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Cartan's theorem on the highest weight vector. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c0205701.png" /> be a complex semi-simple Lie algebra, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c0205702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c0205703.png" />, be canonical generators of it, that is, linearly-independent generators for which the following relations hold:
+
{{TEX|done}}
 
+
Cartan's theorem on the highest weight vector. Let $  \mathfrak g $
<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/c/c020/c020570/c0205704.png" /></td> </tr></table>
+
be a complex semi-simple Lie algebra, let $  e _{i} ,\  f _{i} ,\  h _{i} $ ,  
 
+
$  i = 1 \dots r $ ,  
<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/c/c020/c020570/c0205705.png" /></td> </tr></table>
+
be canonical generators of it, that is, linearly-independent generators for which the following relations hold: $$
 
+
[ e _{i} ,\  f _{j} ]  = 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c0205706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c0205707.png" /> are non-positive integers when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c0205708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c0205709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057010.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057011.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057012.png" /> be the Cartan subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057013.png" /> which is the linear span of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057014.png" />. Also let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057015.png" /> be a linear representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057016.png" /> in a complex finite-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057017.png" />. Then there exists a non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057018.png" /> for which
+
\delta _{ij} h _{i} , 
 
+
[ h _{i} ,\  e _{j} ]  =
<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/c/c020/c020570/c02057019.png" /></td> </tr></table>
+
a _{ij} e _{j} , 
 
