No abstract

We construct a method to obtain the algebraic classification of Poisson algebras defined on a commutative associative algebra, and we apply it to obtain the classification of the 3-dimensional Poisson algebras. In addition, we study the geometric classification, the graph of degenerations and the closures of the orbits of the variety of 3-dimensional Poisson algebras. Finally, we also study the algebraic classification of the Poisson algebras defined on a commutative associative null-filiform or filiform algebra and, to enrich this classification, we study the degenerations between these particular Poisson algebras.

Using the algebraic classification of all 2-dimensional algebras, we give the algebraic classification of all 2-dimensional rigid, conservative and terminal algebras over an algebraically closed field of characteristic 0. We have the geometric classification of the variety of 2-dimensional terminal algebras, and based on the geometric classification of these algebras we formulate some open problems.

We give the classification of all n-dimensional anticommutative complex algebras with (n − 3)dimensional annihilator. Namely, we describe all central extensions of all 3-dimensional anticommutative complex algebras.

The algebraic and geometric classification of all complex 3-dimensional transposed Poisson algebras is obtained. Also we discuss special 3-dimensional transposed Poisson algebras.