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

A topological Maslov index for 3-graded Lie groups 25
IV. Evaluating the index map
In the preceding section we have reduced the problem to calculate the index
function µG: S 3 ⊤ → π1(G 0 ) to triples of the form (e, −e, σ) with σ 2 = −e in the
unital Jordan algebra (V, e). The next step is to calculate the index function on
these triples explicitly by showing that µG(e, −e, σ) is represented by the group
homomorphism
χσ: T ∼ = R/Z → G 0 , t + Z ↦→ exp G(πt[σ, τ.e]).
Applying the representation ρV : G 0 → GL(V ), this leads to the loop
T ∼ = R/Z → G 0 , t + Z ↦→ e πt2L(σ) = P(e πtσ ).
To obtain the explicit formula for the index, we first investigate functoriality
properties of the index and then calculate it explicitly for the group
SL2(C)/{±1}.
Remark IV.1. (a) Let U and G be 3-graded Lie groups and ϕ: U → G a
homomorphism of Lie groups compatible with the 3-grading.
We then have
ϕ(U ± ) = ϕ(exp u±) = expL(ϕ)u± ⊆ exp g± = G ±
and ϕ(U 0 0) ⊆ G 0 0.
For a subset M ⊆ G we write CG(M) for the centralizer of M in G and
for a subset M ⊆ Aut(g) we write CG(M) := Ad −1 (C Aut(g)(M)) for the set of
all those elements g ∈ G for which Ad(g) commutes with M . This means that
for a grading element E ∈ g0 we have G 0 = CG(ad E). If there is a grading
element EU ∈ u0 for which EG := L(ϕ)EU is a grading element of g, then we
thus obtain
ϕ(U 0 ) = ϕ(CU(ad EU)) ⊆ CG(ad EG) = G 0 .
Then ϕ induces a map U + U 0 U − → G + G 0 G − compatible with the projection
maps p G j : G+ G 0 G − → G j in the sense that
p G j ◦ ϕ = ϕ ◦ pU j
, j = +, 0, −.
For z ∈ u+ and w ∈ u− the condition expw exp z ∈ U + U 0 U − therefore
implies
(expL(ϕ)w)(expL(ϕ)z) ∈ G + G 0 G − ,
which shows that L(ϕ) preserves quasi-invertibility, and for such pairs we have
ϕ ◦ p U 0 (exp w exp z) = pG 0 ((expL(ϕ)w)(expL(ϕ)z)).