nLab Lie triple system

Idea

In any Lie algebra we can define the triple commutator of three elements by

$$[a,b,c] \;\coloneqq\; \big[[a,b],c\big] \mathrlap{\,,}$$

and this obeys axioms which form the definition of a Lie triple system. Furthermore, the space of elements of odd degree in any $\mathbb{Z}/2$-graded Lie algebra forms a Lie triple system, and thus so does the tangent space of a symmetric space at any point.

Definition

A Lie triple system is a vector space $V$ equipped with a trilinear map $[\cdot,\cdot,\cdot] \colon V \times V \times V \to V$ obeying the following three identities:

$$[u,v,w] = -[v,u,w]$$

$$[u,v,w] + [w,u,v] + [v,w,u] = 0$$

$$[u,v,[w,x,y]] = [[u,v,w],x,y] + [w,[u,v,x],y] + [w,x,[u,v,y]].$$

(Jacobson 51, Eq. (1.7)-(1.11), Smirnov 09, 2.1)

The first two identities abstract the skew symmetry and Jacobi identity for the triple commutator in a Lie algebra. The third identity, which also holds for the triple commutator in a Lie algebra, says that the linear map $L_{u,v} \colon L \to L$ defined by $L_{u,v}(w) = [u,v,w]$, is a derivation of of the triple product, in the following sense:

$$L_{u,v}[w,x,y] = [L_{u,v} w, x, y] + [w, L_{u,v} x, y] + [w, x, L_{u,v} y] .$$

The identity also implies that the space of linear operators

$$\mathfrak{g}_0 = \text{span}\{L_{u,v}: u, v \in V \}$$

is closed under commutators, hence a Lie algebra. (Smirnov 09, 2.3)

Relation to $\mathbb{Z}/2$-graded Lie algebras

Given any Lie algebra $\mathfrak{g}$ and any linear subspace $V \subseteq \mathfrak{g}$ closed under the triple commutator

$$[u,v,w] = [[u,v], w] ,$$

$V$ becomes a Lie triple system. Thus, given a $\mathbb{Z}/2$-graded Lie algebra (not a Lie superalgebra, an ordinary Lie algebra with a $\mathbb{Z}/2$ grading), say $\mathfrak{g} = \mathfrak{g}_0 \oplus \mathfrak{g}_1$, the space $\mathfrak{g}_1$ of odd elements becomes a Lie triple system.

Conversely, given any Lie triple system $V$, if we define

$$\mathfrak{g}_0 = \text{span}\{L_{u,v}: u, v \in V \}$$

$$\mathfrak{g}_1 = V$$

then $\mathfrak{g} = \mathfrak{g}_0 \oplus \mathfrak{g}_1$ becomes a $\mathbb{Z}/2$-graded Lie algebra with bracket given by

$$[(L,u),(M,v)] = ([L,M]+L_{u,v}, L(v) - M(u))$$

for $L,M \in \mathfrak{g}_0, u,v \in \mathfrak{g}_1$. (Smirnov 09, Corollary 3.4)

Let $\mathbf{LA}$ be the category of Lie algebras, $\mathbf{LA}_{\mathbb{Z}/2}$ be the category of $\mathbb{Z}/2$-graded Lie algebras and $\mathbf{LTS}$ be the category of Lie triple systems. The first construction (with $V=\mathfrak{g}$) defines a forgetful functor $\mathcal{T}\colon\mathbf{LA}\rightarrow\mathbf{LTS}$, which is a right adjoint functor. (Jacobson 51, Smirnov 09, 2.2.) The first construction (with $V=\mathfrak{g}_1$) also defines a functor $\mathcal{R}\colon\mathbf{LA}_{\mathbb{Z}/2}\rightarrow\mathbf{LTS}$. The second contruction defines a faithful functor $\mathfrak{A}\colon\mathbf{LTS}\rightarrow\mathbf{LA}$ and even a fully faithful functor $\mathfrak{A}\colon\mathbf{LTS}\rightarrow\mathbf{LA}_{\mathbb{Z}/2}$, which gives an adjunction:

$$\mathfrak{A} \dashv\mathcal{R}.$$

(Smirnov 09, 4.1)

This adjunction is not an equivalence of categories, since any abelian $\mathbb{Z}/2$-graded Lie algebra (hence with vanishing Lie bracket) converted into a Lie triple system and back into a $\mathbb{Z}/2$-graded Lie algebra has a zero-dimensional space of even elements. Thus, the counit $\mathfrak{A}\mathcal{R}\Rightarrow Id$ is not a natural isomorphism. But since $\mathfrak{A}$ is fully faithful, a suitable subcategory of $\mathbf{LA}_{\mathbb{Z}/2}$ makes the adjunction an equivalence of categories, which is that of the $\mathbb{Z}/2$-graded Lie algebras $\mathfrak{g}=\mathfrak{g}_0\oplus\mathfrak{g}_1$, which are $0$-centrally closed (meaning every central $0$-extension of it splits) and generated by $\mathfrak{g}_1$ under the Lie bracket (meaning $\mathfrak{g}_0=[\mathfrak{g}_1,\mathfrak{g}_1]$). (Smirnov 09, Crl. 4.7 & 4.8)

Related entries

• Jordan triple system

References

• Nathan Jacobson: Lie and Jordan triple systems, American Journal of Mathematics 71 (1949) 149–170 [jstor:2372102] also in: Nathan Jacobson, Collected Mathematical Papers, Contemporary Mathematicians. Birkhäuser Boston (1989) [doi:10.1007/978-1-4612-3694-8_2rbrack;

• Nathan Jacobson: General representation theory of Jordan algebras, Trans. Amer. Math. Soc. 70 (1951) 509–530 [doi:10.1007/978-1-4612-3694-8_9, doi:10.2307/1990612, jstor:1990612]

• Oleg Smirnov: Imbedding of Lie triple systems into Lie algebras, Journal of Algebra 341 1 (2011) 1-12 [arxiv:0906.1170, doi:10.1016/j.jalgebra.2011.06.011]

In the context of Snyder spaces:

• Florian Girelli: Snyder space-time: K-loop and Lie triple system, SIGMA Symmetry Integrability Geom. Methods Appl. 6 074 (2010) [sigma:2010/074, pdf, arxiv/1009.4762]