On the Structure of Low-Dimensional Poisson Algebras over Arbitrary Fields

Abstract

We investigate the structure of Poisson algebras of dimensions at most three over an arbitrary field. Our approach is based on the internal structure of the associated commutative associative and Lie algebras, with particular emphasis on the associative square $P^2$, the derived Lie algebra $[P,P]$, the Lie center, the associative annihilator, idempotents, ideals and decomposability. We obtain a complete classification in dimensions one and two and give a structural classification in dimension three. In dimension two, we prove that the associative and Lie multiplications cannot be simultaneously non-zero. In dimension three, the classification is organized according to the dimension and position of the derived Lie algebra and, in the case of trivial Lie multiplication, according to the dimension of $P^2$. The arbitrary-field setting leads to phenomena which do not occur over the complex field. In particular, quadratic and cubic field extensions appear naturally in the classification, some families depend on equivalence classes of symmetric bilinear forms and on the structure of three-dimensional Lie algebras over the ground field, and characteristic $2$ gives an additional family of Poisson algebras with both multiplications non-zero. Over the complex field, the resulting classification specializes, up to changes of basis and notation, to the known classifications in dimensions at most three.