132
P. Moylan
For SO 0 (1, 4), its representations are described in terms of various realizations
or parallelizations [1]. The spherical, flat, and hyperbolic parallelizations are as
follows. Let KAN be the Iwasawa decomposition of SO 0 (1, 4), and let M be
the centralizer of A in K, then the “compact picture” [2] of its representations
(spherical parallelization) is given by vector-valued fields over K/M ∼ = S 3 , with
S 3 being the three sphere. The Bruhat decomposition SO 0 (1, 4) ∼ =
NMAN leads
to the non-compact picture (or flat parallelization) which describes representations
in terms of vector-valued functions on
N ∼ = R 3 . (By ∼ =
we mean isomorphic to
a dense open subset of SO 0 (1, 4).) Finally the Hannabuss decomposition, which is
SO 0 (1, 4) ∼ =
H AN, where H is SO(1, 3), gives the hyperbolic picture (hyperbolic
parallelization) describing SO 0 (1, 4) representations on SO 0 (1, 3)/M ∼ = T 3 , the
two-sheeted momentum hyperboloid. Now unitary representations of the inhomogeneous Euclidean group in four dimensions, the inhomogeneous Galilean group,
and the Poincaré group have realizations in terms of vector-valued fields on S 3 ,
R 3 , and T 3 , respectively, and the relationship via contraction and deformation of
these representations to representations of SO 0 (1, 4), at least for the case of unitary
principal series representations of SO 0 (1, 4), is very well understood. We would like
to obtain an analogous description of this situation for SO 0 (2, 3) and it is to this goal
that the current article contributes. Compared to SO 0 (1, 4) the situation with respect
to SO 0 (2, 3) is much more complicated: instead of S 3 we now have S 1 × S 2 ; there
are more Bruhat-like decompositions, so there are more non-compact pictures; and
finally there are two analogs of the Hannabuss decomposition [1].
2 SO 0 (2, 3), Sp(2, R), the Poincaré Group and Their Lie
Algebras
Let β 0 = diag(1, 1, − 1, − 1, − 1), where the right-hand side of this equation
denotes a diagonal matrix with diagonal entries as shown inside the parentheses.
SO 0 (2, 3) is the component connected to identity of the group
SO(2, 3) = {g ∈ SL(5, R) | g β 0 g
†
= β 0 }.
( † denotes transpose of a matrix.) Denote by so(2, 3) the Lie algebra of
SO 0 (2, 3). A realization of so(2, 3) is provided by the set of all matrices
(a i,j ) (−1 i, j 3) such that a i,i
=
0 (−1 i 3),
a i,j = − a j,i (1 i j 3 ), a 0,j = a j,0 (1 j 3 ),
a −1,j = a j,−1 (1 j 3 ) and a −1,0 = − a 0,−1 . Let E i,j be the matrix
such that the (i, j ) component is equal to 1 and the other components are all equal
to 0. Let L −1,0 = − E −1,0 + E 0,−1 , L i,j = E i,j − E j,i (1 i, j 3, i = j),
L 0,i = E i,0 + E 0,i (1 i 3 ), L −1,i = E i,−1 + E −1,i (1 i 3 ). The L a,b
(a, b = −1, 0, 1, 2, 3), viewed abstractly, are a basis for so(2, 3). The commutation
relations of the L a,b are
P. Moylan
For SO 0 (1, 4), its representations are described in terms of various realizations
or parallelizations [1]. The spherical, flat, and hyperbolic parallelizations are as
follows. Let KAN be the Iwasawa decomposition of SO 0 (1, 4), and let M be
the centralizer of A in K, then the “compact picture” [2] of its representations
(spherical parallelization) is given by vector-valued fields over K/M ∼ = S 3 , with
S 3 being the three sphere. The Bruhat decomposition SO 0 (1, 4) ∼ =
NMAN leads
to the non-compact picture (or flat parallelization) which describes representations
in terms of vector-valued functions on
N ∼ = R 3 . (By ∼ =
we mean isomorphic to
a dense open subset of SO 0 (1, 4).) Finally the Hannabuss decomposition, which is
SO 0 (1, 4) ∼ =
H AN, where H is SO(1, 3), gives the hyperbolic picture (hyperbolic
parallelization) describing SO 0 (1, 4) representations on SO 0 (1, 3)/M ∼ = T 3 , the
two-sheeted momentum hyperboloid. Now unitary representations of the inhomogeneous Euclidean group in four dimensions, the inhomogeneous Galilean group,
and the Poincaré group have realizations in terms of vector-valued fields on S 3 ,
R 3 , and T 3 , respectively, and the relationship via contraction and deformation of
these representations to representations of SO 0 (1, 4), at least for the case of unitary
principal series representations of SO 0 (1, 4), is very well understood. We would like
to obtain an analogous description of this situation for SO 0 (2, 3) and it is to this goal
that the current article contributes. Compared to SO 0 (1, 4) the situation with respect
to SO 0 (2, 3) is much more complicated: instead of S 3 we now have S 1 × S 2 ; there
are more Bruhat-like decompositions, so there are more non-compact pictures; and
finally there are two analogs of the Hannabuss decomposition [1].
2 SO 0 (2, 3), Sp(2, R), the Poincaré Group and Their Lie
Algebras
Let β 0 = diag(1, 1, − 1, − 1, − 1), where the right-hand side of this equation
denotes a diagonal matrix with diagonal entries as shown inside the parentheses.
SO 0 (2, 3) is the component connected to identity of the group
SO(2, 3) = {g ∈ SL(5, R) | g β 0 g
†
= β 0 }.
( † denotes transpose of a matrix.) Denote by so(2, 3) the Lie algebra of
SO 0 (2, 3). A realization of so(2, 3) is provided by the set of all matrices
(a i,j ) (−1 i, j 3) such that a i,i
=
0 (−1 i 3),
a i,j = − a j,i (1 i j 3 ), a 0,j = a j,0 (1 j 3 ),
a −1,j = a j,−1 (1 j 3 ) and a −1,0 = − a 0,−1 . Let E i,j be the matrix
such that the (i, j ) component is equal to 1 and the other components are all equal
to 0. Let L −1,0 = − E −1,0 + E 0,−1 , L i,j = E i,j − E j,i (1 i, j 3, i = j),
L 0,i = E i,0 + E 0,i (1 i 3 ), L −1,i = E i,−1 + E −1,i (1 i 3 ). The L a,b
(a, b = −1, 0, 1, 2, 3), viewed abstractly, are a basis for so(2, 3). The commutation
relations of the L a,b are