PreprintCC BY 4.0arXiv
On the Isotropy Groups of Locally Finite Derivations of Laurent Polynomial Algebras
Abstract
Let An=k[x1±1,…,xn±1] be a Laurent polynomial algebra over a field k of characteristic zero. We determine explicitly the automorphisms commuting with each locally finite derivation Dμ=∑i=1nμi xi∂/(∂ xi), μ=(μ1,…,μn)T∈Bbbkn. These are precisely the automorphisms ρ(xi)=λi xmi, where λi∈k∗, the exponent vectors mi are the columns of a matrix Mρ∈GLn(ℤ), and μTMρ=μT. We also give an explicit parametrization in dimension two and relate these automorphisms to symmetries of diagonal flows on algebraic tori.
Related
BibTeX
@misc{porto2026isotropy,
title = {{On the Isotropy Groups of Locally Finite Derivations of Laurent Polynomial Algebras}},
author = {Anderson Porto and Marcelo Veloso},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12777},
eprint = {2610.12777},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12777},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Isotropy Groups of Locally Finite Derivations of Laurent Polynomial Algebras
AU - Anderson Porto
AU - Marcelo Veloso
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12777
UR - https://arxiv.org/abs/2610.12777
AB - Let A_n=k[x_1^±1,…,x_n^±1] be a Laurent polynomial algebra over a field k of characteristic zero. We determine explicitly the automorphisms commuting with each locally finite derivation D_μ=∑_i=1^nμ_i x_i∂/(∂ x_i), μ=(μ_1,…,μ_n)^T∈Bbbk^n. These are precisely the automorphisms ρ(x_i)=λ_i x^m_i, where λ_i∈k^∗, the exponent vectors m_i are the columns of a matrix M_ρ∈GL_n(ℤ), and μ^TM_ρ=μ^T. We also give an explicit parametrization in dimension two and relate these automorphisms to symmetries of diagonal flows on algebraic tori.
ER -