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

A topological Maslov index for 3-graded Lie groups 11
Proof.
0
(1) For I =
−1
1
0
and its square is I . Therefore the order of Ce is at most 8 and we have
C 2 e = η G
e
0
−1
1 π
= exp ad(e − f) .
0 2
the matrix 1
√ 2 (1 + I) ∈ SL2(R) is of order 8
If e is invertible, then g is a direct sum of trivial and 3-dimensional
simple sl2(R)-modules (Remark I.10(b)). On both types of modules the matrix
0 1
∈ SL2(R) acts like an involution, so that C
−1 0
2 e is an involution and
therefore C4 e = idg.
(2) From the decomposition
√
1 1 1 1 1 2 0 1 0
√ =
1
2 −1 1 1 0 0 √2 −1 1
in SL2(R) we derive in Aut(g) the decomposition
(2.1) Ce = exp(ade) exp (log √ 2) adh exp(−adf).
Since exp(ade) exp (log √ 2) adh ∈ Ad(P + ) acts as an affine map on V ⊆ X ,
we see that Ce(v) ∈ V is equivalent to exp(−τ.e).v = exp(−f).v ∈ V , which
means that (e, v) is quasi-invertible (Definition I.5, Lemma I.3). If this is the
case, then
exp(−f).v = B(v, e) −1 .(v − Q(v).e)
([BN04a, 2.8]). In the Jordan algebra V (e) we have Q(v).e = Q(v)Q(e).e =
P(v).e = v 2 and
B(v, e) = idV −2L(v) + P(v),
and in the unital Jordan algebra V (e) × R with the identity 1 := (0, 1) we have
1 − 2L(x) + P(x) = P(1,1) − 2P(1, x) + P(x, x) = P(1 − x),
i.e., the quasi-invertibility of (x, e) is equivalent to the quasi-invertibility of x in
the Jordan algebra V (e) . In this algebra we have for any quasi-invertible pair
(v, e):
exp(−f).v = P(1 − v) −1 .(v − v 2 ) = (1 − v) −1 v.
For any element v in the unital Jordan algebra (V2, e), the Cayley transform
therefore takes the form
Ce(v) = e + 2(e − v) −1 v = (e − v + 2v)(e − v) −1 = (e + v)(e − v) −1 .
(3) We have Ce(−e) = (e − e)(e − (−e)) −1 = 0 and Ce(0) = e.
We further have in V , as a subset of X , the relation τX(e) = e ♯ = e
(Proposition B.2), which leads to
exp(−ad f).e = exp(−ad τ.e).e = τX exp(−ade)τX.e = τX exp(−ade).e = τX.0 = τX.f− = f+,