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

Further
A topological Maslov index for 3-graded Lie groups 43
cτ(x) := (τe ad x )−1,−1 = τ 1
2 (adx)2 |g−1
= −τQ(x)τ |g−1 .
This shows that τF.x ∈ V is equivalent to x ∈ V × . Eventually the fact that τ
reverses the grading implies τ.E + E ∈ z(g), so that
We conclude that
nτ(x) = (e − ad x τ.E)1 = (e − ad x .(−E))1 = [x, E] = −x.
τF(x) = dτ(x) −1 nτ(x) = −Q(x) −1 .(−x) = Q(x) −1 .x = x ♯ .
Remark B.3. Let x ∈ V = g1. Then the pairs (f+, f−) and (f+, e ad x .f−)
are transversal, so that the triple (f+, f−, e ad x .f−) is transversal if and only if
e ad x .f− is transversal to f−, i.e., τF(e ad x .f−) is transversal to f+ = τF(f−). In
view of Proposition B.2, this is equivalent to x ∈ V × .
Appendix C. Tripotents and the Peirce decomposition
In this appendix we briefly discuss the Peirce decomposition of a Jordan
triple with respect to a tripotent e and the representation of the corresponding
sl2-subalgebra on g.
Lemma C.1. (Peirce decomposition) For each tripotent e ∈ V the operator
2ee is diagonalizability with eigenvalues in {0, 1, 2}, and for the corresponding
eigenspaces Vα we have
(C.1) {Vα, Vβ, Vγ} ⊆ Vα−β+γ.
The tripotent e is invertible if and only if V = V2.
Proof. We put D := ee. First Lemma A.2(4) leads to the relation
and hence to
2(ee) 2 − Q(e) 2 = e(Q(e)e) = ee,
(C.2) 2D 2 − D = Q(e) 2 .
On the other hand, Lemma A.2(2) yields Q(e) = DQ(e) = Q(e)D, so that
multiplication of (C.2) with D entails
and further
2D 3 − D 2 = DQ(e) 2 = Q(e) 2 = 2D 2 − D,
0 = 2D 3 − 3D 2 + D = D(D − 1)(2D − 1).
Since the three roots of this polynomial are different, D is diagonalizable with
eigenvalues in {0, 1,
1}. The relation (C.1) is a consequence of the fact that D
2
is a Lie triple derivation by (JT2).
If e is invertible, then Q(e) 2 = 2D2 −D = D(2D −1) is invertible, so that
D = idV , i.e., V = V2. If, conversely, V = V2, i.e., D = 1, then Q(e) 2 = idV
implies that Q(e) is an involution, hence invertible.
In the following g denotes a 3-graded Lie algebra with involution τ reversing
the grading and V = g1 carries the Jordan triple structure from Theorem A.8.