nLab Jordan pair

Idea

As a further development of the concept of Jordan algebra, a Jordan triple system axiomatizes the properties of the Jordan triple product

$$\{a,b,c\} \;=\; a b c + c b a \,.$$

of self-adjoint complex matrices. However, this triple product also makes sense when $a$ and $c$ are of a different type than $b$. For example, it makes sense when $a$ and $c$ are $m \times n$ matrices and $b$ is an $n \times m$ matrix, or vice versa. Thus, it turns out to be useful to generalize Jordan triple systems to Jordan pairs. The theory of bounded symmetric domains has been developed by Ottmar Loos using Jordan pairs (Loos77).

Definition

A Jordan pair is a pair of vector spaces $V_+$ and $V_-$ together with a pair of trilinear maps

$$\{\cdot,\cdot,\cdot\}_+ \colon V_+ \times V_- \times V_+ \to V_+ \, ,$$

$$\{\cdot,\cdot,\cdot\}_- \colon V_- \times V_+ \times V_- \to V_- \, .$$

obeying two symmetry axioms

$$\{u,v,w\}_\pm = \{w,v,u\}_\pm$$

and two Jacobi-like axioms

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

This definition works well over any field or even commutative ring that is not of characteristic 2 or 3.

The trilinear maps are often reformulated as quadratic maps

$$Q_+ \colon V_+ \to \text{Hom}(V_-, V_+)$$

$$Q_- \colon V_- \to \text{Hom}(V_+, V_-) .$$

For a simple example of a Jordan pair, let $V_+$ be a finite-dimensional vector space and $V_-$ the dual of that vector space, with the quadratic maps given by

$$Q_+(v)(f) = f(v) \,v$$

$$Q_-(f)(v) = f(v) \, f.$$

for $v \in V_+, f \in V_-$.

Advantages

Jordan pairs are more general than Jordan algebras. Further, the connection between Jordan structures, 3-graded Lie algebras and bounded symmetric domains is arguably clearer in terms of Jordan pairs than using Jordan algebras or Jordan triple systems (Loos77).

Isomorphisms of Jordan pairs are more general than isomorphisms of Jordan algebras or Jordan triple systems: they generalize the so-called “isotopies” of these. An isotopy $f \colon V \to V'$ of Jordan triple systems is a pair of bijective linear maps $f_+, f_- \colon V \to V'$ such that

$$f_+(\{u,v,w\}) = \{f_+(u), f_-(v), f^+(w) \}$$

for all $u,v,w \in V$. The use of two separate linear maps is somewhat odd here, but natural for Jordan pairs. An isomorphism from a Jordan pair $V_\pm$ to a Jordan pair $V'_\pm$ is a pair of bijective linear maps $f_\pm \colon V_\pm \to V'_\pm$ such that

$$f_\pm\bigl(\{u,v,w\}_\pm\bigr) = \{f_\pm(u), f_\mp(v), f_\pm(w)\}_\pm$$

for all $u,w \in V_\pm, v \in V_\mp$.

Unlike for Jordan algebras and Jordan triple systems, there is a natural way to define inner automorphisms of Jordan pairs (Loos75). (See below for inner derivations.)

Furthermore, Jordan pairs always contain sufficiently many idempotents (which generalize the projection operators familiar in the case of self-adjoint complex matrices) (Loos75). It may happen even in a finite-dimensional simple Jordan algebra that the unit element cannot be written as the sum of orthogonal division idempotents (the algebra need not have “capacity”). In a Jordan triple system the relevant generalization of an idempotent is a tripotent, an element $p$ with $\{p,p,p\} = p$, but if we make $\mathbb{R}$ into a Jordan triple system with $\{a,b,c\} = -(a b c + c b a)$ it has no nonzero tripotents. In the case of a Jordan pair $(V_+, V_-)$, we define an idempotent to be a pair of elements $p_+ \in V_+, p_- \in V_-$ such that

$$\{p_\pm, p_\mp, p_\pm\}_\pm = p_{\pm} \, .$$

Relation to 3-graded Lie algebras

A 3-graded Lie algebra is a $\mathbb{Z}$-graded Lie algebra that vanishes except in grades -1, 0 and 1, say

$$\mathfrak{g} = \mathfrak{g}_{-1} \oplus \mathfrak{g}_0\oplus \mathfrak{g}_1 \, .$$

Given any 3-graded Lie algebra, the pair of vector spaces $(\mathfrak{g}_{1}, \mathfrak{g}_{-1})$ becomes a Jordan pair with brackets

$$\{a,b,c\}_\pm = [[a,b],c]$$

for $a,c \in \mathfrak{g}_{\pm 1}$ and $b \in \mathfrak{g}_{\mp 1}$.