+
[ h _{i} ,\  f _{j} ]  =
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057020.png" /> are certain numbers. This theorem was established by E. Cartan [[#References|[1]]]. The vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057021.png" /> is called the highest weight vector of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057022.png" /> and the linear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057024.png" /> defined by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057026.png" />, is called the highest weight of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057027.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057028.png" />. The ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057029.png" /> is called the set of numerical marks of the highest weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057030.png" />. Cartan's theorem gives a complete classification of irreducible finite-dimensional linear representations of a complex semi-simple finite-dimensional Lie algebra. It asserts that each finite-dimensional complex irreducible representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057031.png" /> has a unique highest weight vector (up to proportionality), and that the numerical marks of the corresponding highest weight are non-negative integers. Two finite-dimensional irreducible representations are equivalent if and only if the corresponding highest weights are the same. Any set of non-negative integers is the set of numerical marks of the highest weight of some finite-dimensional complex irreducible representation.
+
- a _{ij} f _{j} ,
 +
$$
 +
$$
 +
[ h _{i} ,\  h _{j} ]  =   0 ,
 +
$$
 +
where $  a _{ii} = 2 $ ,  
 +
$  a _{ij} $
 +
are non-positive integers when $  i \neq j $ ,
 +
$  i ,\  j = 1 \dots r $ ,  
 +
$  a _{ij} = 0 $
 +
implies $  a _{ji} = 0 $ ,  
 +
and let $  \mathfrak t $
 +
be the Cartan subalgebra of $  \mathfrak g $
 +
which is the [[linear span]] of $  h _{1} \dots h _{r} $ .  
 +
Also let $  \rho $
 +
be a linear representation of $  \mathfrak g $
 +
in a complex finite-dimensional space $  V $ .  
 +
Then there exists a non-zero vector $  v \in V $
 +
for which $$
 +
\rho ( e _{i} ) v  =  0 , 
 +
\rho ( h _{i} ) v  =   k _{i} v , 
 +
i = 1 \dots r ,
 +
$$
 +
where the $  k _{i} $
 +
are certain numbers. This theorem was established by E. Cartan [[#References|[1]]]. The vector $  v $
 +
is called the highest weight vector of the representation $  \rho $
 +
and the linear function $  \Lambda $
 +
on $  \mathfrak t $
 +
defined by the condition $  \Lambda ( h _{i} ) = k _{i} $ ,  
 +
$  i = 1 \dots r $ ,  
 +
is called the highest weight of the representation $  \rho $
 +
corresponding to $  v $ .  
 +
The ordered set $  ( k _{1} \dots k _{r} ) $
 +
is called the set of numerical marks of the highest weight $  \Lambda $ .  
 +
Cartan's theorem gives a complete classification of irreducible finite-dimensional linear representations of a complex semi-simple finite-dimensional Lie algebra. It asserts that each finite-dimensional complex irreducible representation of $  \mathfrak g $
 +
has a unique highest weight vector (up to proportionality), and that the numerical marks of the corresponding highest weight are non-negative integers. Two finite-dimensional irreducible representations are equivalent if and only if the corresponding highest weights are the same. Any set of non-negative integers is the set of numerical marks of the highest weight of some finite-dimensional complex irreducible representation.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Cartan,   "Les tenseurs irréductibles et les groupes linéaires simples et semi-simples" ''Bull. Sci. Math.'' , '''49''' (1925) pp. 130–152 {{MR|}} {{ZBL|51.0322.01}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Jacobson,   "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) {{MR|0148716}} {{MR|0143793}} {{ZBL|0121.27504}} {{ZBL|0109.26201}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> , ''Theórie des algèbres de Lie. Topologie des groupes de Lie'' , ''Sem. S. Lie'' , '''Ie année 1954–1955''' , Ecole Norm. Sup. (1955) {{MR|}} {{ZBL|0068.02102}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D.P. Zhelobenko,   "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) {{MR|0473097}} {{MR|0473098}} {{ZBL|0228.22013}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J. Dixmier,   "Enveloping algebras" , North-Holland (1977) (Translated from French) {{MR|0498737}} {{MR|0498740}} {{MR|0498742}} {{ZBL|0346.17010}} {{ZBL|0339.17007}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A. Borel (ed.) R. Carter (ed.) C.W. Curtis (ed.) N. Iwahori (ed.) T.A. Springer (ed.) R. Steinberg (ed.) , ''Seminar on algebraic groups and related finite groups'' , ''Lect. notes in math.'' , '''131''' , Springer (1970)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Cartan, "Les tenseurs irréductibles et les groupes linéaires simples et semi-simples" ''Bull. Sci. Math.'' , '''49''' (1925) pp. 130–152 {{MR|}} {{ZBL|51.0322.01}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) {{MR|0148716}} {{MR|0143793}} {{ZBL|0121.27504}} {{ZBL|0109.26201}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> , ''Theórie des algèbres de Lie. Topologie des groupes de Lie'' , ''Sem. S. Lie'' , '''Ie année 1954–1955''' , Ecole Norm. Sup. (1955) {{MR|}} {{ZBL|0068.02102}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) {{MR|0473097}} {{MR|0473098}} {{ZBL|0228.22013}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J. Dixmier, "Enveloping algebras" , North-Holland (1977) (Translated from French) {{MR|0498737}} {{MR|0498740}} {{MR|0498742}} {{ZBL|0346.17010}} {{ZBL|0339.17007}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A. Borel (ed.) R. Carter (ed.) C.W. Curtis (ed.) N. Iwahori (ed.) T.A. Springer (ed.) R. Steinberg (ed.) , ''Seminar on algebraic groups and related finite groups'' , ''Lect. notes in math.'' , '''131''' , Springer (1970) {{MR|}} {{ZBL|0192.36201}} </TD></TR></table>
  
  
Line 20: Line 55:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.E. Humphreys,   "Introduction to Lie algebras and representation theory" , Springer (1972) {{MR|0323842}} {{ZBL|0254.17004}} </TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) {{MR|0323842}} {{ZBL|0254.17004}} </TD></TR></table>
 
 
Cartan's theorem in the theory of functions of several complex variables. These are the so-called theorems A and B on coherent analytic sheaves on Stein manifolds, first proved by H. Cartan [[#References|[1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057032.png" /> be the sheaf of germs of holomorphic functions on a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057033.png" />. A sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057035.png" />-modules on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057036.png" /> is called a coherent analytic sheaf if there exists in a neighbourhood of each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057037.png" /> an exact sequence of sheaves
 
 
 
<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/c/c020/c020570/c02057038.png" /></td> </tr></table>
 
 
 
for some natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057039.png" />. Examples are all locally finitely-generated subsheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057040.png" />.
 
 
 
Theorem A. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057041.png" /> be a coherent analytic sheaf on a [[Stein manifold|Stein manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057042.png" />. Then there exists for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057043.png" /> a finite number of global sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057045.png" /> such that any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057046.png" /> of the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057047.png" /> is representable 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/c/c020/c020570/c02057048.png" /></td> </tr></table>
 
 
 
with all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057049.png" />. (In other words, locally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057050.png" /> is finitely generated over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057051.png" /> by its global sections.)
 
