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

A topological Maslov index for 3-graded Lie groups 31
Proposition IV.8. Suppose that g is a complex Lie algebra and that τ
is antilinear. Then (V, e, Q(e)) is a complex unital Jordan algebra and the
involution Q(e)v = v ∗ is antilinear. For each hermitian projection p = p ∗ =
p 2 ∈ V+ let
γp: R/Z → G 0 , t + Z ↦→ exp(2πit[p, τ.p])
denote the corresponding projection loop, which is a group homomorphism. We
then have for the involution σ = e − 2p the projection loop formula
µG(e, −e, −iσ) = µG(e, −e, −ie) − [γp]
Proof. We have V− = iV+, so that every unitary element in V− is of the form
iσ, where σ ∈ V+ is a hermitian involution. Then p := 1
2 (e − σ) is a hermitian
idempotent in the Jordan algebra (V, e) with σ = e − 2p.
The index µG(e, −e, −iσ) can be calculated directly from the real τ -
invariant subalgebra generated by e and −iσ, which is isomorphic to sl2(C). As
we have seen in Theorem IV.6, this leads to the one-parameter subgroup T → G0 corresponding to the element
π[iσ, τ.e] ∈ exp −1 (1).
In particular, the index µG(e, −e, −ie) corresponds to the element
and the difference is the element
π[ie, τ.e] = πi[e, τ.e] ∈ exp −1 (1),
(4.2) π[ie − iσ, τ.e] = πi[e − σ, τ.e] = 2πi[p, τ.e] = 2πi[p, τ.p],
which belongs to the Lie algebra gp := spanC {p, τ.p, [p, τ.p]} ∼ = sl2(C) (cf.
1 0
Definition I.9), where h := [p, τ.p] corresponds to
∈ sl2(C) which
0 −1
satisfies exp(2πih) = 1. From (4.2) we now derive the projection loop formula
because [e, τ.e] is central in g0 (Remark I.11).
V. The Maslov index for some examples
In this section we give more concrete formulas for the index function for
several classes of hermitian Banach-∗-algebras and discuss the case of finitedimensional
bounded symmetric domains.
Example V.1. We take a closer look at the index function for the case G =
GL2(A)/{±1} for a hermitian Banach-∗-algebra.