Product and complex structures on 3-Bihom-Lie algebras

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.