 
 
Theorem B. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057052.png" /> be a coherent analytic sheaf on a Stein manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057053.png" />. Then all cohomology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057054.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057055.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057056.png" /> are trivial:
 
 
 
<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/c/c020/c020570/c02057057.png" /></td> </tr></table>
 
 
 
These Cartan theorems have many applications. From Theorem A, various theorems can be obtained on the existence of global analytic objects on Stein manifolds. The main corollary of Theorem B is the solvability of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057059.png" />-problem: On a Stein manifold, the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057060.png" /> with the compatibility condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057061.png" /> is always solvable.
 
 
 
The scheme of application of Theorem B is as follows: If
 
 
 
<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/c/c020/c020570/c02057062.png" /></td> </tr></table>
 
 
 
is an exact sequence of sheaves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057063.png" />, then the sequence
 
  
<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/c/c020/c020570/c02057064.png" /></td> </tr></table>
+
Cartan's theorem in the theory of functions of several complex variables. These are the so-called theorems A and B on coherent analytic sheaves on Stein manifolds, first proved by H. Cartan [[#References|[1]]]. Let  $  {\mathcal O} $
 +
be the sheaf of germs of holomorphic functions on a complex manifold  $  X $ .
 +
A sheaf  $  {\mathcal S} $
 +
of  $  {\mathcal O} $ -
 +
modules on  $  X $
 +
is called a coherent analytic sheaf if there exists in a neighbourhood of each point  $  x \in X $
 +
an exact sequence of sheaves $$
 +
{\mathcal O} ^{p}  \rightarrow 
 +
{\mathcal O} ^{q}  \rightarrow 
 +
{\mathcal S}  \rightarrow  0
 +
$$
 +
for some natural numbers  $  p ,\  q $ .  
 +
Examples are all locally finitely-generated subsheaves of  $  {\mathcal O} ^{p} $ .
  
<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/c/c020/c020570/c02057065.png" /></td> </tr></table>
 
  
is also exact. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057066.png" /> is a Stein manifold, then
+
Theorem A. Let  $  {\mathcal S} $
 +
be a coherent analytic sheaf on a [[Stein manifold|Stein manifold]]  $  X $ .
 +
Then there exists for each point  $  x \in X $
 +
a finite number of global sections  $  s _{1} \dots s _{N} $
 +
of  $  {\mathcal S} $
 +
such that any element  $  s $
 +
of the fibre  $  {\mathcal S} _{x} $
 +
is representable in the form $$
 +
s  =  h _{1} ( s _{1} ) _{x} + \dots + h _{N} ( s _{N} ) _{x} ,
 +
$$
 +
with all  $  h _{j} \in {\mathcal O} _{x} $ .
 +
(In other words, locally  $  {\mathcal S} $
 +
is finitely generated over  $  {\mathcal O} $
 +
by its global sections.)
  
<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/c/c020/c020570/c02057067.png" /></td> </tr></table>
+
Theorem B. Let  $  {\mathcal S} $
 +
be a coherent analytic sheaf on a Stein manifold  $  X $ .
 +
Then all cohomology groups of  $  X $
 +
of order  $  p \geq 1 $
 +
with coefficients in  $  {\mathcal S} $
 +
are trivial: $$
 +
H ^{p} ( X ,\  {\mathcal S} )  =
 +
0    \textrm{ for }  p \geq 1 .
 +
$$
 +
These Cartan theorems have many applications. From Theorem A, various theorems can be obtained on the existence of global analytic objects on Stein manifolds. The main corollary of Theorem B is the solvability of the  $  \overline \partial  $ -
 +
problem: On a Stein manifold, the equation  $  \overline \partial  $
 +
with the compatibility condition  $  \overline \partial  f = g $
 +
is always solvable.
  
