---
title: Lie–Leibniz Triple Systems
url: https://www.emergentmind.com/topics/lie-leibniz-triple
type: topic
---

# Lie–Leibniz Triple Systems

Searching arXiv for the cited papers and closely related work on Leibniz triple systems and Lie-type triple structures.
arxiv_search query: "Leibniz triple systems split Leibniz triple systems Lie triple systems relative Rota-Baxter 1411.6693 1106.5033"
arxiv_search results requested for:
- "Leibniz triple systems" [1106.5033]
- "On split Leibniz triple systems" [1411.6693]
- "On the structures of split Leibniz triple systems" [1411.7026]
- "On the structure of graded Leibniz triple systems" [1603.08426]
- "Lie Admissible Triple Algebras: The Connection Algebra of Symmetric Spaces" [2306.15582]
- "Generalized metric n-Leibniz algebras and generalized orthogonal representation of metric Lie algebras" [1809.00565]
- "Lie triple system central extensions of Lie algebras" [1010.1760]
- "Notes on Cohomologies of Lie Triple Systems" [1401.7725]
- "Cohomology of Leibniz Triple Systems and its applications" [2108.03996]
- "k-Leibniz algebras from lower order ones: from Lie triple to Lie l-ple systems" [1304.0885]
“Lie–Leibniz triple” (Editor's term) usefully denotes the ternary framework in which **Lie triple systems**, **Leibniz triple systems**, and closely related constructions are studied together. In the formal literature, the central object is the **Leibniz triple system**: a vector space endowed with a trilinear product satisfying the two identities
$$
\{a,\{b,c,d\},e\}=\{\{a,b,c\},d,e\}-\{\{a,c,b\},d,e\}-\{\{a,d,b\},c,e\}+\{\{a,d,c\},b,e\},
$$
$$
\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},
$$
obtained functorially from the identities of Lie triple systems by the Kolesnikov–Pozhidaev algorithm [1106.5033]. Every Lie triple system is a Leibniz triple system, and every Leibniz algebra yields a Leibniz triple system by the iterated bracket formula \(\{x,y,z\}=[[x,y],z]\) [1106.5033]. This suggests that the Lie–Leibniz triple perspective is best understood not as a single additional axiom system, but as a common ternary domain in which Lie-type symmetry, Leibniz-type derivation identities, root decompositions, cohomology, and higher-arity generalizations are organized.

## 1. Basic ternary structures

A **Lie triple system** is a vector space with a trilinear product \([x,y,z]\) satisfying skew-symmetry in the first two variables, the cyclic identity
$$
[x,y,z]+[y,z,x]+[z,x,y]=0,
$$
and the derivation identity
$$
[x,y,[u,v,w]]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]].
$$
A **Leibniz triple system** drops the antisymmetry requirements but retains the Leibniz-type ternary identities displayed above [1106.5033].

Two canonical constructions situate Leibniz triple systems between Lie triple systems and Leibniz algebras. First, if \(L\) is a Leibniz algebra, then \(\{x,y,z\}=[[x,y],z]\) defines a Leibniz triple system structure on \(L\). Second, every Lie triple system automatically satisfies the Leibniz triple identities, so Lie triple systems embed as the antisymmetric sector of the Leibniz theory [1411.7026].

A fundamental internal invariant is the ideal
$$
J=\operatorname{span}\{\{a,b,c\}-\{a,c,b\}+\{b,c,a\}:a,b,c\in T\}.
$$
This ideal satisfies
$$
\{T,T,J\}=\{T,J,T\}=0,
$$
and \(T\) is a Lie triple system if and only if \(J=0\) [1411.7026]. Accordingly, \(J\) measures the defect from Lie-type antisymmetry. In the same spirit, a Leibniz triple system is called **simple** when its product is nonzero and its only ideals are \(\{0\}\), \(J\), and \(T\); in the Lie triple case this reduces to the usual notion because \(J=0\) [1411.7026].

