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

A topological Maslov index for 3-graded Lie groups 29
Definition IV.5. In the following we write η G σ : SL2(C)/{±1} → G for the
unique morphism of 3-graded involutive Lie groups with L(η G σ ) = ηg σ whose
existence follows from the simple connectedness of SL2(C) and Lemma IV.3.
According to Remark IV.1, we have
µG(e, −e, σ) = π1(η G σ )µ SL2(C)/{±1}(1, −1, i).
Therefore the calculation of the index map is essentially reduced to the calculation
of the single case µ SL2(C)/{±1}(1, −1, i).
The next proposition provides the index function for SL2(C)/{±1}.
Proposition IV.6. We consider the 3-graded involutive Lie group G :=
SL2(C)/{±1} which satisfies (A1-3) by Theorem II.7. We have an isomorphism
ρ: G 0
= ±
z 0
0 z−1
: z ∈ C ×
→ C ×
z 0
, ±
0 z−1
↦→ z 2
and identify π1(G 0 ) accordingly with π1(C × ) ∼ = Z, where we use p C ×: C →
C × , z ↦→ e 2πiz as the universal covering map. In these terms we have
µG(1, −1, ±i) = ∓1.
Proof. In the following we shall use the explicit formulas from the discussion
of hermitian Banach algebras in Example II.6. We have
which leads to
B SL2(C)(z, w) =
1 − zw 0
0 (1 − zw) −1
,
BG(z, w) = (1 − zw) 2
in terms of our identification of G 0 with C × . From that we further obtain for
quasi-invertible triples (z1, z2, z3):
dG(z1, z2, z3) = (1 − z1z2) 2 (1 − z3z2) −2 (1 − z3z1) 2 (1 − z2z1) −2 (1 − z2z3) 2 (1 − z1z3) −2
We obtain in particular
1 −
z1z2 2
1 −
z3z1 2
1 − z2z3
=
1 − z2z1 1 − z1z3 1 − z3z2
1 − z1z2
dG(z1, z2, 0) =
1 − z2z1
For the curve
2
2
.
1 −
z3 2
1 + z3
and dG(1, −1, z3) =
1 − z3 1 + z3
α1: [0, 1] → C 3 ⊤ , t ↦→ (t, −t, 0)
2
.