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In [[mathematics]], in the field of [[representation theory]], the '''Borel–Weil theorem''', named after [[Armand Borel]] and [[André Weil]], provides a concrete model for [[irreducible representation]]s of [[compact Lie group]]s and [[complex numbers|complex]] [[semisimple Lie group]]s. These representations are realized in the spaces of global [[Section (fiber bundle)|sections]] of [[holomorphic line bundle]]s on the [[flag manifold]] of the group. Its generalization to higher cohomology spaces is called the [[Borel–Weil–Bott theorem]].
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== Statement of the theorem ==
The theorem can be stated either for a complex semisimple Lie group ''G'' or for its compact form ''K''. Let ''G'' be a [[connected space|connected]] complex semisimple Lie group, ''B'' a [[Borel subgroup]] of ''G'', and ''X''=''G''/''B'' the [[flag variety]]. In this picture, ''X'' is a complex manifold and a nonsingular algebraic ''G''-variety. The flag variety can also be described as a compact [[homogeneous space]] ''K''/''T'', where ''T''=''K''&cap;''B'' is a (compact) [[Cartan subgroup]] of ''K''. An [[Weight_(representation_theory)#Integral_weight|integral weight]] ''λ''
determines a ''G''-equivariant holomorphic line bundle ''L''<sub>''λ''</sub> on ''X'' and the group ''G'' acts on its space of global sections,
 
:<math>\Gamma(G/B,L_\lambda).\ </math>
 
The Borel–Weil theorem states that if ''λ'' is a ''dominant'' integral weight then this representation is an irreducible [[highest weight representation]] of ''G'' with highest weight ''λ''. Its restriction to ''K'' is an irreducible unitary representation of ''K'' with highest weight ''λ'', and each irreducible unitary representations of ''K'' is obtained in this way for a unique value of ''λ''.
 
== Concrete description ==
The weight ''λ'' gives rise to a character (one-dimensional representation) of the Borel subgroup ''B'', which is denoted ''&chi;''<sub>''λ''</sub>. Holomorphic sections of the holomorphic line bundle ''L''<sub>''λ''</sub> over ''G''/''B'' may be described more concretely as [[holomorphic map]]s
 
:<math> f: G\to \mathbb{C}_{\lambda}: f(gb)=\chi_{\lambda}(b)f(g)</math>
 
for all ''g''&isin;''G'' and ''b''&isin;''B''.  
 
The action of ''G'' on these sections is given by
: <math>g\cdot f(h)=f(g^{-1}h)</math>  
 
for ''g'',''h''&isin;''G''.
 
== Example ==
Let ''G'' be the complex [[special linear group]] ''SL''(2,'''C'''), with a Borel subgroup consisting of upper triangular matrices with determinant one. Integral weights for ''G'' may be identified with [[integer]]s, with dominant weights corresponding to nonnegative integers, and the corresponding characters ''&chi;''<sub>''n''</sub> of ''B'' have the form
 
: <math> \chi_n
\begin{pmatrix}
a & b\\
0 & a^{-1}
\end{pmatrix}=a^n.
</math>
 
The flag variety ''G''/''B'' may be identified with the complex projective line '''P'''<sup>1</sup> with homogeneous coordinates ''X'', ''Y'' and the space of the global sections of the line bundle ''L''<sub>''n''</sub> is identified with the space of homogeneous polynomials of degree ''n'' on '''C'''<sup>''2''</sup>. For ''n''&ge;0, this space has dimension ''n''+1 and forms an irreducible representation under the standard action of ''G'' on the polynomial algebra '''C'''[''X'',''Y'']. Weight vectors are given by monomials
 
: <math> X^i Y^{n-i}, \quad 0\leq i\leq n </math>
 
of weights 2''i''&minus;''n'', and the highest weight vector ''X''<sup>''n''</sup> has weight ''n''.
 
==History==
The theorem dates back to the early 1950s and can be found in {{harvtxt|Serre|1995}} and {{harvtxt|Tits|1955}}.
 
==References==
*{{citation
| last = Serre | first = Jean-Pierre | authorlink = Jean-Pierre Serre
| title = Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts (d'après Armand Borel et André Weil)
| journal = Séminaire Bourbaki
| volume = 2 | issue = 100 | pages = 447–454 | publisher = Soc. Math. France | location = Paris | year = 1995}}. In French; translated title: “Linear representations and Kähler homogeneous spaces of compact Lie groups (after Armand Borel and André Weil.”
 