## 2. Envelopes, embeddings, and binary realization

The main binary realization of a Leibniz triple system is its **universal Leibniz envelope**. If \(T\) is a Leibniz triple system, one forms the free Leibniz algebra \(A(T)_L\) on the underlying vector space and quotients by the ideal generated by \((x\cdot y)\cdot z-\{x,y,z\}\). The resulting Leibniz algebra
$$
U(T)=A(T)_L/I(T)
$$
realizes the ternary product as an iterated Leibniz bracket [1106.5033]. Explicitly, as a vector space,
$$
U(T)=T\oplus(T\otimes T),
$$
with multiplication determined by
$$
[a,b]=a\otimes b,\qquad [a\otimes b,c]=\{a,b,c\},
$$
$$
[a,b\otimes c]=\{a,b,c\}-\{a,c,b\},\qquad
[a\otimes b,c\otimes d]=\{a,b,c\}\otimes d-\{a,b,d\}\otimes c
$$
[1407.3978]. The envelope carries a natural \(\mathbb{Z}_2\)-grading, and the associated involutive automorphism \(\theta\) satisfies
$$
T=\{x\in U(T):\theta(x)=-x\},
$$
so \(T\) appears as the \((-1)\)-eigenspace of \(\theta\) [1407.3978].

A different but related construction, crucial for structure theory, is the **standard embedding**. For a Leibniz triple system \(T\), one considers the two-graded right Leibniz algebra
$$
L=L_0\oplus L_1,\qquad L_1:=T,\qquad L_0:=\operatorname{span}\{x\otimes y:x,y\in T\},
$$
with the bracket on homogeneous elements given by
$$
[(x\otimes y,z),(u\otimes v,w)]
=
(\{x,y,u\}\otimes v-\{x,y,v\}\otimes u+z\otimes w,\;
\{x,y,w\}+\{z,u,v\}-\{z,v,u\}).
$$
This embedding supplies the ambient Leibniz algebra used to define root spaces, maximal abelian subalgebras, and split decompositions [1411.6693].

At the level of identities, the enveloping approach is exact: every polynomial identity satisfied by the iterated Leibniz bracket \(((a,b),c)\) in every Leibniz algebra is a consequence of the defining identities of Leibniz triple systems [1106.5033]. This makes the Leibniz triple system axioms the intrinsic ternary identity theory of iterated Leibniz brackets.

## 3. Split and graded structure theory

The most developed structural results concern **split Leibniz triple systems**. Let \(H\) be a maximal abelian subalgebra of the degree-zero part of the standard embedding. For \(\alpha\in H_0^*\), the root space of \(T\) is
$$
T_\alpha=\{t\in T:[t,h]=\alpha(h)t\text{ for all }h\in H\},
$$
and the system is called split if
$$
T=T_0\oplus\bigoplus_{\alpha\in\Lambda}T_\alpha,
$$
with \(\{T_0,T_0,T_0\}=0\) and \(\{T_\alpha,T_{-\alpha},T_0\}=0\) for every \(\alpha\in\Lambda\) [1411.6693]. The root system is **symmetric** when \(\alpha\in\Lambda\) implies \(-\alpha\in\Lambda\).

The triple product is root-additive: whenever \(\{T_\alpha,T_\beta,T_\gamma\}\neq 0\),
$$
\alpha+\beta+\gamma\in\Lambda\cup\{0\},
\qquad
\{T_\alpha,T_\beta,T_\gamma\}\subseteq T_{\alpha+\beta+\gamma}.
$$
Likewise, root-space brackets in the embedding satisfy the corresponding additive rules [1411.6693]. This permits a connectivity calculus on the root system. Two roots are said to be connected if they can be joined by a finite chain of root sums satisfying the partial-sum conditions of the theory; the resulting equivalence classes define subsystems \(\Lambda_\alpha\) and, from them, ideals \(I_{[\alpha]}\) [1411.6693].

The fundamental decomposition theorem states that for a split Leibniz triple system with symmetric root system,
$$
T=U+\sum_{[j]\in\Lambda/\sim} I_{[j]},
$$
where \(U\subseteq T_0\) is a complement of the span of degree-zero triple products and each \(I_{[j]}\) is an ideal attached to a root-connectivity class [1411.6693]. Distinct classes are mutually annihilating:
$$
\{I_{[j]},T,I_{[k]}\}
=
\{I_{[j]},I_{[k]},T\}
=
\{T,I_{[j]},I_{[k]}\}
=0
\quad\text{if }[j]\neq[k].
$$
If the annihilator vanishes and \(\{T,T,T\}=T\), the decomposition becomes direct:
$$
T=\bigoplus_{[j]\in\Lambda/\sim} I_{[j]}
$$
[1411.6693].

