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A special case of an [[Analytic polyhedron|analytic polyhedron]]. A bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w0976101.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w0976102.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w0976103.png" /> is said to be a Weil domain if there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w0976104.png" /> functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w0976105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w0976106.png" />, holomorphic in a fixed neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w0976107.png" /> of the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w0976108.png" />, such that
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1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w0976109.png" />;
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2) the faces of the Weil domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w09761010.png" />, i.e. the sets
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A special case of an [[Analytic polyhedron|analytic polyhedron]]. A bounded domain  $  D $
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in  $  n $-
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dimensional space  $  \mathbf C  ^ {n} $
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is said to be a Weil domain if there exist  $  N \geq  n $
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functions  $  f _ {i} ( z) $,
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$  i= 1 \dots N $,  
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holomorphic in a fixed neighbourhood  $  U ( \overline{D}\; ) $
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of the closure  $  \overline{D}\; $,
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such that
  
<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/w/w097/w097610/w09761011.png" /></td> </tr></table>
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1)  $  D= \{ {z } : {| f _ {i} ( z) | < 1,  i = 1 \dots N,  z \in U ( \overline{D}\; ) } \} $;
  
have dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w09761012.png" />;
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2) the faces of the Weil domain  $  D $,
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i.e. the sets
  
3) the edges of the Weil domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w09761013.png" />, i.e. the intersections of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w09761014.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w09761015.png" />) different faces, have dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w09761016.png" />.
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$$
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\sigma _ {i}  = \{ {z \in D } : {
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| f _ {i} ( z) | = 1 ,\
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| f _ {j} ( z) | \leq  1 ,\
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j \neq i } \}
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,
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$$
  
The totality of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097610/w09761017.png" />-dimensional edges of a Weil domain is called the skeleton of the domain. The [[Bergman–Weil representation|Bergman–Weil representation]] applies to Weil domains. These domains are named for A. Weil [[#References|[1]]], who obtained the first important results for these domains.
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have dimension  $  2n - 1 $;
 +
 
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3) the edges of the Weil domain  $  D $,
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i.e. the intersections of any  $  k $(
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$  2 \leq  k \leq  n $)
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different faces, have dimension  $  \leq  2n - k $.
 +
 
 +
The totality of all $  n $-
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dimensional edges of a Weil domain is called the skeleton of the domain. The [[Bergman–Weil representation|Bergman–Weil representation]] applies to Weil domains. These domains are named for A. Weil [[#References|[1]]], who obtained the first important results for these domains.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Weil,  "L'intégrale de Cauchy et les fonctions de plusieurs variables"  ''Math. Ann.'' , '''111'''  (1935)  pp. 178–182</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Weil,  "L'intégrale de Cauchy et les fonctions de plusieurs variables"  ''Math. Ann.'' , '''111'''  (1935)  pp. 178–182</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.A. Fuks,  "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.M. [G.M. Khenkin] Henkin,  J. Leiterer,  "Theory of functions on complex manifolds" , Birkhäuser  (1984)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.A. Fuks,  "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.M. [G.M. Khenkin] Henkin,  J. Leiterer,  "Theory of functions on complex manifolds" , Birkhäuser  (1984)</TD></TR></table>

Latest revision as of 08:29, 6 June 2020


A special case of an analytic polyhedron. A bounded domain $ D $ in $ n $- dimensional space $ \mathbf C ^ {n} $ is said to be a Weil domain if there exist $ N \geq n $ functions $ f _ {i} ( z) $, $ i= 1 \dots N $, holomorphic in a fixed neighbourhood $ U ( \overline{D}\; ) $ of the closure $ \overline{D}\; $, such that

1) $ D= \{ {z } : {| f _ {i} ( z) | < 1, i = 1 \dots N, z \in U ( \overline{D}\; ) } \} $;

2) the faces of the Weil domain $ D $, i.e. the sets

$$ \sigma _ {i} = \{ {z \in D } : { | f _ {i} ( z) | = 1 ,\ | f _ {j} ( z) | \leq 1 ,\ j \neq i } \} , $$

have dimension $ 2n - 1 $;

3) the edges of the Weil domain $ D $, i.e. the intersections of any $ k $( $ 2 \leq k \leq n $) different faces, have dimension $ \leq 2n - k $.

The totality of all $ n $- dimensional edges of a Weil domain is called the skeleton of the domain. The Bergman–Weil representation applies to Weil domains. These domains are named for A. Weil [1], who obtained the first important results for these domains.

References

[1] A. Weil, "L'intégrale de Cauchy et les fonctions de plusieurs variables" Math. Ann. , 111 (1935) pp. 178–182
[2] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)
[3] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)

Comments

References

[a1] B.A. Fuks, "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1963) (Translated from Russian)
[a2] G.M. [G.M. Khenkin] Henkin, J. Leiterer, "Theory of functions on complex manifolds" , Birkhäuser (1984)
How to Cite This Entry:
Weil domain. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weil_domain&oldid=49200
This article was adapted from an original article by M. Shirinbekov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article