whose dual vector space is a Zariski-dense coadjoint orbit, through the triviality of the annihilator of an irreducible representation.
ER -