Pre-Leibniz algebras

Abstract

The notion of pre-Leibniz algebras was recently introduced in the study of Rota-Baxter operators on Leibniz algebras. In this paper, we first construct a graded Lie algebra whose Maurer-Cartan elements are pre-Leibniz algebras. Using this characterization, we define the cohomology of a pre-Leibniz algebra with coefficients in a representation. This cohomology is shown to split the Loday-Pirashvili cohomology of Leibniz algebras. As applications of our cohomology, we study formal and finite order deformations of a pre-Leibniz algebra. Finally, we define homotopy pre-Leibniz algebras and classify some special types of homotopy pre-Leibniz algebras.

KEYWORDS:

• Cohomology
• deformations
• homotopy pre-Leibniz algebras
• Maurer-Cartan elements
• pre-Leibniz algebras

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17A32
• 17A99
• 17B56
• 16S80
• 18G55

Acknowledgments

The author would like to thank Indian Institute of Technology (IIT) Kharagpur for providing the beautiful academic environment where the research has been carried out.