Product and Complex Structures on 3-Bihom-Lie Algebras

Abstract

In this paper, we introduce the notion of a product structure on a 3-Bihom-Lie algebra, which is a Nijenhuis operator with some conditions. We prove 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 under each of which a 3-Bihom-Lie algebra has a special decomposition. Similarly, we introduce a complex structure on a 3-Bihom-Lie algebra and there are also four types of special complex structures. Finally, we establish the relation between a complex structure and a product structure.