A topological Maslov index for 3-graded Lie groups - ResearchGate

A topological Maslov index for 3-graded Lie groups 37
Proposition A.5. (a) Let (V, {·, ·, ·}) be a Jordan triple system and e ∈ S
an invertible tripotent. Then
ab := {a, e, b}, a ∗ := {e, a, e}
defines on V the structure of an involutive Jordan algebra and the Jordan triple
structure can be reconstructed from (V, e, ∗) by
{x, y, z} = (xy ∗ )z + x(y ∗ z) − y ∗ (xz), x, y, z ∈ V.
The set S of involutions of the Jordan triple V coincides with the set
S = {v ∈ V : v ∗ = v −1 }
of unitary elements of the unital involutive Jordan algebra (V, e, ∗).
(b) If (V, e, ∗) is a unital involutive Jordan algebra, then
{x, y, z} := (xy ∗ )z + x(y ∗ z) − y ∗ (xz), x, y, z ∈ V
defines a Jordan triple structure on V with
ab = {a, e, b} and a ∗ = {e, a, e}.
Proof. (a) It follows from Theorem A.3 that ab := {a, e, b} defines on V a
Jordan algebra structure with multiplication maps L(a) = ae. In particular
L(e) = ee = idV , so that e is an identity of V . Moreover, Q(e) 2 = idV follows
from Lemma A.4(3). Next
(a 2 ) ∗ = Q(e)Q(a)e = Q(e)Q(a)Q(e)e = Q(Q(e)a)e = Q(a ∗ )e = (a ∗ ) 2 ,
and polarization leads to (ab) ∗ = a ∗ b ∗ for a, b ∈ V.
Finally Theorem A.3(b) entails
(xy ∗ )z + x(y ∗ z) − y ∗ (xz) = {x, Q(e)y ∗ , z} = {x, Q(e) 2 y, z} = {x, y, z}.
The condition z ∈ S means z ♯ = z, so that the description of the set S
in terms of the involutive Jordan algebra follows from (z ♯ ) ∗ = Q(e)Q(z) −1 z =
P(z) −1 z = z −1 .
Remark A.6. If a ∈ V × is invertible, then xy := {x, a ♯ , y} defines on V
the structure of a Jordan algebra with identity a because L(a) = aa ♯ = idV
(Lemma A.4).