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TY - JOUR
T1 - Abelian subalgebras and ideals of maximal dimension in Poisson algebras
AU - Ouaridi, Amir
AU - Navarro, Rosa
AU - Towers, David
PY - 2024/12/15
Y1 - 2024/12/15
N2 - This paper studies the abelian subalgebras and ideals of maximal dimension ofPoisson algebras P of dimension n. We introduce the invariants α and β for Poisson algebras, which correspond to the dimension of an abelian subalgebra and ideal of maximal dimension, respectively. We prove that these invariants coincide if α(P) = n−1. We characterize the Poisson algebras with α(P) = n − 2 over arbitrary fields. In particular, we characterize Lie algebras L with α(L) = n − 2. We also show that α(P) = n − 2 for nilpotent Poisson algebras implies β(P) = n − 2. Finally, we study these invariants for various distinguished Poisson algebras, providing us with several examples and counterexamples.
AB - This paper studies the abelian subalgebras and ideals of maximal dimension ofPoisson algebras P of dimension n. We introduce the invariants α and β for Poisson algebras, which correspond to the dimension of an abelian subalgebra and ideal of maximal dimension, respectively. We prove that these invariants coincide if α(P) = n−1. We characterize the Poisson algebras with α(P) = n − 2 over arbitrary fields. In particular, we characterize Lie algebras L with α(L) = n − 2. We also show that α(P) = n − 2 for nilpotent Poisson algebras implies β(P) = n − 2. Finally, we study these invariants for various distinguished Poisson algebras, providing us with several examples and counterexamples.
U2 - 10.1016/j.jalgebra.2024.07.032
DO - 10.1016/j.jalgebra.2024.07.032
M3 - Journal article
VL - 660
SP - 680
EP - 704
JO - Journal of Algebra
JF - Journal of Algebra
SN - 0021-8693
ER -