An analogous result holds for **graded Leibniz triple systems** \(E=\bigoplus_{g\in G}E_g\) over an arbitrary abelian group \(G\). Writing
$$
\Sigma^1=\{g\in G\setminus\{1\}:E_g\neq 0\},
$$
one defines connectivity on \(\Sigma^1\), obtains equivalence classes \([g]\), and constructs ideals \(I_{[g]}\) such that
$$
E=U+\sum_{[g]\in\Sigma^1/\sim} I_{[g]},
$$
with the same annihilation relations between distinct components [1603.08426]. Under tightness and \(\operatorname{Ann}(E)=0\), this also becomes a direct sum decomposition.

## 4. Ideals, simplicity, radicals, and extension theory

Root connectivity constrains ideals very strongly. In the maximal-length setting, where each nonzero root space is one-dimensional, the split theory becomes sharper. A split Leibniz triple system is **of maximal length** when \(\dim T_\alpha=1\) for every \(\alpha\in\Lambda\), and it is **root-multiplicative** when all triple products allowed by the root-addition rules are nonzero in the prescribed sense [1411.7026]. Under the hypotheses
- \(T=\{T,T,T\}\),
- \(\operatorname{Ann}_{\operatorname{Lie}}(T)=0\),
- root-multiplicativity,
- \(\dim L_\alpha=1\) for all \(\alpha\in\Lambda\),
- symmetric \(\Lambda^J\) and \(\Lambda^{\neg J}\),
- and vanishing conditions involving \(T_0\) and \(\Lambda^{\neg J}\),

simplicity is characterized by primeness together with global \(\neg J\)-connectivity of both \(\Lambda^J\) and \(\Lambda^{\neg J}\) [1411.7026]. In less refined form, simplicity forces all nonzero roots to lie in a single connectivity class [1411.6693].

The intrinsic structure theory also parallels the binary Leibniz theory via the universal envelope. The **solvable radical** \(R(T)\) of a Leibniz triple system is the maximal solvable ideal, and the ternary kernel
$$
\operatorname{Ker}(T)=\operatorname{span}\{\{a,b,c\}-\{a,c,b\}+\{b,c,a\}\}
$$
satisfies \(\operatorname{Ker}(T)\subseteq R(T)\) and \(T/\operatorname{Ker}(T)\) is a Lie triple system [1407.3978]. The relation with the envelope is precise:
$$
R(T)=\operatorname{Rad}(U(T))\cap T,
\qquad
N(T)=\operatorname{Nil}(U(T))\cap T,
$$
where \(N(T)\) is the nilpotent radical [1407.3978]. A Leibniz triple analogue of Levi’s theorem holds:
$$
T=S\dotplus R(T),
$$
with \(S\) a semisimple subsystem that is in fact a semisimple Lie triple system [1407.3978].

At the extension level, a perfect Lie algebra \(\mathfrak g\) may be viewed simultaneously as a Lie algebra, a Leibniz algebra, and a Lie triple system via \(\{x,y,z\}=[x,[y,z]]\). Its universal central extension in the category of Lie triple systems is isomorphic to the universal central extension in the Lie category when \(\operatorname{char}F\neq 2\), and to the universal central extension in the Leibniz category when \(\operatorname{char}F=2\) [1010.1760]. This places Lie triple extensions directly between the Lie and Leibniz extension theories.

## 5. Cohomology, deformations, and operator methods

For **Lie triple systems**, Yamaguti cohomology can be reconstructed from Leibniz cohomology. If \(T\) is a Lie triple system, the fundamental set \(L=T\otimes T\) carries a Leibniz algebra structure, and the Yamaguti complex of \(T\) with coefficients in a module \(V\) coincides with the Loday–Pirashvili cohomology complex of the Leibniz algebra \(L\) with coefficients in \(\operatorname{Hom}(T,V)\) [1401.7725]. In this framework, \(H^3(T,V)\) classifies abelian extensions, and **Nijenhuis operators** describe trivial deformations [1401.7725].

For **Leibniz triple systems**, the relevant low-degree cohomology is concentrated in odd degrees. The first coboundary
$$
\delta f(x_1,x_2,x_3)
=
r(x_2,x_3)f(x_1)+m(x_1,x_3)f(x_2)+l(x_1,x_2)f(x_3)-f(\{x_1,x_2,x_3\})
$$
shows that 1-cocycles in the adjoint module are derivations [2108.03996]. The third cohomology group \(H^3(L,V)\) classifies central extensions of Leibniz triple systems by \(V\), and this classification is implemented explicitly through cocycle-defined extensions on \(L\oplus V\) [2108.03996]. The same paper develops \(T^*\)-extensions: if \(\theta\in Z^3(L,L^*)\), then
$$
T_\theta L=L\oplus L^*
$$
carries a Leibniz triple system structure, and every even-dimensional quadratic Leibniz triple system is isomorphic to a \(T^*\)-extension under a suitable isotropic-ideal hypothesis [2108.03996]. A quadratic Leibniz triple system \((L,B)\) admits a symplectic form if and only if there exists an invertible derivation \(D\) skew-symmetric with respect to \(B\) [2108.03996].

