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

A topological Maslov index for 3-graded Lie groups 39
and therefore L(v −1 ) = Q(v) −1 L(v) (cf. [Jac68, Ch. I, Sect. 11, Th. 13]). Note
that the Jordan identity [L(v), L(v 2 )] = 0 means that Q(v) = P(v) commutes
with L(v).
To derive the second identity, we first calculate
P(e + v −1 )P(v) = P(e) + 2P(e, v −1 ) + P(v −1 ) P(v) = P(v) + 2L(v −1 )P(v) + idV
= P(v) + 2L(v) + idV = P(e + v).
Now we consider the unital Jordan algebra (V, w) with the isotopic product
a ∗w b := {a, w −1 , b} and the quadratic representation
P(v) = Q(v)Q(w) −1 = P(v)P(w) −1
(Theorem A.3, Lemma A.4). Then we obtain with the formula in the preceding
paragraph and
P(w).v −1 = P(w)P(v) −1 .v = P(v) −1 .v
the relation
P(v −1 + w −1 ) = P P(w) −1 .(P(w).v −1 + w) = P(w) −1 P(P(w).v −1 + w)P(w) −1
= P(w) −1 P(w + P(w).v −1 ) = P(w) −1 P(w + P(v) −1 .v)
= P(w) −1 P(w + v) P(v) −1 = P(w) −1 P(w + v)P(w) −1 P(w)P(v) −1
= P(w) −1 P(w + v)P(v) −1 .
Lemma A.10. For invertible elements x, y in the Jordan triple V we have
(1) Q(x)Q(x ♯ + y ♯ )Q(y) = Q(x + y) and
(2) B(x, y ♯ ) = Q(x − y)Q(y) −1 .
Proof. (1) We consider on V the unital Jordan algebra structure defined by
ab := {a, x ♯ , b} with unit x (Remark A.6). Then the quadratic representation of
the unital Jordan algebra (V, x) is given by P(v) = Q(v)Q(x) −1 and the Jordan
inversion by v −1 = P(v) −1 .v = Q(x)v ♯ (Theorem A.3). Hence Lemma A.9 leads
to
Q(x)Q(x ♯ + y ♯ )Q(x) = Q(Q(x)x ♯ + Q(x)y ♯ ) = Q(x + y ♯ ) = P(x + y ♯ )Q(x)
= P(x + y)P(y ♯ )Q(x) = P(x + y)P(y) −1 Q(x) = Q(x + y)Q(y) −1 Q(x).
This completes the proof.
(2) In view of Lemma A.2(8) and (1), assertion (2) follows from
B(x, y ♯ ) = Q(x − Q(x)y ♯ )Q(x) −1 = Q(x)Q(x ♯ − y ♯ )Q(x)Q(x) −1
= Q(x)Q(x ♯ − y ♯ ) = Q(x − y)Q(y) −1 .