O-operators on Lie triple systems

Abstract

The purpose of this paper is to study cohomology and deformations of O-operators on Lie triple systems. We define a cohomology of an O-operator T as the Lie-Yamaguti cohomology of a certain Lie triple system induced by T with coefficients in a suitable representation. Then we consider infinitesimal and formal deformations of O-operators from cohomological viewpoint. Moreover we provide relationships between O-operators on Lie algebras and associated Lie triple systems.

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

“…ρ( x, y, z ) = [D ρ,µ (x, y), ρ(z)], (7) µ(z, w)µ(x, y) − µ(y, w)µ(x, z) − µ(x, y, z, w ) + D ρ,µ (y, z)µ(x, w) = 0, (8) µ( x, y, z , w) + µ(z, x, y, w ) = [D ρ,µ (x, y), µ(z, w)], (9) where the bilinear map D ρ,µ : ⊗ 2 g → gl(V) is given by…”

“…As stated before, a Lie triple system is a spacial case of a Lie-Yamaguti algebra, so the conclusions in the present paper can also be adapted to the Lie triple systems context. See [8] for more details about cohomologies and deformations of relative Rota-Baxter operators on Lie triple systems. However, unlike other algebras such as Lie algebras or Leibniz algebras, there does not exist a suitable graded Lie algebra whose Maurer-Cartan elements are just the Lie-Yamaguri algebra structure by now, thus we do not find a suitable L ∞ -algebra that controls the deformations of relative Rota-Baxter operators.…”

Relative Rota-Baxter operators of nonzero weights on Lie Triple Systems

“…In the sequel, we give the relationship between relative Rota-Baxter operators of weight λ and Nijenhuis operators. Recall from [6] that a Nijenhuis operator on a Lie triple system…”

“…Lie triple systems have important applications in physics, such as in quantum mechanics and numerical analysis of differential equations. For more results see [6,16,27]. People often refer to Lie algebras when considering the related problems of Lie triple systems, since Lie triple systems are closely related to Lie algebras.…”

“…In [17], the author showed the concept of Ooperators (relative Rota-Baxter operator) on Lie algebras. Then O-operators have been used to define cohomology on 3-Lie algebras, Lie triple system and Lie-Yamaguti algebras, see [6,21,23].…”