Conversely, there is a functorial way to build a 3-graded Lie algebra from a Jordan pair. Start with a Jordan pair $(V_+, V_-)$. We construct a 3-graded Lie algebra $\mathfrak{g} = \mathfrak{g}_{-1} \oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1$ as follows. First, set

$$\mathfrak{g}_1 = V_+, \qquad \mathfrak{g}_{-1} = V_-$$

and let $\mathfrak{g}_0$ consist of the inner derivations of the Jordan pair. These are linear maps $D \colon V_+ \oplus V_- \to V_+ \oplus V_-$ that are linear combinations of those of the form

$$D(x, y) = (\{x, y, \cdot\}_+, -\{y, x, \cdot\}_-)$$

for $x \in V_+, y \in V_-$. We write $D(x,y) = (D_+(x,y), D_-(x,y))$.

Next, define a Lie bracket on $\mathfrak{g}$ as follows:

1) $[\mathfrak{g}_1, \mathfrak{g}_{-1}] \to \mathfrak{g}_0$: For $x \in V_+, y \in V_-$, set

$$[x, y] = D(x, y) = (Q(x, y, \cdot), -Q(y, x, \cdot))$$

2) $[\mathfrak{g}_0, \mathfrak{g}_{\pm 1}] \to \mathfrak{g}_{\pm 1}$: For $D = (D_+, D_-) \in \mathfrak{g}_0$, set

$$[D, x] = D_+(x) \; \text{ for } \; x \in V_+, \qquad [D, y] = D_-(y) \; \text{ for } \; y \in V_-$$

3) $[\mathfrak{g}_0, \mathfrak{g}_0] \to \mathfrak{g}_0$: Here we use the usual commutator of linear operators:

$$[(D_+, D_-), (E_+, E_-)] = ([D_+, E_+], [D_-, E_-]) \, .$$

4) $[\mathfrak{g}_i, \mathfrak{g}_j] = 0$ when $|i + j| \gt 1$.

Relation to Jordan triple systems

There is a way to see a Jordan triple system as a Jordan pair with extra structure, and vice versa (McCrimmon78). This suggests that there are adjoint functors between the category of Jordan triple systems and the category of Jordan pairs.

Jordan triple systems as Jordan pairs with involutions

Define a Jordan pair with involution to be a Jordan pair $(V_\pm, \{\cdot, \cdot, \cdot\}_\pm)$ with a vector space isomorphism $\alpha \colon V_+ \to V_-$ such that

$$Q_-(\alpha(v)) = \alpha \circ Q_+(v) \alpha$$

for all $v \in V_+$, where $Q_\pm$ are the quadratic maps mentioned above.

The category of Jordan triple systems is equivalent to the category of Jordan pairs with involution (Loos75). In this equivalence, a Jordan pair with involution gives a Jordan triple system $(V, \{\cdot,\cdot,\cdot\})$ with $V = V_+$ and

$$\{u,v,w\} = \{u,\alpha(v),w\}_+ \, .$$

Conversely, a Jordan triple system $(V, \{\cdot, \cdot, \cdot \})$ gives a Jordan pair $(V_\pm, \{\cdot, \cdot, \cdot\}_\pm)$ with $V_+ = V_- = V$ and

$$\{u,v,w\}_+ = \{u,v,w\} \qquad for \; all \; u,w \in V_+, v \in V_-$$

$$\{u,v,w\}_- = \{u,v,w\} \qquad for \; all \; u,w \in V_-, v \in V_- \, .$$

This has an involution given by $\alpha = 1_V$.

Jordan pairs as polarized Jordan triple systems

Define a polarized Jordan triple system to be a Jordan triple system $(V, \{\cdot, \cdot, \cdot\})$ together with a direct sum decomposition $V = V_+ \oplus V_-$ such that

$$\{u,v,w\}_\pm \in V_\pm$$

for all $u,w \in V_\pm, v \in V_\mp$ and

$$\{u,v,w\}_\pm = 0$$

for all other possibilities, i.e. for all $u, v \in V_\pm$, $w \in V_\mp$, all $v, w \in V_\pm$, $u \in V_\mp$, and all $u, v, w \in V_\pm$.

The category of Jordan pairs is equivalent to the category of polarized Jordan triple systems (Loos75).

Related concepts

• Jordan algebra

• quadratic Jordan algebra

• Jordan triple system

• exceptional Jordan algebra/Albert algebra

References

• Ottmar Loos: Jordan Pairs, Lecture Notes in Mathematics 460 Springer (1975) [ISBN:978-3-540-37499-2, doi:10.1007/BFb0080843, ark:/13960/t9774fs7f]

• Ottmar Loos: Bounded Symmetric Domains and Jordan Pairs, Lecture notes, University of California (1977)

• Kevin McCrimmon, Review: Ottmar Loos, Jordan pairs, Bull. Amer. Math. Soc. 84 4 (1978) 685-690 [euclid]