Noncommutative Leibniz–Poisson Algebras

Abstract

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).

Key Words:

• Associative algebras
• Hochschild cohomology
• Leibniz algebras
• Leibniz cohomology
• Poisson algebras

2000 Mathematics Subject Classification:

• 17A32
• 17G60
• 17B63
• 18D50

ACKNOWLEDGMENTS

The authors express their gratitude to T. Pirashvili for giving the idea to consider these questions and for helpful suggestions. The first author was supported by MCYT, Grant BFM2003-04686-C02-02 (European FEDER support included). The second author is grateful to the University of Vigo for financial support and hospitality.

Notes

Communicated by V. Artamonov.