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

A topological Maslov index for 3-graded Lie groups 35
Problem V. (a) Is µG a cocycle in the sense that
µG(z1, z2, z3) = µG(z1, z2, z4) + µG(z2, z3, z4) + µG(z1, z4, z3)?
(b) Is the index function invariant under the full group G τ ? This would
follow if G 0 acts trivially on π1(G 0 ), but this is certainly not always the case
because G may be of the form G = G1 × G2 with G2 ⊆ G 0 and G2 can be any
Lie group.
If A is a hermitian Banach-∗-algebra, then GL2(A) 0 ∼ = A × × A × . In
this case the problem from above leads to the question whether π0(A × ) act
trivially on π1(A × ). This is not always the case, as we see for A = M2(R)
with the involution a ↦→ a ⊤ . In this case π0(A × ) = π0(GL2(R)) ∼ = Z/2Z and
π1(A × ) = π1(GL2(R)) = π1(SL2(R)) ∼ = Z, where the group π0(A × ) acts by
inversion on π1(A × ).
Appendix A. Jordan triple systems and Jordan algebras
In this appendix we collect some basic facts on Jordan algebras and Jordan
triples over a field K with 2, 3 ∈ K × .
Definition A.1. (a) A vector space V over a field K is said to be a Jordan
triple system (JTS) if it is endowed with a trilinear map {·}: V × V × V → V
satisfying:
(JT1) {x, y, z} = {z, y, x}.
(JT2) {a, b, {x, y, z}} = {{a, b, x}, y, z} − {x, {b, a, y}, z} + {x, y, {a, b, z}} for all
a, b, x, y, z ∈ V .
For x, y ∈ V we define operators xy, Q(x) and Q(x, z) on V by
(xy).z := {x, y, z}, Q(x)(y) := {x, y, x}, Q(x, z)(y) := {x, y, z}.
The Bergman operator of V is defined by
B(x, y) := 1 − 2xy + Q(x)Q(y).
We define the set of invertible elements of V by V × := {v ∈ V : Q(v) ∈
GL(V )} and the inversion map by V × → V × , v ↦→ v ♯ := Q(v) −1 .v. The elements
of the set
S := {v ∈ V × : v ♯ = v} = {v ∈ V × : {v, v, v} = v}
are called involutions, resp., invertible tripotents.
Lemma A.2. If 3 ∈ K × and (V, {·, ·, ·}) is a Jordan triple system, then the
following formulas hold for x, y, z ∈ V :
(1) Q(x).{y, x, z} = {Q(x).y, z, x} = {x, y, Q(x).z}.
(2) Q(x)(yx) = (xy)Q(x) = Q(Q(x).y, x).