Degenerations of Poisson algebras
Hani Abdelwahab, Amir Fernández Ouaridi, Cándido Martı́n González
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 [Formula: see text]-dimensional Poisson algebras. In addition, we study the geometric classification, the graph of degenerations and the closures of the orbits of the variety of [Formula: see text]-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.
Topics & Concepts
MathematicsAssociative propertyPoisson distributionCommutative propertyPoisson algebraPure mathematicsAssociative algebraAlgebra representationVariety (cybernetics)Algebraic varietyAlgebra over a fieldSubalgebraAlgebraic structureAlgebraic geometryAlgebraic numberPoisson bracketDivision algebraLie algebraMathematical analysisStatisticsAdvanced Topics in AlgebraHomotopy and Cohomology in Algebraic TopologyAlgebraic structures and combinatorial models