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A left-invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m0629001.png" />-form on a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m0629002.png" />, i.e. a differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m0629003.png" /> of degree 1 on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m0629004.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m0629005.png" /> for any left translation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m0629006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m0629007.png" />. The Maurer–Cartan forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m0629008.png" /> are in one-to-one correspondence with the linear forms on the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m0629009.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290010.png" />; specifically, the mapping which sends each Maurer–Cartan form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290011.png" /> to its value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290012.png" /> is an isomorphism of the space of Maurer–Cartan forms onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290013.png" />. The differential of a Maurer–Cartan form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290014.png" /> is a left-invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290015.png" />-form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290016.png" />, defined by the formula
+
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$#C+1 = 50 : ~/encyclopedia/old_files/data/M062/M.0602900 Maurer\ANDCartan form
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290018.png" /> are arbitrary left-invariant vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290019.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290020.png" /> is a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290021.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290023.png" />, be Maurer–Cartan forms such that
+
A left-invariant $  1 $-
 +
form on a Lie group  $  G $,
 +
i.e. a differential form  $  \omega $
 +
of degree 1 on  $  G $
 +
satisfying the condition  $  l _ {g}  ^  \star  \omega = \omega $
 +
for any left translation  $  l _ {g} : x \rightarrow gx $,
 +
$  g, x \in G $.  
 +
The Maurer–Cartan forms on  $  G $
 +
are in one-to-one correspondence with the linear forms on the tangent space  $  T _ {e} ( G) $
 +
at the point  $  e $;
 +
specifically, the mapping which sends each Maurer–Cartan form  $  \omega $
 +
to its value  $  \omega _ {e} \in T _ {e} ( G)  ^  \star  $
 +
is an isomorphism of the space of Maurer–Cartan forms onto  $  T _ {e} ( G)  ^  \star  $.
 +
The differential of a Maurer–Cartan form  $  \omega $
 +
is a left-invariant  $  2 $-
 +
form on  $  G $,
 +
defined by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290024.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
d \omega ( X, Y)  = - \omega ([ X, Y]),
 +
$$
 +
 
 +
where  $  X, Y $
 +
are arbitrary left-invariant vector fields on  $  G $.
 +
Suppose that  $  X _ {1} \dots X _ {n} $
 +
is a basis in  $  T _ {e} ( G) $
 +
and let  $  \omega _ {i} $,
 +
$  i = 1 \dots n $,
 +
be Maurer–Cartan forms such that
 +
 
 +
$$
 +
( \omega _ {i} ) _ {e} ( X _ {j} )  = \delta _ {ij} ,\ \
 +
j = 1 \dots n.
 +
$$
  
 
Then
 
Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
d \omega_{i}  = - \sum_{j,k=1} ^ { n }  c_{jk}  ^ {i} \omega_{j} \wedge \omega_k ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290026.png" /> are the structure constants of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290028.png" /> consisting of the left-invariant vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290029.png" />, with respect to the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290030.png" /> determined by
+
where $  c _ {jk}  ^ {i} $
 +
are the structure constants of the Lie algebra $  \mathfrak g $
 +
of $  G $
 +
consisting of the left-invariant vector fields on $  G $,  
 +
with respect to the basis $  \widetilde{X}  _ {1} \dots \widetilde{X}  _ {n} $
 +
determined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290031.png" /></td> </tr></table>
+
$$
 +
( \widetilde{X}  _ {i} ) _ {e}  = X _ {i} ,\ \
 +
i = 1 \dots n.
 +
$$
  
The equalities (2) (or (1)) are called the Maurer–Cartan equations. They were first obtained (in a different, yet equivalent form) by L. Maurer [[#References|[1]]]. The forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290032.png" /> were introduced by E. Cartan in 1904 (see [[#References|[2]]]).
+
The equalities (2) (or (1)) are called the Maurer–Cartan equations. They were first obtained (in a different, yet equivalent form) by L. Maurer [[#References|[1]]]. The forms $  \omega _ {i} $
 +
were introduced by E. Cartan in 1904 (see [[#References|[2]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290033.png" /> be the canonical coordinates in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290034.png" /> determined by the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290035.png" />. Then the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290036.png" /> are written in the form
+
Let $  x _ {1} \dots x _ {n} $
 +
be the canonical coordinates in a neighbourhood of the point $  e \in G $
 +
determined by the basis $  X _ {1} \dots X _ {n} $.  
 +
Then the forms $  \omega _ {i} $
 +
are written in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290037.png" /></td> </tr></table>
+
$$
 +
\omega _ {i}  = \sum _ {j=1} ^ { n }  A _ {ij} ( x _ {1} \dots x _ {n} )  dx _ {j} ,
 +
$$
  
 
in which the matrix
 
in which the matrix
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290038.png" /></td> </tr></table>
+
$$
 +
A( x _ {1} \dots x _ {n} )  = \
 +
( A _ {ij} ( x _ {1} \dots x _ {n} ))
 +
$$
  
 
is calculated by the formula
 
is calculated by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290039.png" /></td> </tr></table>
+
$$
 +
A( x _ {1} \dots x _ {n} )  = \
 +
 
 +
\frac{1- e ^ {- \mathop{\rm ad}  X } }{ \mathop{\rm ad}  X }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290041.png" /> is the adjoint representation of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290042.png" />.
+
where $  X = \sum_{i=1}^ {n} x _ {i} \widetilde{X}  _ {i} $
 +
and $  \mathop{\rm ad} $
 +
is the adjoint representation of the Lie algebra $  \mathfrak g $.
  
