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Ciro Santilli
@cirosantilli
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Lie algebra of
O
(
n
)
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Lie algebra of a isometry group
Updated
2025-04-24
+
Created
1970-01-01
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We
can
almost
reach the
Lie algebra
of any
isometry group
in
a
single go. For every
X
in the
Lie algebra
we
must have:
∀
v
,
w
∈
V
,
t
∈
R
(
e
tX
v
∣
e
tX
w
)
=
(
v
∣
w
)
(1)
because
e
tX
has to be in the
isometry group
by
definition
as
shown at
Section "Lie algebra of a matrix Lie group"
.
Then:
d
t
d
(
e
tX
v
∣
e
tX
w
)
0
=
0
⟹
(
X
e
tX
v
∣
e
tX
w
)
+
(
e
tX
v
∣
X
e
tX
w
)
0
=
0
⟹
(
X
v
∣
w
)
+
(
v
∣
Xw
)
=
0
(2)
so
we
reach:
∀
v
,
w
∈
V
(
X
v
∣
w
)
=
−
(
v
∣
Xw
)
(3)
With this relation,
we
can easily determine the
Lie algebra
of common
isometries
:
Lie algebra of
O
(
n
)
Bibliography:
An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee (2011)
page 151
Total
articles
:
1