nLab Jordan triple system

Idea

Jordan algebras were invented to axiomatize some properties of the Jordan product $a \circ b = (a b + b a)/2$ of self-adjoint complex matrices, which serve as observables in quantum mechanics. As work proceeded, some researchers found it convenient to focus on another binary operation on self-adjoint matrices,

$$U_a (b) = 2 a \circ (a \circ b) - (a \circ a) \circ b = a b a \, .$$

Axiomatizing the properties of this operation led to the definition of a quadratic Jordan algebra. However, it is often convenient to replace this operation $U_a(b)$ by a trilinear operation, obtained by polarization as follows:

$$\{a,b,c\} \;=\; U_{a+c}(b) - U_a(b) - U_c(b) = a b c + c b a \,.$$

From this triple product we can easily recover the original Jordan product in the presence of a unit, at least if we can divide by 2:

$$\frac{1}{2} \{a,b,1\} = \frac{1}{2}(a b + b a) = a \circ b \, .$$

A Jordan triple system axiomatizes the properties of this triple product of self-adjoint complex matrices.

Later it was discovered that there are close relations between Jordan triple systems and Lie triple systems. Just as Lie triple systems naturally give $\mathbb{Z}/2$-graded Lie algebras, Jordan triple systems naturally give a certain class of so-called 3-graded Lie algebras, which are $\mathbb{Z}$-graded Lie algebras concentrated in degrees $-1,0$ and $1$. And just as the tangent space of any point in a symmetric space is naturally a Lie triple system, the tangent space of any point in a hermitian symmetric space is naturally a Jordan triple system.

A further development of the Jordan algebra concept, in some sense based on Jordan triple systems, is that of a Jordan pair.

Definition

A Jordan triple system is a vector space $V$ equipped with a trilinear map $\{\cdot, \cdots, \cdot \} \colon V \times V \times V \to V$ obeying two axioms:

$$\{a,b,c\} = \{c,b,a\} \,,$$

$$\big\{a,b,\{c,d,e\}\big\} - \big\{c,d,\{a,b,e\}\big\} \;=\; \big\{\{a,b,c\},d,e\big\} - \big\{c,\{b,a,d\},e\big\} \,.$$

Any subspace of an associative algebra closed under the operation $\{a,b,c\} = a b c + c b a$ obeys these axioms, and the first axiom captures the symmetry of this operation under switching the first and last arguments. The second, subtler axiom implies that the operations $L_{a,b} \colon V \to V$ given by $L_{a,b}(c) = \{a,b,c\}$ form a Lie algebra under commutators.

Jordan triple systems from Jordan algebras

Any Jordan algebra gives a Jordan triple system by

$$\{a,b,c\} \;=\; a \circ (b \circ c) + (a \circ b) \circ c - b \circ (a \circ c).$$

There are however other Jordan triple systems. For example, let $M(m,n)$ be the space of $m \times n$ matrices with entries in a field, and given $a \in M(m,n)$ let $a^T \in M(n,m)$ be its transpose. We can make $M(m,n)$ into a Jordan triple system by defining

$$\{a,b,c\} = a b^t c - c b^t a$$

or in the case of matrices over $\mathbb{C}$ or any other star-ring

$$\{a,b,c\} = a b^\ast c - c b^\ast a \, .$$

Lie triple systems from Jordan triple systems

Any Jordan triple system gives a Lie triple system with Lie triple product given by

$$[x,y,z] = \{x,y,z\} - \{y,x,z\} .$$

Relation to 3-graded Lie algebras

Given any 3-graded Lie algebra $\mathfrak{g}$ with an anti-graded involution $\varepsilon \colon \mathfrak{g} \rightarrow \mathfrak{g}$ (restricting to $\mathfrak{g}_{-1} \rightarrow \mathfrak{g}_1$ and $\mathfrak{g}_0 \rightarrow \mathfrak{g}_0$) and any linear subspace $V \subseteq \mathfrak{g}$ closed under the triple commutator

$$\{u,v,w\} = \big[ [u,\varepsilon(v)], w \big] ,$$

$V$ becomes a Jordan triple system. Thus, given a 3-graded Lie algebra with an anti-graded involution, say $\mathfrak{g} = \mathfrak{g}_{-1} \oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1$, the space $\mathfrak{g}_1$ of odd elements becomes a Jordan triple system. (For $u,v,w\in\mathfrak{g}_1$, one has $\varepsilon(v)\in\mathfrak{g}_{-1}$ and therefore $[u,\varepsilon(v)]\in\mathfrak{g}_0$, which results in $\{u,v,w\}\in\mathfrak{g}_1$.)

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

$$\mathfrak{g}_{-1} = V$$

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

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

then $\mathfrak{g} = \mathfrak{g}_{-1} \oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1$ becomes a 3-graded Lie algebra with bracket given by

$$\big[ (a,L,u),(b,M,v) \big] \;=\; \big( L(b) - M(a), [L,M]+L_{a,v}-L_{b,u}, L(v) - M(u) \big) \,,$$

for $L,M \in \mathfrak{g}_0, u,v \in \mathfrak{g}_1$. (Caveny & Smirnov 11 Thm. 5.3)

Let $\mathbf{LA}_{3gr}^\varepsilon$ be the category of 3-graded Lie algebras with an anti-graded involution and let $\mathbf{JTS}$ the category of Jordan triple systems. The first construction (with $V=\mathfrak{g}_1$) gives a forgetful functor $U\colon\mathbf{LA}_{3gr}^\varepsilon\rightarrow\mathbf{JTS}$, while the second gives a fully faithful functor $F\colon\mathbf{JTS} \rightarrow\mathbf{LA}_{3gr}^\varepsilon$. There is an adjunction:

$$F \dashv U$$

(Caveny & Smirnov 2011 6.1).

This adjunction is not an equivalence of categories, since the counit $F \circ U \Rightarrow Id$ is not a natural isomorphism. But since $F$ is fully faithful, the unit $Id \Rightarrow U \circ F$ is a natural isomorphism and $\mathbf{JTS}$ is a coreflective subcategory of $\mathbf{LA}_{3gr}^\varepsilon$. Thus, restricting the adjunction to suitable full subcategory of $\mathbf{LA}_{3gr}^\varepsilon$ gives an equivalence of categories. This subcategory is that of 3-graded Lie algebras $\mathfrak{g}=\mathfrak{g}_{-1}\oplus\mathfrak{g}_0\oplus\mathfrak{g}_1$ with involution that are centrally 0-closed (meaning every central 0-extension of it splits uniquely) and 0-perfect (meaning $\mathfrak{g}_0=[\mathfrak{g}_{-1},\mathfrak{g}_1]$) (Caveny & Smirnov 11 Crl. 6.6).

Relation to Jordan pairs

There is a way to turn a Jordan triple system into a Jordan pair, and vice versa. For details see Jordan pair.

Related concepts

• Jordan algebra

• quadratic Jordan algebra

• Jordan pair

• exceptional Jordan algebra/Albert algebra

• Lie triple system

References

• Ottmar Loos: Jordan triple systems, $R$-spaces, and bounded symmetric domains, Bulletin of the American Mathematical Society 77 (1971) 558–561 [doi:10.1090/S0002-9904-1971-12753-2, pdf]

• 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_2]

• Deanna Caveny and Oleg Smirnov: Categories of Jordan structures and graded Lie algebras (2011), [arxiv:1106.2447]