and hence, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057068.png" /> is mapping onto and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057070.png" />, are isomorphisms.
+
The scheme of application of Theorem B is as follows: If $  \overline \partial  g = 0 $
 +
is an exact sequence of sheaves on  $$
 +
0  \rightarrow  {\mathcal S}  \rightarrow 
 +
F  \rightarrow  G  \rightarrow  0,
 +
$$
 +
then the sequence $  X $
 +
$$
 +
\dots \rightarrow  H ^{p} ( X ,\  {\mathcal S} )  \rightarrow 
 +
H ^{p} ( X ,\  F \  )    \stackrel{ {\phi _{p}}} \rightarrow   
 +
H ^{p} ( X ,\  G )  \rightarrow
 +
$$
 +
is also exact. If  $$
 +
\rightarrow 
 +
H ^{p+1} ( X ,\  {\mathcal S} )  \rightarrow \dots
 +
$$
 +
is a Stein manifold, then $  X $
 +
and hence, $$
 +
H ^{p} ( X ,\  {\mathcal S} )  =   0 ,  p \geq 1 ,
 +
$$
 +
is mapping onto and the $  \phi _{0} $ ,  
 +
$  \phi _{p} $ ,  
 +
are isomorphisms.
  
Theorem B is best possible: If on a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057071.png" /> the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057072.png" /> for every coherent analytic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057073.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020570/c02057074.png" /> is a Stein manifold. Theorems A and B together with their numerous corollaries constitute the so-called Oka–Cartan theory of Stein manifolds. A corollary of these theorems is the solvability on Stein manifolds of all the classical problems of multi-dimensional complex analysis, such as the Cousin problem, the [[Levi problem|Levi problem]], the Poincaré problem and others. Theorems A and B generalize verbatim to Stein spaces (cf. [[Stein space|Stein space]]).
+
Theorem B is best possible: If on a complex manifold $  p \geq 1 $
 +
the group $  X $
 +
for every coherent analytic sheaf $  H ^{1} ( X ,\  {\mathcal S} ) = 0 $ ,  
 +
then $  {\mathcal S} $
 +
is a Stein manifold. Theorems A and B together with their numerous corollaries constitute the so-called Oka–Cartan theory of Stein manifolds. A corollary of these theorems is the solvability on Stein manifolds of all the classical problems of multi-dimensional complex analysis, such as the Cousin problem, the [[Levi problem|Levi problem]], the Poincaré problem and others. Theorems A and B generalize verbatim to Stein spaces (cf. [[Stein space|Stein space]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Cartan,   "Variétés analytiques complexes et cohomologie" R. Remmert (ed.) J.-P. Serre (ed.) , ''Collected works'' , Springer (1979) pp. 669–683 {{MR|0064154}} {{ZBL|0053.05301}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R.C. Gunning,   H. Rossi,   "Analytic functions of several complex variables" , Prentice-Hall (1965) {{MR|0180696}} {{ZBL|0141.08601}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L. Hörmander,   "An introduction to complex analysis in several variables" , North-Holland (1973) {{MR|0344507}} {{ZBL|0271.32001}} </TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Cartan, "Variétés analytiques complexes et cohomologie" R. Remmert (ed.) J.-P. Serre (ed.) , ''Collected works'' , Springer (1979) pp. 669–683 {{MR|0064154}} {{ZBL|0053.05301}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) {{MR|0180696}} {{ZBL|0141.08601}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) {{MR|0344507}} {{ZBL|0271.32001}} </TD></TR></table>
  
 
''E.M. Chirka''
 
''E.M. Chirka''
Line 71: Line 143:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.M. [G.M. Khenkin] Henkin,   J. Leiterer,   "Theory of functions on complex manifolds" , Birkhäuser (1984) (Translated from Russian) {{MR|0795028}} {{MR|0774049}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Grauert,   R. Remmert,   "Theory of Stein spaces" , Springer (1977) (Translated from German) {{MR|0513229}} {{ZBL|0379.32001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S.G. Krantz,   "Function theory of several complex variables" , Wiley (1982) pp. Sect. 7.1 {{MR|0635928}} {{ZBL|0471.32008}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> R.M. Range,   "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. VI, Par. 6 {{MR|0847923}} {{ZBL|}} </TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> G.M. [G.M. Khenkin] Henkin, J. Leiterer, "Theory of functions on complex manifolds" , Birkhäuser (1984) (Translated from Russian) {{MR|0795028}} {{MR|0774049}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1977) (Translated from German) {{MR|0513229}} {{ZBL|0379.32001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Sect. 7.1 {{MR|0635928}} {{ZBL|0471.32008}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. VI, Par. 6 {{MR|0847923}} {{ZBL|}} </TD></TR></table>

