Noncommutative Leibniz-Poisson algebras

摘要

We define noncommutative Leibniz-Poisson algebras (NLP-algebras) and their dual algebras, construct free objects in the corresponding categories, and relate free objects on one element set with the set of planar binary rooted trees. We define actions, crossed modules, representations, extensions, and cohomology of NLP-algebras and study their properties; in particular we derive a long exact sequence, relating this cohomology with Hochschild and Leibniz cohomologies, and with the cohomology of algebras with bracket defined in Casas and Pirashvili (2006).