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The K-loops of hyperbolic spaces
The K-loops of hyperbolic spaces
Torben Steckelberg, University of Hamburg
August 22, 2007
The K-loops of hyperbolic spaces
Contents
1 Absolute Spaces
2 Hyperbolic spaces
3 The K-loops L(d)
4 Connection between L(d) and hyperbolic spaces
The K-loops of hyperbolic spaces
Absolute Spaces
Order in absolute spaces
a b c
b is between a and c
a c b
b is not between a and c
The K-loops of hyperbolic spaces
Absolute Spaces
Order in absolute spaces
a b c
b is between a and c
a c b
b is not between a and c
The K-loops of hyperbolic spaces
Absolute Spaces
Congruence in absolute spaces
a b c d
(a, b) and (c , d) are congruent: (a, b) ≡ (c , d)
a b c d
(a, b) and (c , d) are not congruent
The K-loops of hyperbolic spaces
Absolute Spaces
Congruence in absolute spaces
a b c d
(a, b) and (c , d) are congruent: (a, b) ≡ (c , d)
a b c d
(a, b) and (c , d) are not congruent
The K-loops of hyperbolic spaces
Absolute Spaces
The euclidean spaces
Example
Let F be an euclidean field. For d ∈ N, d ≥ 2 let
F d
be the set of points. As set of lines we define the well-knowneuclidean lines:
G :={
G = {a + λb | λ ∈ F} with a ∈ F d , b ∈ F d\ {0}}
.
The K-loops of hyperbolic spaces
Hyperbolic spaces
H-parallel
G
a
P P ∩ G = ∅
H H ∩ G = ∅
The K-loops of hyperbolic spaces
Hyperbolic spaces
H-parallel
G
aP P ∩ G = ∅
H H ∩ G = ∅
The K-loops of hyperbolic spaces
Hyperbolic spaces
H-parallel
G
aP P ∩ G = ∅
H H ∩ G = ∅
The K-loops of hyperbolic spaces
Hyperbolic spaces
Klein-spaces
Example
Let F be an euclidean field. For d ∈ N, d ≥ 2 let
K (F )d := Kd :=
{(x1, . . . , xd) ∈ F d |
d∑i=1
x2i < 1
}.
As set of lines we define the set of all secants of the unit ball:
The K-loops of hyperbolic spaces
The K-loops L(d)
Definition of K-loops
Definition
A loop (L, ·) is called a K-loop, if
(K1) For all a, b, c ∈ L is a(b · ac) = (a · ba)c .
(K2) For all a, b ∈ L exists a unique inverse and (ab)−1 = a−1b−1
is true.
The K-loops of hyperbolic spaces
The K-loops L(d)
Helpful definitions
From now on let F be an euclidean field that is (d + 1)−real withd ∈ N≥2:
Definition
An euclidean field F is called (d + 1)−real if the characteristicpolynomial of every symmetric positive definite matrixA ∈ F (d+1)×(d+1) splits into linear factors.
Definition
Let J := diag(Ed ,−1) =
1 0 . . . 0
0. . .
. . ....
.... . . 1 0
0 . . . 0 −1
∈ GL(d + 1,F ).
The K-loops of hyperbolic spaces
The K-loops L(d)
Helpful definitions
From now on let F be an euclidean field that is (d + 1)−real withd ∈ N≥2:
Definition
An euclidean field F is called (d + 1)−real if the characteristicpolynomial of every symmetric positive definite matrixA ∈ F (d+1)×(d+1) splits into linear factors.
Definition
Let J := diag(Ed ,−1) =
1 0 . . . 0
0. . .
. . ....
.... . . 1 0
0 . . . 0 −1
∈ GL(d + 1,F ).
The K-loops of hyperbolic spaces
The K-loops L(d)
Definition of the K-loops L(d)
Definition
For d ≥ 2 let
L(d) :={M ∈ GL(d + 1,F ) | M p.def.,MtJM = J ∧ det(M) = 1
}From [Kiechle, Theory of K-Loops] is known:
Theorem
Let d ≥ 2. With a special binary operation
◦ : L(d)× L(d) → L(d)
(L(d), ◦) is a K-loop.
The K-loops of hyperbolic spaces
The K-loops L(d)
Definition of the K-loops L(d)
Definition
For d ≥ 2 let
L(d) :={M ∈ GL(d + 1,F ) | M p.def.,MtJM = J ∧ det(M) = 1
}From [Kiechle, Theory of K-Loops] is known:
Theorem
Let d ≥ 2. With a special binary operation
◦ : L(d)× L(d) → L(d)
(L(d), ◦) is a K-loop.
