The Maurer-Cartan form is a Lie algebra valued differential form on Lie groups that is frequently used in differential geometry and mathematical physics . It is named after the German mathematician and university professor Ludwig Maurer and the French mathematician Élie Cartan .
definition
Be a Lie group, its Lie algebra . For induces the left multiplication
G
{\ displaystyle G}
G
=
T
e
G
{\ displaystyle {\ mathfrak {g}} = T_ {e} G}
G
∈
G
{\ displaystyle g \ in G}
L.
G
-
1
:
G
→
G
{\ displaystyle L_ {g ^ {- 1}}: G \ rightarrow G}
L.
G
-
1
(
H
)
: =
G
-
1
H
{\ displaystyle L_ {g ^ {- 1}} (h): = g ^ {- 1} h}
the differential
(
D.
L.
G
-
1
)
G
:
T
G
G
→
T
e
G
=
G
{\ displaystyle (DL_ {g ^ {- 1}}) _ {g}: T_ {g} G \ rightarrow T_ {e} G = {\ mathfrak {g}}}
.
The mason cartan shape is defined by
ω
∈
Ω
1
(
G
,
G
)
{\ displaystyle \ omega \ in \ Omega ^ {1} (G, {\ mathfrak {g}})}
ω
(
v
)
: =
(
D.
L.
G
-
1
)
G
(
v
)
{\ displaystyle \ omega (v): = (DL_ {g ^ {- 1}}) _ {g} (v)}
for .
v
∈
T
G
G
,
G
∈
G
{\ displaystyle v \ in T_ {g} G, g \ in G}
Mason-Cartan equation
The Maurer-Cartan form satisfies the equation
d
ω
+
1
2
[
ω
,
ω
]
=
0
{\ displaystyle d \ omega + {\ frac {1} {2}} \ left [\ omega, \ omega \ right] = 0}
.
Here the commutator is Lie algebra valued differential forms by
[
ω
∧
η
]
(
v
1
,
v
2
)
=
[
ω
(
v
1
)
,
η
(
v
2
)
]
-
[
ω
(
v
2
)
,
η
(
v
1
)
]
{\ displaystyle [\ omega \ wedge \ eta] (v_ {1}, v_ {2}) = [\ omega (v_ {1}), \ eta (v_ {2})] - [\ omega (v_ {2 }), \ eta (v_ {1})]}
and the outer derivative through
d
ω
{\ displaystyle d \ omega}
d
ω
(
X
,
Y
)
=
X
(
ω
(
Y
)
)
-
Y
(
ω
(
X
)
)
-
ω
(
[
X
,
Y
]
)
{\ displaystyle d \ omega (X, Y) = X (\ omega (Y)) - Y (\ omega (X)) - \ omega ([X, Y])}
Are defined.
Individual evidence
↑ Jeffrey M. Lee, Manifolds and differential geometry . American Mathematical Society, Providence, RI 2009, ISBN 0-8218-4815-1 , Chapter: 5.6 The Maurer Cartan Form.
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