Generalized Derivations of Lie triple systems

In this paper, we present some basic properties concerning the derivation algebra ${\rm Der}(T)$, the quasiderivation algebra ${\rm QDer}(T)$ and the generalized derivation algebra ${\rm GDer}(T)$ of a Lie triple system $T$, with the relationship ${\rm Der}(T)\subseteq {\rm QDer}(T)\subseteq {\rm GDer}(T)\subseteq {\rm End}(T)$. Furthermore, we completely determine those Lie triple systems $T$ with condition ${\rm QDer}(T)={\rm End}(T)$. We also show that the quasiderivations of $T$ can be embedded as derivations in a larger Lie triple system.


Introduction
Lie triple systems arose initially in Cartan's study of Riemannian geometry. Jacobson [8] first introduced them in connection with problems from Jordan theory and quantum mechanics, viewing Lie triple systems as subspaces of Lie algebras that are closely related to the ternary product. Lister gave the structure theory of Lie triple systems of characteristic 0 in [14]. Hopkin introduced the concepts of nilpotent ideals and the nil-radical of Lie triple systems, she successfully generalized Engel's theorem to Lie triple systems in characteristic zero [7]. More recently, Lie triple systems have been connected with the study of the Yang-Baxter equations [10].
As is well known, derivation and generalized derivation algebras are very important subjects in the research of Lie algebras. In the study of Levi factors in derivation algebras of nilpotent Lie algebras, the generalized derivations, quasiderivations, centroids and quasicentroids play key roles [1]. In [15], Melville dealt particularly with the centroids of nilpotent Lie algebras. The most important and systematic research on the generalized derivation algebras of a Lie algebra and their subalgebras was due to Leger and Luks. In [11], some nice properties of the quasiderivation algebras and of the centroids have been obtained. In particular, they investigated the structure of the generalized derivation algebras and characterized the Lie algebras satisfying certain conditions. Meanwhile, they also pointed that there exist some connections between quasiderivations and cohomology of Lie algebras. For the generalized derivation algebras of more general nonassociative algebras, the readers will be referred to the papers [2-6, 9, 15, 18].
In this paper, we generalize some beautiful results in [11] to Lie triple system. In particular, we seek to understand the structure of the generalized derivation algebras of a Lie triple system or conversely, we want to characterize the Lie triple systems for which the generalized derivation algebras or their Lie subalgebras satisfy some special conditions. This paper is organized as follows. Section 2 contains some elementary observations about generalized derivations, quasiderivations, centroids and quasicentroids, some of which are technical results to be used in the sequel. In Section 3, we characterize completely those Lie triple systems T for which QDer.T / D End.T /. Such

Generalized derivation algebras and their subalgebras
First, we give some basic properties of center derivation algebra, quasiderivation algebra and the generalized derivation algebra of a Lie triple system. Proposition 3.1. Let T be a Lie triple system. Then the following statements hold: (1) GDer.T /; QGer.T / and C.T / are subalgebras of End.T /.
We define .T˝T˝T / C WD hx˝y˝zCy˝x˝z j x; y; z 2 T i and .T˝T˝T / WD hx˝y˝z y˝x˝z j x; y; z 2 T i, then both .T˝T˝T / C and .T˝T˝T / are subspaces of T˝T˝T . It is easy to check that  .x˝y˝z Cy˝x˝z/ in V for some x; y; z 2 T . A direct computation shows that all elements in .T˝T˝T / C are obtainable by repeated application of elements of End.T / to P .x˝y˝z C y˝x˝z/ and formation of linear combinations. Hence V is .T˝T˝T / C itself. Thus, .T˝T˝T / C as an End.T /-module is irreducible. On the other hand, the converse of Theorem 4.3 is also valid. One can prove the following theorem.

The quasiderivations of Lie triple systems
In this section, we will prove that the quasiderivations of T can be embedded as derivations in a larger Lie triple system and obtain a direct sum decomposition of Der(T ) when the center Z.T / is equal to zero. In this section, t is an indeterminate and we consider the linear space FOEt =t 4 . For convenience, let t i denote the congruence class of t i in FOEt=t 4 .

Conclusion
In fact, definitions of GDer.T /, QDer.T /, QC.T / and C.T / of a Lie triple system T could be extended to n-Lie algebras (n 3) and some other types of algebras such as superalgebras, color algebras, non-commutative algebras and Hom-type algebras. Take an n-Lie algebra L for example, we could define GDer.L/ as bellow. Let .L; OE ; ; / be a n-Lie algebra. An endomorphism D 1 2 End.L/ is said to be a generalized derivation on . It is easy to verify that GDer.L/ is a Lie algebra called a generalized derivation algebra.
Other definitions on QDer.L/, QC.L/ and C.L/ of L could be presented in a similar way. The discussion in Section 2, 3, 4 in this paper could probably be carried to the case of n-Lie algebra L. We could take the same or similar discussions about some other types of algebras mentioned above. Some of the results have been generalized in recent years and the case of other algebras could be left to the authors.