O-operators and related structures on Leibniz algebras

Abstract

An $\mathcal{O}$-operator has been used to extend a Leibniz algebra by its representation. In this paper, we investigate several structures related to $\mathcal{O}$-operators on Leibniz algebras and introduce (dual) $\mathcal{O}$N-structures on Leibniz algebras associated to their representations. It is proved that $\mathcal{O}$-operators and dual $\mathcal{O}$N-structures generate each other under certain conditions. It is also shown that a solution of the strong Maurer-Cartan equation on the twilled Leibniz algebra gives rise to a dual $\mathcal{O}$N-structure. Finally, r – n structures, RBN-structures and $\mathcal{B}N$-structures on Leibniz algebras are thoroughly studied and their interdependent relations are also studied.

Keywords:

• $\mathcal{B}N$-structure
• (dual) $\mathcal{O}$ N-structure
• Leibniz algebra
• Maurer-Cartan equation
• $\mathcal{O}$-operator
• r – n structure
• RBN-structure

2020 Mathematics Subject Classification:

• Primary: 13D03, 16T10
• Secondary: 16S80

Acknowledgments

The work is supported by the National Natural Science Foundation of China (grant no. 12171303), Simons Foundation (grant no. 523868), the Natural Science Foundation of Zhejiang Province of China (grant no. LY19A010001) and the Science and Technology Planning Project of Zhejiang Province (2022C01118).

Additional information

Funding

The work is supported by the National Natural Science Foundation of China (grant no. 12171303), Simons Foundation (grant no. 523868), the Natural Science Foundation of Zhejiang Province of China (grant no. LY19A010001) and the Science and Technology Planning Project of Zhejiang Province (2022C01118).