*{{citation
  | last = Tits | first = Jacques | authorlink = Jacques Tits
| title = Sur certaines classes d'espaces homogènes de groupes de Lie
| series = Acad. Roy. Belg. Cl. Sci. Mém. Coll.
| volume = 29 | year = 1955}} In French.
 
*{{citation
| last = Sepanski | first = Mark R.
| title = Compact Lie groups.
| series = Graduate Texts in Mathematics | volume = 235
| publisher = Springer | location = New York | year = 2007}}.
 
*{{citation
| last = Knapp | first = Anthony W.
| title = Representation theory of semisimple groups: An overview based on examples
| series = Princeton Landmarks in Mathematics
| publisher = Princeton University Press | location = Princeton, NJ | year = 2001}}. Reprint of the 1986 original.
 
{{DEFAULTSORT:Borel-Weil theorem}}
[[Category:Representation theory of Lie groups]]
[[Category:Theorems in representation theory]]

Latest revision as of 19:27, 3 January 2015

I don't generally mention myself. I take pleasure in sharing fascinating articles with many others. I also wish to check out the internet due to the fact I do think you will find a lot to discover available. When I'm on the web.
|
My name: Alta Vigna
My age: 39 years old
Country: Denmark
City: Kobenhavn K
Postal code: 1206
Street: Toftvej 59
|
My name is Alta Vigna. I life in Kobenhavn K (Denmark).
|
My name is Alta from Kobenhavn K studying Business. I did my schooling, secured 91% and hope to find someone with same interests in Surfing.
|
I like Surfing.
I to learn Danish in my free time.
|
My name is Alta (27 years old) and my hobbies are Jukskei and Poker.
|
Hello, dear friend! I am Alta. I smile that I could join to the entire globe. I live computer store in palatine Denmark, in the REGION SJALLAND region. I dream to check out the various countries, to obtain acquainted with intriguing individuals.
|
I'm Alta (23) from Kobenhavn K, Denmark.
I'm learning Danish literature at a local university and I'm just about to graduate.
I have a part time job in a the office.
|
Hi!
My name is Alta and I'm a 21 years old boy from Denmark.
|
Hi, everybody! My name is Alta.
It is a little about myself: I live in Denmark, my city of Kobenhavn K.
It's called often Eastern or cultural capital of REGION SJALLAND. I've married 3 years ago.
I have 2 children - a son (Kathaleen) and the daughter (Madison). We all like Surfing.
|
Hello!
I'm Danish female :).
I really like Surfing!
|
Hello from Denmark. I'm glad to be here. My first name is Alta.
I live in a small city called Kobenhavn K in nothern Denmark.
I was also born in Kobenhavn K 33 years ago. Married in September 2003. I'm working at the backery.
|
My name is Alta Vigna but everybody calls me Alta. I'm from Denmark. I'm studying at the high school (1st year) and I play the French Horn for 5 years. Usually I choose songs from my famous films :D.
I have two brothers. I love Programming, watching movies and Crocheting.
|
I'm a 30 years old, married and work at the high school (Business).
In my free time I learn Danish. I have been twicethere and look forward to go there sometime in the future. I like to read, preferably on my beloved Kindle. I really love to watch Arrested Development and American Dad as well as documentaries about nature. I love Surfing.
|
I'm Alta and I live in Kobenhavn K.
I'm interested in Business, Surfing and Danish art. I like to travel and watching Modern Family.
|
I'm Alta and I live with my husband and our 3 children in Kobenhavn K, in the REGION SJALLAND south part. My hobbies are Photography, Basketball and Antiquing.
|
Hello, I'm Alta, a 30 year old from Kobenhavn K, Denmark.
My hobbies include (but are not limited to) Travel, Gaming and watching Modern Family.
|
Hi there! :) My name is Alta, I'm a student studying Business from Kobenhavn K, Denmark.
|
I'm Alta and I live in a seaside city in northern Denmark, Kobenhavn K. I'm 37 and I'm will soon finish my study at Business.
|
I am Alta from Kobenhavn K. I am learning to play the French Horn. Other hobbies are Surfing.
|
I am Alta and was born on 22 September 1979. My hobbies are Air sports and Singing.
|
My name is Alta and I am studying Graduate School and Continuing Education and Summer Sessions at Kobenhavn K / Denmark.
}



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