Nijenhuis operators, product structures and complex structures on Lie–Yamaguti algebras

Abstract

In this paper, first, we study linear deformations of a Lie–Yamaguti algebra and introduce the notion of a Nijenhuis operator. Then we introduce the notion of a product structure on a Lie–Yamaguti algebra, which is a Nijenhuis operator [Formula: see text] satisfying [Formula: see text]. There is a product structure on a Lie–Yamaguti algebra if and only if the Lie–Yamaguti algebra is the direct sum of two subalgebras (as vector spaces). There are some special product structures, each of which corresponds to a special decomposition of the original Lie–Yamaguti algebra. In the same way, we introduce the notion of a complex structure on a Lie–Yamaguti algebra. Finally, we add a compatibility condition between a product structure and a complex structure to introduce the notion of a complex product structure on a Lie–Yamaguti algebra.

Cohomology and deformations of Relative Rota-Baxter operators on Lie-Yamaguti algebras

“…We may prove Theorem 3.4 by checking that (g; ̺, ̟) satisfies the conditions in Definition 2.5, but here we will prove it by another way. In order to do this, we should go back some notions in [25]. Recall that a Nijenhuis operator on a Lie-Yamaguti algebra (g, […”

“…Deformations and extensions of Lie-Yamaguti algebras were examined in [19,20,34,35]. Sheng, the first author, and Zhou analyzed product structures and complex structures on Lie-Yamaguti algebras by means of Nijenhuis operators on Lie-Yamaguti algebras in [25]. Takahashi studied modules over quandles by the mean of representations of Lie-Yamaguti algebras in [27].…”

Cohomology and relative Rota-Baxter-Nijenhuis structures on LieYRep pairs

“…Yamaguti introduced representations and established cohomology theory of Lie-Yamaguti algebras in [26,27] during from 1950's to 1960's. The first author and Sheng focused on deformations, Nijenhuis operators, and relative Rota-Baxter operators on Lie-Yamaguti algebras in [22,23]. Recently, we studied cohomology and deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras ( [30]).…”

Relative Rota-Baxter operators and symplectic structures on Lie-Yamaguti algebras

“…Lie-Yamaguti algebras were widely studied recently. In particular, irreducible Lie-Yamaguti algebras and their relations to orthogonal Lie algebras were deeply studied in [8,9,10,11]; Deformations and extensions of Lie-Yamaguti algebras were studied in [19,31]; Product structures and complex structures on Lie-Yamaguti algebras were studied in [25] using Nijenhuis operators on Lie-Yamaguti algebras. In [26], the author studied modules over quandles using representations of Lie-Yamaguti algebras.…”