[Paper Review] Generalized Derivations of Lie triple systems

Jia Zhou, Liangyun Chen+1 more|arXiv (Cornell University)|Dec 25, 2014

Advanced Topics in Algebra3 citations

TL;DR

This paper generalizes the theory of derivations, quasiderivations, and generalized derivations from Lie algebras to Lie triple systems. It establishes a hierarchy of derivation algebras, characterizes Lie triple systems for which quasiderivations span the full endomorphism algebra, and constructs a larger Lie triple system where quasiderivations embed as derivations, with a semidirect decomposition when the center is trivial.

ABSTRACT

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

Motivation & Objective

• To extend the theory of generalized derivations from Lie algebras to Lie triple systems.
• To characterize Lie triple systems for which the quasiderivation algebra equals the full endomorphism algebra.
• To embed quasiderivations of a Lie triple system as derivations in a larger Lie triple system.
• To analyze the structure of the derivation algebra of the extended system, particularly when the center is trivial.

Proposed method

• Define derivation, quasiderivation, generalized derivation, centroid, and quasicentroid in the context of Lie triple systems.
• Establish the inclusion chain: central derivations ⊆ derivations ⊆ quasiderivations ⊆ generalized derivations ⊆ endomorphisms.
• Construct a new Lie triple system $\breve{T} = Tt \oplus Ut^3 \oplus [T,T,T]t^3$ using a graded tensor product with a polynomial ring.
• Define a map $\varphi: \mathrm{QDer}(T) \to \mathrm{End}(\breve{T})$ that lifts quasiderivations to linear maps on $\breve{T}$.
• Prove that $\varphi$ is injective and that $\varphi(D) \in \mathrm{Der}(\breve{T})$ for all $D \in \mathrm{QDer}(T)$.
• Show that when $\mathrm{Z}(T) = \{0\}$, the derivation algebra of $\breve{T}$ decomposes as a semidirect sum: $\mathrm{Der}(\breve{T}) = \varphi(\mathrm{QDer}(T)) \dotplus \mathrm{ZDer}(\breve{T})$.

Experimental results

Research questions

• RQ1 Which Lie triple systems satisfy $\mathrm{QDer}(T) = \mathrm{End}(T)$?
• RQ2 Can quasiderivations of a Lie triple system be embedded as derivations in a larger algebraic structure?
• RQ3 What is the structure of the derivation algebra of the extended Lie triple system $\breve{T}$?
• RQ4 How does the center of $T$ affect the decomposition of $\mathrm{Der}(\breve{T})$?
• RQ5 What is the relationship between generalized derivations, quasiderivations, and centroids in Lie triple systems?

Key findings

• The paper completely characterizes Lie triple systems for which $\mathrm{QDer}(T) = \mathrm{End}(T)$, including all two-dimensional simple Lie triple systems and all commutative Lie triple systems.
• The map $\varphi: \mathrm{QDer}(T) \to \mathrm{End}(\breve{T})$ is injective and maps quasiderivations into $\mathrm{Der}(\breve{T})$, establishing an embedding of quasiderivations as derivations in $\breve{T}$.
• For any Lie triple system $T$, the quasiderivations can be embedded as derivations in the extended Lie triple system $\breve{T} = Tt \oplus Ut^3 \oplus [T,T,T]t^3$, where $U$ is a complement to $[T,T,T]$.
• When $\mathrm{Z}(T) = \{0\}$, the derivation algebra of $\breve{T}$ admits a semidirect decomposition: $\mathrm{Der}(\breve{T}) = \varphi(\mathrm{QDer}(T)) \dotplus \mathrm{ZDer}(\breve{T})$.
• The construction ensures that $\varphi(D)$ is independent of the choice of $D'$ in the quasiderivation condition, making the embedding well-defined.
• The extended system $\breve{T}$ is closed under the ternary product and satisfies the axioms of a Lie triple system, confirming the validity of the construction.