Relative Rota–Baxter Leibniz algebras, their characterization and cohomology

Abstract

Recently, relative Rota–Baxter (Lie/associative) algebras are extensively studied in the literature from cohomological points of view. In this paper, we consider relative Rota–Baxter Leibniz algebras (rRB Leibniz algebras) as the object of our study. We construct an $L_{\mathrm{\infty }}$-algebra that characterizes rRB Leibniz algebras as its Maurer–Cartan elements. Then we define representations of an rRB Leibniz algebra and introduce cohomology with coefficients in a representation. As applications of cohomology, we study deformations and abelian extensions of rRB Leibniz algebras.

KEYWORDS:

• Leibniz algebras
• (relative) Rota–Baxter Leibniz algebras
• $L_{\mathrm{\infty }}$-algebras
• cohomology
• deformations

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 17A32
• 17A36
• 17B56
• 16S80

Acknowledgments

The author would like to thank the esteemed referees for their useful comments on the earlier version of the manuscript. Some parts of the work was carried out when the author was a postdoctoral fellow at IIT Kanpur. The author also thanks IIT Kharagpur for the beautiful atmosphere where the final parts of the paper has been done.

Disclosure statement

No potential conflict of interest was reported by the author(s).