The deformation theory of operators on Leibniz triple systems has also been developed. For a representation \((l,m,r)\) of a Leibniz triple system \(\mathfrak L\) on \(V\), a **relative Rota–Baxter operator**
$$
T:V\to\mathfrak L
$$
is characterized by
$$
\{Tu,Tv,Tw\}
=
T\big(l(Tu,Tv)w+m(Tu,Tw)v+r(Tv,Tw)u\big).
$$
Such an operator induces a Leibniz triple system structure on \(V\), generates its own cohomology theory, and satisfies the usual deformation pattern: \(H^1_T(V,\mathfrak L)\) controls infinitesimal deformations, while \(H^3_T(V,\mathfrak L)\) carries the obstruction classes for extending order-\(n\) deformations [2210.04589]. The same work shows that cohomology of relative Rota–Baxter operators on Leibniz algebras maps naturally to the associated Leibniz triple system cohomology.

## 6. Related frameworks and higher generalizations

Several adjacent theories enlarge the Lie–Leibniz triple landscape without collapsing back to the basic Leibniz triple axioms. One direction is the theory of **Lie admissible triple algebras**. Here an algebra \((\mathcal A,\rhd)\) carries a derived triple bracket
$$
[x,y,z]=\operatorname{ass}(x,y,z)-\operatorname{ass}(y,x,z),
\qquad
\operatorname{ass}(x,y,z)=x\rhd(y\rhd z)-(x\rhd y)\rhd z,
$$
satisfying
$$
[x,y,z]+[y,z,x]+[z,x,y]=0
$$
and
$$
w\rhd[x,y,z]-[w\rhd x,y,z]-[x,w\rhd y,z]-[x,y,w\rhd z]=0.
$$
The connection algebra \((\mathfrak X_M,\rhd)\) of a manifold is such a structure if and only if \((M,\nabla)\) is a locally symmetric space; its triple bracket is then a Lie triple system, and the whole structure embeds canonically into a post-Lie algebra [2306.15582]. This is Lie-type in the ternary bracket and Leibniz-type in the derivation rule for \(\rhd\).

Another direction is metric higher-arity theory. A **generalized metric \(n\)-Leibniz algebra** \((V,[\cdot,\dots,\cdot],S)\) is in one-to-one correspondence with a **Lie triple data**
\((\mathfrak g,V,\rho)\), where \((\mathfrak g,[\cdot,\cdot],\omega)\) is a metric Lie algebra, \(S\) is a symmetric non-degenerate \((n-1)\)-tensor on \(V\), and \(\rho\) is a faithful generalized orthogonal representation [1809.00565]. This correspondence identifies the \(n\)-Leibniz bracket with the action of a Lie algebra of inner derivations and extends to generalized orthogonal derivations and automorphisms [1809.00565].

Higher-order Leibniz constructions also produce generalized Lie-type triple systems. Starting from a metric \(n\)-Leibniz algebra, one may construct a \((2n-3)\)-Leibniz algebra via the Kasymov trace form; imposing appropriate symmetry conditions yields **Lie \(n\)-ple systems** and **Lie \(\ell\)-ple systems**, both reducing to ordinary Lie triple systems when \(\ell=3\) [1304.0885]. At the bialgebraic level, the notion of **3-Leibniz bialgebra** extends Leibniz bialgebras to ternary brackets, and there is a theorem relating a 3-Leibniz bialgebra \((A,\gamma)\) to a binary Leibniz bialgebra \((A\otimes A,\delta)\) on its associated Leibniz algebra [1604.04475].

Taken together, these developments support a precise reading of the Lie–Leibniz triple theme. Lie triple systems provide the antisymmetric benchmark; Leibniz triple systems provide the fundamental non-antisymmetric ternary generalization; split, graded, cohomological, metric, and higher-arity theories show that much of Lie-theoretic structure survives after antisymmetry is relaxed, but typically survives in modified form through the ideals \(J\), root connectivity, enveloping Leibniz algebras, and duality with associated binary Leibniz structures [1411.6693].

Source: https://www.emergentmind.com/topics/lie-leibniz-triple