Constructions of harmonic maps between Hadamard manifolds

CHAPTER 3
Dirichlet problem at infinity for harmonic maps
between Damek-Ricci spaces
3.1 Damek-Ricci spaces
3.1.1 Generalized Heisenberg algebra
We start this section with a brief review of generalized Heisenberg algebras due to Kaplan
[21].
Let (n, 〈 , 〉 Ò
) be a 2-step nilpotent Lie algebra with inner product 〈 , 〉 Ò
, z its center
and v the orthogonal complement of z in n with respect to 〈 , 〉 Ò
. Since n is 2-step nilpotent,
ad v| Ú
is a map from v to z for any v ∈ v. We set k v := {u ∈ v | ad v(u) =[v, u] =0} and
consider the orthogonal decomposition v = k v ⊕ v v with respect to 〈 , 〉 Ò
. We call (n, 〈 , 〉 Ò
)a
generalized Heisenberg algebra if ad v| Úv : v v → z is a surjective isometry for any unit vector
v ∈ v.
Generalized Heisenberg algebras can be constructed systematically in the following fashion:
Let (u, 〈 , 〉 Ù
) and (v, 〈 , 〉 Ú
) be real vector spaces equipped with inner product. Let
µ : u × v → v be a composition of quadratic forms, that is, µ is a bilinear map satisfying
|µ(u, v)| Ú
= |u| Ù
|v| Ú
for all u ∈ u and v ∈ v. Note that µ : u × v → v satisfies µ(u 0 ,v)=v
for some u 0 ∈ u. Indeed, for any given u 0 ∈ u, set T : v → v by T (v) :=µ(u 0 ,v). Then
µ ′ (u, v) :=µ(u, T −1 (v)) satisfies µ ′ (u 0 ,v)=v. Hence we may suppose µ(u 0 ,v)=v. Define
φ : v × v → u by
〈u, φ(v, v ′ )〉 Ù
= 〈µ(u, v),v ′ 〉 Ú
.
Let z be the orthogonal complement of Ru 0 = {ru 0 | r ∈ R} in u, π: u → z the orthogonal
projection and n the direct sum n := v ⊕ z of v and z with the natural inner product. Define
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