Lie Triple Systems Research Articles

Cohomology and homotopy of Lie triple systems

In this paper, first we give the controlling algebra of Lie triple systems. In particular, the cohomology of Lie triple systems can be characterized by the controlling algebra. Then using controlling algebras, we introduce the notions of homotopy Nambu algebras and homotopy Lie triple systems. We show that 2-term homotopy Lie triple systems is equivalent to Lie triple 2-systems, and the latter is the categorification of a Lie triple system. Finally we study skeletal and strict Lie triple 2-systems. We show that skeletal Lie triple 2-systems can be classified the third cohomology group, and strict Lie triple 2-systems are equivalent to crossed modules of Lie triple systems.

The role of derivations in the nilpotence and semisimplicity of Lie triple systems

We explore the role derivations of a Lie triple system play in the nilpotence and semisimplicity of T. In particular analogous to two theorems of Jacobson for Lie algebras, (by giving a sufficient condition for the nilpotence of T in terms of the existence of an invertible derivation, and in terms of the existence of certain automorphisms), are stated. The connections between the Lie algebra of derivations of and the semisimplicity of T are also studied.

Cohomology and Deformations of Compatible Lie Triple Systems

Cohomology and Deformations of Compatible Lie Triple Systems

Automorphisms of extensions of Lie-Yamaguti algebras and inducibility problem

Lie-Yamaguti algebras generalize both the notions of Lie algebras and Lie triple systems. In this paper, we consider the inducibility problem for automorphisms of Lie-Yamaguti algebra extensions. More precisely, given an abelian extension [Display omitted] of a Lie-Yamaguti algebra L, we are interested in finding the pairs (ϕ,ψ)∈Aut(V)×Aut(L), which are inducible by an automorphism in Aut(L˜). We connect the inducibility problem to the (2,3)-cohomology of Lie-Yamaguti algebra. In particular, we show that the obstruction for a pair of automorphisms in Aut(V)×Aut(L) to be inducible lies in the (2,3)-cohomology group H(2,3)(L,V). We develop the Wells exact sequence for Lie-Yamaguti algebra extensions, which relates the space of derivations, automorphism groups, and (2,3)-cohomology groups of Lie-Yamaguti algebras. As an application, we describe certain automorphism groups of semi-direct product Lie-Yamaguti algebras. In a sequel, we apply our results to discuss inducibility problem for nilpotent Lie-Yamaguti algebras of index 2. We give examples of infinite families of such nilpotent Lie-Yamaguti algebras and characterize the inducible pairs of automorphisms for extensions arising from these examples. Finally, we write an algorithm to find out all the inducible pairs of automorphisms for extensions arising from nilpotent Lie-Yamaguti algebras of index 2.

Maurer-Cartan characterizations and cohomologies of crossed homomorphisms on Lie triple systems

In this paper, first we give the notion of a crossed homomorphism on a Lie triple system with respect to an action on another Lie triple system. We also construct an -algebra using the higher derived brackets, whose Maurer-Cartan elements are crossed homomorphisms on Lie triple systems. Consequently, we obtain the twisted -algebra that controls deformations of a given crossed homomorphism on Lie triple systems. Next we construct a cohomology theory for a crossed homomorphism on Lie triple systems and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Moreover we consider the relationship between cohomology groups of crossed homomorphisms on Lie algebras and associated Lie triple systems.

Cohomologies and deformations of O-operators on Lie triple systems

In this paper, first, we provide a graded Lie algebra whose Maurer–Cartan elements characterize Lie triple system structures. Then, we use it to study cohomology and deformations of O-operators on Lie triple systems by constructing a Lie 3-algebra whose Maurer–Cartan elements are O-operators. Furthermore, 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. Therefore, we consider infinitesimal and formal deformations of O-operators from a cohomological viewpoint. Moreover, we provide relationships between O-operators on Lie algebras and associated Lie triple systems.

Cohomologies of Rota-Baxter Lie triple systems and applications

The purpose of the present paper is to investigate cohomologies of Rota-Baxter Lie triple systems and applications. First, we consider representations of Rota-Baxter Lie triple systems. Furthermore, we study cohomologies of Rota-Baxter operators and Rota-Baxter Lie triple systems. Abelian extensions of Rota-Baxter Lie triple systems are discussed using cohomology groups. Finally, we study -extensions of Rota-Baxter Lie triple systems.

Cohomologies and deformations of Lie triple systems with derivations

The aim of this paper is to study cohomological theory of Lie triple systems with derivations. We introduce the notion of a LtsDer pair and its representation. We also introduce the matched pair of LtsDer pairs. Moreover, we characterize the relation between the cohomologies of a LtsDer pair and its associated LeibDer pair. The deformation and [Formula: see text]-operator of LtsDer pairs are also discussed. We show that the representation of the given LtsDer pair becomes a new LtsDer pair and the former LtsDer pair becomes the representation of the new LtsDer pair, naturally to investigate cohomologies of the new LtsDer pair. Finally, we consider abelian extensions of LtsDer pairs.

Graded systems of quotients and Martindale-like quotients of graded Lie triple systems

The paper is devoted to introducing graded systems of quotients of graded Lie triple systems, graded weak systems of quotients of graded Lie triple systems and graded systems of Martindale-like quotients of graded Lie triple systems with respect to power graded filters of sturdy graded ideals. And then we investigate certain vital properties of graded Lie triple systems, which can be lifted from a graded Lie triple system to its graded systems of quotients. Furthermore, the connections among graded systems of quotients, graded weak systems of quotients and graded systems of Martindale-like quotients are surveyed. What’s more, the relationship between graded systems of quotients of graded Lie triple systems and systems of quotients of Lie triple systems is highlighted. Besides, the relationship between graded systems of quotients of a graded Lie triple system and graded algebras of quotients of the graded Lie algebra L(T) (the standard embedding of a graded Lie triple system T) is considered, too.