Latest revision as of 20:04, 27 February 2021

Cartan's theorem on the highest weight vector. Let $ \mathfrak g $ be a complex semi-simple Lie algebra, let $ e _{i} ,\ f _{i} ,\ h _{i} $ , $ i = 1 \dots r $ , be canonical generators of it, that is, linearly-independent generators for which the following relations hold: $$ [ e _{i} ,\ f _{j} ] = \delta _{ij} h _{i} , [ h _{i} ,\ e _{j} ] = a _{ij} e _{j} , [ h _{i} ,\ f _{j} ] = - a _{ij} f _{j} , $$ $$ [ h _{i} ,\ h _{j} ] = 0 , $$ where $ a _{ii} = 2 $ , $ a _{ij} $ are non-positive integers when $ i \neq j $ , $ i ,\ j = 1 \dots r $ , $ a _{ij} = 0 $ implies $ a _{ji} = 0 $ , and let $ \mathfrak t $ be the Cartan subalgebra of $ \mathfrak g $ which is the linear span of $ h _{1} \dots h _{r} $ . Also let $ \rho $ be a linear representation of $ \mathfrak g $ in a complex finite-dimensional space $ V $ . Then there exists a non-zero vector $ v \in V $ for which $$ \rho ( e _{i} ) v = 0 , \rho ( h _{i} ) v = k _{i} v , i = 1 \dots r , $$ where the $ k _{i} $ are certain numbers. This theorem was established by E. Cartan [1]. The vector $ v $ is called the highest weight vector of the representation $ \rho $ and the linear function $ \Lambda $ on $ \mathfrak t $ defined by the condition $ \Lambda ( h _{i} ) = k _{i} $ , $ i = 1 \dots r $ , is called the highest weight of the representation $ \rho $ corresponding to $ v $ . The ordered set $ ( k _{1} \dots k _{r} ) $ is called the set of numerical marks of the highest weight $ \Lambda $ . Cartan's theorem gives a complete classification of irreducible finite-dimensional linear representations of a complex semi-simple finite-dimensional Lie algebra. It asserts that each finite-dimensional complex irreducible representation of $ \mathfrak g $ has a unique highest weight vector (up to proportionality), and that the numerical marks of the corresponding highest weight are non-negative integers. Two finite-dimensional irreducible representations are equivalent if and only if the corresponding highest weights are the same. Any set of non-negative integers is the set of numerical marks of the highest weight of some finite-dimensional complex irreducible representation.

References

[1] E. Cartan, "Les tenseurs irréductibles et les groupes linéaires simples et semi-simples" Bull. Sci. Math. , 49 (1925) pp. 130–152 Zbl 51.0322.01
[2] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) MR0148716 MR0143793 Zbl 0121.27504 Zbl 0109.26201
[3] , Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Ecole Norm. Sup. (1955) Zbl 0068.02102
[4] D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013
[5] J. Dixmier, "Enveloping algebras" , North-Holland (1977) (Translated from French) MR0498737 MR0498740 MR0498742 Zbl 0346.17010 Zbl 0339.17007
[6] A. Borel (ed.) R. Carter (ed.) C.W. Curtis (ed.) N. Iwahori (ed.) T.A. Springer (ed.) R. Steinberg (ed.) , Seminar on algebraic groups and related finite groups , Lect. notes in math. , 131 , Springer (1970) Zbl 0192.36201


Comments

References

[a1] J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) MR0323842 Zbl 0254.17004

Cartan's theorem in the theory of functions of several complex variables. These are the so-called theorems A and B on coherent analytic sheaves on Stein manifolds, first proved by H. Cartan [1]. Let $ {\mathcal O} $ be the sheaf of germs of holomorphic functions on a complex manifold $ X $ . A sheaf $ {\mathcal S} $ of $ {\mathcal O} $ - modules on $ X $ is called a coherent analytic sheaf if there exists in a neighbourhood of each point $ x \in X $ an exact sequence of sheaves $$ {\mathcal O} ^{p} \rightarrow {\mathcal O} ^{q} \rightarrow {\mathcal S} \rightarrow 0 $$ for some natural numbers $ p ,\ q $ . Examples are all locally finitely-generated subsheaves of $ {\mathcal O} ^{p} $ .


