134
P. Moylan
We have the short exact sequence
Z 2 → Sp(2, R) → SO 0 (2, 3).
(2)
An explicit isomorphism ϕ of the respective Lie algebras is given by
ϕ(L −1,0 ) =
1
2
0 −I 2
I 2 0
, ϕ(L 1,2 ) =
1
2
σ 2 0
σ 2
,
ϕ(L 1,3 ) =
1
2
0 −σ 1
σ 1 0
, ϕ(L 2,3 ) =
1
2
0 −σ 3
σ 3 0
,
ϕ(L −1,1 ) =
1
2
0 σ 1
σ 1 0
, ϕ(L −1,2 ) =
1
2
0 σ 3
σ 3 0
, ϕ(L −1,3 ) =
1
2
I 2 0
0 −I 2
,
ϕ(L 0,1 ) =
1
2
−σ 1 0
0 σ 1
, ϕ(L 0,2 ) =
1
2
−σ 3 0
0 σ 3
, ϕ(L 03 ) =
1
2
0 I 2
I 2 0
where I 2 =
1 0
0 1
, σ 1 =
0 1
1 0
, σ 2 =
0 −1
1 0
, and σ 3 =
1 0
0 −1
. The
linear span of the four matrices in the top two rows of the above equation is k, the
Lie algebra of the maximal compact subgroup K of Sp(2, R). With θ the Cartan
involution on sp(2, R) specified by θ(X) = −X † for X ∈ sp(2, R), we have that
sp(2, R) = k ⊕ ˜
p where ˜
p is the real linear span of the six matrices in the bottom
two rows of the above equation.
A maximal parabolic subgroup P = MAN of Sp(2, R) is: (t, y 0 , y 1 , y 2 ∈ R)
M =
m =
(σ 3 )
0
0 (σ 3 ) † −1
=
a b
c d
∈ SL(2, R) , , ∈ {0, 1}
;
A =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
a(t) =
⎛
⎜
⎜
⎝
t 0 0 0
0 t 0 0
0 0 t −1 0
0 0 0 t −1
⎞
⎟
⎟
⎠
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
; N =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
n(t) =
⎛
⎜
⎜
⎝
1
0
00
0
1
00
y 0 + y 2 y 1 1 0
y 1 y 0 − y 2 0 1
⎞
⎟
⎟
⎠
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
.
Then P = MAN is a maximal proper parabolic subgroup of Sp(2, R).
3 Algebraic Results
We define commutative algebraic extensions of D(p) and D(g) as [3–6]:
D(p) =
a + bY
a, b ∈ D(p)
, where Y commutes with all elements of D(p)
and satisfies the equation Y 2 = P 2 ;
D(g) =
a + b
Y + c
Y 2 + d
Y 3
a, b, c, d ∈