The K-loops of hyperbolic spaces
The K-loops L(d)
A bijection ϕ : L(d) → F d1
For a matrix A ∈ F (d+1)×(d+1) is
A ∈ L(d) ⇔ A =
(Ed+ γ2
1+γ·aat γa
γat γ
)with γ := 1√
1−ataand a ∈ F d
1 arbitrary, so we know:
Theorem
ϕ : L(d) → F d1 :
(Ed+ γ2
1+γ·aat γa
γat γ
)7→ a
is a bijection.
The K-loops of hyperbolic spaces
The K-loops L(d)
A bijection ϕ : L(d) → F d1
For a matrix A ∈ F (d+1)×(d+1) is
A ∈ L(d) ⇔ A =
(Ed+ γ2
1+γ·aat γa
γat γ
)with γ := 1√
1−ataand a ∈ F d
1 arbitrary, so we know:
Theorem
ϕ : L(d) → F d1 :
(Ed+ γ2
1+γ·aat γa
γat γ
)7→ a
is a bijection.
The K-loops of hyperbolic spaces
The K-loops L(d)
A fibration F that generates an incidence-structure
Obviously
F := Set of all secants through 0 of F d1
is a Fibration of ϕ (L(d)).
The K-loops of hyperbolic spaces
The K-loops L(d)
Incidence-structure
The following result can be shown:
Theorem
G := ϕ({
A ◦ f | A ∈ L(d), f ∈ ϕ−1 (F)})
is the set of all secantsof F d
1 . So (ϕ (L(d)) ,G) and(L(d), ϕ−1 (G)
)are incidence-spaces
and isomorphic to (Kd ,Gd).
The K-loops of hyperbolic spaces
The K-loops L(d)
Incidence-structure
The following result can be shown:
Theorem
G := ϕ({
A ◦ f | A ∈ L(d), f ∈ ϕ−1 (F)})
is the set of all secantsof F d
1 . So (ϕ (L(d)) ,G) and(L(d), ϕ−1 (G)
)are incidence-spaces
and isomorphic to (Kd ,Gd).
The K-loops of hyperbolic spaces
Connection between L(d) and hyperbolic spaces
Congruence
How to define the congruence, so that L(d) becomes a hyperbolicspace?
Definition
For A,B,C ,D ∈ L(d) we simply define
(A,B) ≡ (C ,D) :⇔ (ϕ(A), ϕ(B)) ≡d (ϕ(C ), ϕ(D)).
The K-loops of hyperbolic spaces
Connection between L(d) and hyperbolic spaces
Congruence
How to define the congruence, so that L(d) becomes a hyperbolicspace?
Definition
For A,B,C ,D ∈ L(d) we simply define
(A,B) ≡ (C ,D) :⇔ (ϕ(A), ϕ(B)) ≡d (ϕ(C ), ϕ(D)).
The K-loops of hyperbolic spaces
Connection between L(d) and hyperbolic spaces
Order
How to define the order, so that L(d) becomes a hyperbolic space?
Definition
For collinear A,B,C ∈ L(d) with A 6= B,C we simply define
(A|B,C ) := (ϕ(A)|ϕ(B), ϕ(C )).
The K-loops of hyperbolic spaces
Connection between L(d) and hyperbolic spaces
Order
How to define the order, so that L(d) becomes a hyperbolic space?
Definition
For collinear A,B,C ∈ L(d) with A 6= B,C we simply define
(A|B,C ) := (ϕ(A)|ϕ(B), ϕ(C )).
The K-loops of hyperbolic spaces
Connection between L(d) and hyperbolic spaces(L(d), ϕ−1 (F) ,≡, ζ
)is a hyperbolic space
Obviously is true:
Theorem
ϕ : L(d) → F d1
is an isomorphism of the incidence-spaces that holds thecongruence and holds the order.Therefore
(L(d), ϕ−1 (F) ,≡, ζ
)is a hyperbolic space for all d ∈ N
with d ≥ 2.
The K-loops of hyperbolic spaces
Connection between L(d) and hyperbolic spaces(L(d), ϕ−1 (F) ,≡, ζ
)is a hyperbolic space
Obviously is true:
Theorem
ϕ : L(d) → F d1
is an isomorphism of the incidence-spaces that holds thecongruence and holds the order.Therefore
(L(d), ϕ−1 (F) ,≡, ζ
)is a hyperbolic space for all d ∈ N
with d ≥ 2.
The K-loops of hyperbolic spaces
Connection between L(d) and hyperbolic spaces
Result
Theorem
Let (H,GH ,≡, ζ) be a hyperbolic space of dimension m ∈ N overan euclidean field F that is (m + 1)−real.Then (H,GH ,≡, ζ) is isomorphic to the incidence-space defined bythe K-loop L(m):
(H,GH ,≡, ζ) ∼=(L(m), ϕ−1 (F) ,≡, ζ
).