Theorem A. Let $ {\mathcal S} $ be a coherent analytic sheaf on a Stein manifold $ X $ . Then there exists for each point $ x \in X $ a finite number of global sections $ s _{1} \dots s _{N} $ of $ {\mathcal S} $ such that any element $ s $ of the fibre $ {\mathcal S} _{x} $ is representable in the form $$ s = h _{1} ( s _{1} ) _{x} + \dots + h _{N} ( s _{N} ) _{x} , $$ with all $ h _{j} \in {\mathcal O} _{x} $ . (In other words, locally $ {\mathcal S} $ is finitely generated over $ {\mathcal O} $ by its global sections.)

Theorem B. Let $ {\mathcal S} $ be a coherent analytic sheaf on a Stein manifold $ X $ . Then all cohomology groups of $ X $ of order $ p \geq 1 $ with coefficients in $ {\mathcal S} $ are trivial: $$ H ^{p} ( X ,\ {\mathcal S} ) = 0 \textrm{ for } p \geq 1 . $$ These Cartan theorems have many applications. From Theorem A, various theorems can be obtained on the existence of global analytic objects on Stein manifolds. The main corollary of Theorem B is the solvability of the $ \overline \partial $ - problem: On a Stein manifold, the equation $ \overline \partial $ with the compatibility condition $ \overline \partial f = g $ is always solvable.

The scheme of application of Theorem B is as follows: If $ \overline \partial g = 0 $ is an exact sequence of sheaves on $$ 0 \rightarrow {\mathcal S} \rightarrow F \rightarrow G \rightarrow 0, $$ then the sequence $ X $ $$ \dots \rightarrow H ^{p} ( X ,\ {\mathcal S} ) \rightarrow H ^{p} ( X ,\ F \ ) \stackrel{ {\phi _{p}}} \rightarrow H ^{p} ( X ,\ G ) \rightarrow $$ is also exact. If $$ \rightarrow H ^{p+1} ( X ,\ {\mathcal S} ) \rightarrow \dots $$ is a Stein manifold, then $ X $ and hence, $$ H ^{p} ( X ,\ {\mathcal S} ) = 0 , p \geq 1 , $$ is mapping onto and the $ \phi _{0} $ , $ \phi _{p} $ , are isomorphisms.

Theorem B is best possible: If on a complex manifold $ p \geq 1 $ the group $ X $ for every coherent analytic sheaf $ H ^{1} ( X ,\ {\mathcal S} ) = 0 $ , then $ {\mathcal S} $ is a Stein manifold. Theorems A and B together with their numerous corollaries constitute the so-called Oka–Cartan theory of Stein manifolds. A corollary of these theorems is the solvability on Stein manifolds of all the classical problems of multi-dimensional complex analysis, such as the Cousin problem, the Levi problem, the Poincaré problem and others. Theorems A and B generalize verbatim to Stein spaces (cf. Stein space).

References

[1] H. Cartan, "Variétés analytiques complexes et cohomologie" R. Remmert (ed.) J.-P. Serre (ed.) , Collected works , Springer (1979) pp. 669–683 MR0064154 Zbl 0053.05301
[2] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) MR0180696 Zbl 0141.08601
[3] L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) MR0344507 Zbl 0271.32001

E.M. Chirka

Comments

In [a1] the theory related to Cartan's Theorems A and B is developed on the basis of integral representations, and not on the basis of sheaves, as in [2] or [a2], or on the basis of the Cauchy–Riemann equations, as in [3].

Generalizations to Stein manifolds are in [a2].

See also Cousin problems. For the Poincaré problem (on meromorphic functions), cf. Stein space and Meromorphic function.

References

[a1] G.M. [G.M. Khenkin] Henkin, J. Leiterer, "Theory of functions on complex manifolds" , Birkhäuser (1984) (Translated from Russian) MR0795028 MR0774049
[a2] H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1977) (Translated from German) MR0513229 Zbl 0379.32001
[a3] S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Sect. 7.1 MR0635928 Zbl 0471.32008
[a4] R.M. Range, "Holomorphic functions and integral representation in several complex variables" , Springer (1986) pp. Chapt. VI, Par. 6 MR0847923
How to Cite This Entry:
Cartan theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cartan_theorem&oldid=21822
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article