The algebraic and geometric classification of nilpotent Lie triple systems up to dimension four

In this paper we generalize the Skjelbred–Sund method, used to classify nilpotent Lie algebras, in order to classify triple systems with non-zero annihilator. We develop this method with the purpose of classifying nilpotent Lie triple systems, obtaining from it the algebraic classification of the nilpotent Lie triple systems up to dimension four. Additionally, we obtain the geometric classification of the variety of nilpotent Lie triple systems up to dimension four.

3-Derivations and 3-Automorphisms on Lie Algebras

In this paper, first we establish the explicit relation between 3-derivations and 3- automorphisms of a Lie algebra using the differential and exponential map. More precisely, we show that the Lie algebra of 3-derivations is the Lie algebra of the Lie group of 3-automorphisms. Then we study the derivations and automorphisms of the standard embedding Lie algebra of a Lie triple system. We prove that derivations and automorphisms of a Lie triple system give rise to derivations and automorphisms of the corresponding standard embedding Lie algebra. Finally we compute the 3-derivations and 3-automorphisms of 3-dimensional real Lie algebras.

Central Extensions and Nijenhuis Operators of Hom- δ -Jordan Lie Triple Systems

In this paper, the equivalence of central extensions and H α , α V 3 T , V is proven in the study in Hom- δ -Jordan Lie triple systems. The concepts of Nijenhuis operators of Hom- δ -Jordan Lie triple systems are given. Moreover, a trivial deformation is got.

On representation theory of Lie triple systems

On representation theory of Lie triple systems

Systems of quotients and Martindale-like quotients of Jordan–Lie triple systems

This article is devoted to introducing the definition of systems of quotients of Jordan–Lie triple systems and surveying the vital properties which can be lifted from a Jordan–Lie triple system to systems of quotients of Jordan–Lie triple system. Furthermore, the notions of systems of Martindale-like quotients of a Jordan–Lie triple system with respect to a power filter of sturdy ideals are introduced and the relationship between the systems of quotients of a Jordan–Lie triple system and the systems of Martindale-like quotients of a Jordan–Lie triple system are investigated. What’s more, main propositions about algebras of quotients of δ Jordan–Lie algebras are presented and the relationship between algebras of quotients of Jordan–Lie algebras and systems of quotients of Jordan–Lie triple systems are highlighted. Besides, some connections between systems of Martindale-like quotients of a Jordan–Lie triple system T and algebras of Martindale-like quotients of its standard embedding Jordan algebra L(T) are considered, too.

Equivariant one-parameter deformations of Lie triple systems

In this article, we introduce equivariant formal deformation theory of Lie triple systems. We introduce an equivariant deformation cohomology of Lie triple systems and using this we study the equivariant formal deformation theory of Lie triple systems.

Lie triple systems and Leibniz algebras

Abstract The present paper deals with the Lie triple systems via Leibniz algebras. A perfect Lie algebra as a perfect Leibniz algebra and as a perfect Lie triple system is considered and the appropriate universal central extensions are studied. Using properties of Leibniz algebras, it is shown that the Lie triple system universal central extension is either the universal central extension of the Leibniz algebra or the universal central extension of the Lie algebra.

Тройные лиевы системы, связанные с (-1,1)-алгебрами

We introduce a Lie triple system associated with the central isotope of (− 1, 1)-algebra. The associator ideal of (−1, 1)-algebra is nilpotent if and only if the Lie triple system is nilpotent. The relationship of the constructed Lie triple system with other known Lie triple systems is discussed.

Veronese powers of operads and pure homotopy algebras

We define the $m$th Veronese power of a weight graded operad $\mathcal{P}$ to be its suboperad $\mathcal{P}^{[m]}$ generated by operations of weight $m$. It turns out that, unlike Veronese powers of associative algebras, homological properties of operads are, in general, not improved by this construction. However, under some technical conditions, Veronese powers of quadratic Koszul operads are meaningful in the context of the Koszul duality theory. Indeed, we show that in many important cases the operads $\mathcal{P}^{[m]}$ are related by Koszul duality to operads describing strongly homotopy algebras with only one nontrivial operation. Our theory has immediate applications to objects as Lie $k$-algebras and Lie triple systems. In the case of Lie $k$-algebras, we also discuss a similarly looking ungraded construction which is frequently used in the literature. We establish that the corresponding operad does not possess good homotopy properties, and that it leads to a very simple example of a non-Koszul quadratic operad for which the Ginzburg--Kapranov power series test is inconclusive.

Cohomology and 1-Parameter Formal Deformations of Hom-$$\delta $$-Lie Triple Systems

In this paper, we study the cohomology theory of Hom- $$\delta $$ -Lie triple systems. We develop the 1-parameter formal deformation theory of Hom- $$\delta $$ -Lie triple systems, and prove that it is governed by the cohomology group.

On split Lie color triple systems

Abstract In order to begin an approach to the structure of arbitrary Lie color triple systems, (with no restrictions neither on the dimension nor on the base field), we introduce the class of split Lie color triple systems as the natural generalization of split Lie triple systems. By developing techniques of connections of roots for this kind of triple systems, we show that any of such Lie color triple systems T with a symmetric root system is of the form T = U + ∑[α]∈Λ1/∼ I[α] with U a subspace of T0 and any I[α] a well described (graded) ideal of T, satisfying {I[α], T, I[β]} = 0 if [α] ≠ [β]. Under certain conditions, in the case of T being of maximal length, the simplicity of the triple system is characterized.