Product and complex structures on 3-Bihom-Lie algebras

TL;DR

This paper introduces and characterizes product and complex structures on 3-Bihom-Lie algebras, providing conditions for their existence and exploring their interrelation.

Contribution

It defines new structures on 3-Bihom-Lie algebras and establishes their equivalence to certain algebraic decompositions, expanding the understanding of these algebras.

Findings

A 3-Bihom-Lie algebra admits a product structure iff it decomposes into two Bihom subalgebras.

Four special conditions lead to specific algebraic decompositions.

The paper explores the relationship between complex and product structures.

Abstract

In this paper, we first introduce the notion of a product structure on a $3$-Bihom-Lie algebra which is a Nijenhuis operator with some conditions. And we provide that a $3$-Bihom-Lie algebra has a product structure if and only if it is the direct sum of two vector spaces which are also Bihom subalgebras. Then we give four special conditions such that each of them can make the $3$-Bihom-Lie algebra has a special decomposition. Similarly, we introduce the definition of a complex structure on a $3$-Bihom-Lie algebra and there are also four types special complex structures. Finally, we show that the relation between a complex structure and a product structure.