Furthermore, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290043.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290044.png" />-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290045.png" />-form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290046.png" /> which assigns to each tangent vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290047.png" /> the unique left-invariant vector field containing this vector (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290048.png" /> is called the canonical left differential form). Then
+
Furthermore, let $  \theta $
 +
be the $  \mathfrak g $-
 +
valued $  1 $-
 +
form on $  G $
 +
which assigns to each tangent vector to $  G $
 +
the unique left-invariant vector field containing this vector ( $  \theta $
 +
is called the canonical left differential form). Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290049.png" /></td> </tr></table>
+
$$
 +
\theta  = \sum_{i=1}^ { n }  \widetilde{X}  _ {i} \omega _ {i}  $$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062900/m06290050.png" /></td> </tr></table>
+
$$
 +
d \theta +
 +
\frac{1}{2}
 +
[ \theta , \theta ]  = 0,
 +
$$
  
 
which is yet another way of writing the Maurer–Cartan equations.
 
which is yet another way of writing the Maurer–Cartan equations.

Latest revision as of 08:11, 9 January 2024


A left-invariant $ 1 $- form on a Lie group $ G $, i.e. a differential form $ \omega $ of degree 1 on $ G $ satisfying the condition $ l _ {g} ^ \star \omega = \omega $ for any left translation $ l _ {g} : x \rightarrow gx $, $ g, x \in G $. The Maurer–Cartan forms on $ G $ are in one-to-one correspondence with the linear forms on the tangent space $ T _ {e} ( G) $ at the point $ e $; specifically, the mapping which sends each Maurer–Cartan form $ \omega $ to its value $ \omega _ {e} \in T _ {e} ( G) ^ \star $ is an isomorphism of the space of Maurer–Cartan forms onto $ T _ {e} ( G) ^ \star $. The differential of a Maurer–Cartan form $ \omega $ is a left-invariant $ 2 $- form on $ G $, defined by the formula

$$ \tag{1 } d \omega ( X, Y) = - \omega ([ X, Y]), $$

where $ X, Y $ are arbitrary left-invariant vector fields on $ G $. Suppose that $ X _ {1} \dots X _ {n} $ is a basis in $ T _ {e} ( G) $ and let $ \omega _ {i} $, $ i = 1 \dots n $, be Maurer–Cartan forms such that

$$ ( \omega _ {i} ) _ {e} ( X _ {j} ) = \delta _ {ij} ,\ \ j = 1 \dots n. $$

Then

$$ \tag{2 } d \omega_{i} = - \sum_{j,k=1} ^ { n } c_{jk} ^ {i} \omega_{j} \wedge \omega_k , $$

where $ c _ {jk} ^ {i} $ are the structure constants of the Lie algebra $ \mathfrak g $ of $ G $ consisting of the left-invariant vector fields on $ G $, with respect to the basis $ \widetilde{X} _ {1} \dots \widetilde{X} _ {n} $ determined by

$$ ( \widetilde{X} _ {i} ) _ {e} = X _ {i} ,\ \ i = 1 \dots n. $$

The equalities (2) (or (1)) are called the Maurer–Cartan equations. They were first obtained (in a different, yet equivalent form) by L. Maurer [1]. The forms $ \omega _ {i} $ were introduced by E. Cartan in 1904 (see [2]).

Let $ x _ {1} \dots x _ {n} $ be the canonical coordinates in a neighbourhood of the point $ e \in G $ determined by the basis $ X _ {1} \dots X _ {n} $. Then the forms $ \omega _ {i} $ are written in the form

$$ \omega _ {i} = \sum _ {j=1} ^ { n } A _ {ij} ( x _ {1} \dots x _ {n} ) dx _ {j} , $$

in which the matrix

$$ A( x _ {1} \dots x _ {n} ) = \ ( A _ {ij} ( x _ {1} \dots x _ {n} )) $$

is calculated by the formula

$$ A( x _ {1} \dots x _ {n} ) = \ \frac{1- e ^ {- \mathop{\rm ad} X } }{ \mathop{\rm ad} X } , $$

where $ X = \sum_{i=1}^ {n} x _ {i} \widetilde{X} _ {i} $ and $ \mathop{\rm ad} $ is the adjoint representation of the Lie algebra $ \mathfrak g $.

Furthermore, let $ \theta $ be the $ \mathfrak g $- valued $ 1 $- form on $ G $ which assigns to each tangent vector to $ G $ the unique left-invariant vector field containing this vector ( $ \theta $ is called the canonical left differential form). Then

$$ \theta = \sum_{i=1}^ { n } \widetilde{X} _ {i} \omega _ {i} $$

and

$$ d \theta + \frac{1}{2} [ \theta , \theta ] = 0, $$

which is yet another way of writing the Maurer–Cartan equations.

References

[1] L. Maurer, Sitzungsber. Bayer. Akad. Wiss. Math. Phys. Kl. (München) , 18 (1879) pp. 103–150
[2] E. Cartan, "Sur la structure des groupes infinis de transformations" Ann. Ecole Norm. , 21 (1904) pp. 153–206
[3] C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946)
[4] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)
[5] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
How to Cite This Entry:
Maurer-Cartan form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maurer-Cartan_form&oldid=17716
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article