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SO5

SO5, or SO(5), denotes the special orthogonal group in five dimensions. It is the group of orientation-preserving isometries of five-dimensional Euclidean space, realized as the subgroup of GL(5, R) consisting of matrices A with A^T A = I and det(A) = 1. As a Lie group, SO(5) is compact and connected, with dimension 10.

The associated Lie algebra is so(5), the set of all 5×5 real skew-symmetric matrices, and it has

The universal cover of SO(5) is Spin(5), which is isomorphic to Sp(2), the compact symplectic group of

The fundamental group satisfies π1(SO(5)) ≅ Z2, consistent with the general property π1(SO(n)) ≅ Z2 for n ≥ 3.

dimension
10.
so(5)
is
a
simple
Lie
algebra;
in
the
classification
of
complex
simple
Lie
algebras
it
corresponds
to
type
B2.
The
group
SO(5)
has
rank
2
and
a
root
system
of
type
B2.
degree
2.
Consequently
Spin(5)
≅
Sp(2)
and
SO(5)
≅
Sp(2)/{±I}.
This
reflects
the
close
relationship
between
the
two
groups:
SO(5)
and
its
double
cover
share
the
same
local
structure,
with
Spin(5)
providing
the
spinor
representations.
The
defining
(fundamental)
representation
of
SO(5)
is
5-dimensional
on
R^5,
and
there
exists
a
4-dimensional
spinor
representation
arising
from
Spin(5).
Applications
of
SO(5)
appear
in
higher-dimensional
geometry,
the
study
of
compact
Lie
groups,
and
areas
of
physics
that
involve
rotations
in
five
dimensions.