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The unique representation of an arbitrary element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530601.png" /> of a non-compact connected semi-simple real [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530602.png" /> as a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530603.png" /> of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530604.png" /> of analytic subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530605.png" />, respectively, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530607.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530608.png" /> are defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i0530609.png" /> be a [[Cartan decomposition|Cartan decomposition]] of the [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306011.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306012.png" /> be the maximal commutative subspace of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306013.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306014.png" /> be a nilpotent Lie subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306016.png" /> is the linear hull of the root vectors in some system of positive roots with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306017.png" />. The decomposition of the Lie algebra as the direct sum of the subalgebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306020.png" /> is called the Iwasawa decomposition [[#References|[1]]] of the semi-simple real Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306021.png" />. The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306024.png" /> are defined to be the analytic subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306025.png" /> corresponding to the subalgebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306028.png" />, respectively. The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306031.png" /> are closed; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306033.png" /> are simply-connected; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306034.png" /> contains the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306035.png" />, and the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306036.png" /> under the adjoint representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306037.png" /> is a maximal compact subgroup of the adjoint group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306038.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306039.png" /> is an analytic diffeomorphism of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306040.png" /> onto the Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306041.png" />. The Iwasawa decomposition plays a fundamental part in the representation theory of semi-simple Lie groups. The Iwasawa decomposition can be defined also for connected semi-simple algebraic groups over a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306042.png" />-adic field (or, more generally, for groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306043.png" />-adic type) (see [[#References|[4]]], [[#References|[5]]]).
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The unique representation of an arbitrary element $  g $
 +
of a non-compact connected semi-simple real [[Lie group|Lie group]] $  G $
 +
as a product $  g = k an $
 +
of elements $  k,\  a,\  n $
 +
of analytic subgroups $  K,\  A,\  N $ ,  
 +
respectively, where $  K $ ,  
 +
$  A $
 +
and $  N $
 +
are defined as follows. Let $  \mathfrak g = \mathfrak k + \mathfrak P $
 +
be a [[Cartan decomposition|Cartan decomposition]] of the [[Lie algebra|Lie algebra]] $  \mathfrak g $
 +
of $  G $ ;  
 +
let $  \mathfrak a $
 +
be the maximal commutative subspace of the space $  \mathfrak P $ ,  
 +
and let $  \mathfrak N $
 +
be a nilpotent Lie subalgebra of $  \mathfrak g $
 +
such that $  \mathfrak N $
 +
is the linear hull of the root vectors in some system of positive roots with respect to $  \mathfrak a $ .  
 +
The decomposition of the Lie algebra as the direct sum of the subalgebras $  \mathfrak k $ ,  
 +
$  \mathfrak a $
 +
and $  \mathfrak N $
 +
is called the Iwasawa decomposition [[#References|[1]]] of the semi-simple real Lie algebra $  \mathfrak g $ .  
 +
The groups $  K $ ,  
 +
$  A $
 +
and $  N $
 +
are defined to be the analytic subgroups of $  G $
 +
corresponding to the subalgebras $  \mathfrak k $ ,  
 +
$  \mathfrak a $
 +
and $  \mathfrak N $ ,  
 +
respectively. The groups $  K $ ,  
 +
$  A $
 +
and $  N $
 +
are closed; $  A $
 +
and $  N $
 +
are simply-connected; $  K $
 +
contains the centre of $  G $ ,  
 +
and the image of $  K $
 +
under the adjoint representation of $  G $
 +
is a maximal compact subgroup of the adjoint group of $  G $ .  
 +
The mapping $  (k,\  a,\  n) \rightarrow kan $
 +
is an analytic diffeomorphism of the manifold $  K \times A \times N $
 +
onto the Lie group $  G $ .  
 +
The Iwasawa decomposition plays a fundamental part in the representation theory of semi-simple Lie groups. The Iwasawa decomposition can be defined also for connected semi-simple algebraic groups over a $  p $ -
 +
adic field (or, more generally, for groups of $  p $ -
 +
adic type) (see [[#References|[4]]], [[#References|[5]]]).
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
An example of an Iwasawa decomposition is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306046.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306048.png" /> the subgroup of diagonal matrices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306050.png" /> a lower triangular matrix with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306051.png" />'s on the diagonal. So, in particular, every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053060/i05306052.png" /> gets written as a product of a special orthogonal matrix and a lower triangular matrix.
+
An example of an Iwasawa decomposition is $  \mathop{\rm SL}\nolimits _{n} ( \mathbf R ) = K A N $
 +
with $  K = \mathop{\rm SO}\nolimits _{n} ( \mathbf R ) $ ,  
 +
$  A $
 +
the subgroup of diagonal matrices of $  \mathop{\rm SL}\nolimits _{n} ( \mathbf R ) $
 +
and $  N $
 +
a lower triangular matrix with $  1 $ '
 +
s on the diagonal. So, in particular, every element of $  \mathop{\rm SL}\nolimits _{n} ( \mathbf R ) $
 +
gets written as a product of a special orthogonal matrix and a lower triangular matrix.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4 {{MR|0754767}} {{ZBL|0543.58001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4 {{MR|0754767}} {{ZBL|0543.58001}} </TD></TR></table>

Revision as of 11:08, 17 December 2019

The unique representation of an arbitrary element $ g $ of a non-compact connected semi-simple real Lie group $ G $ as a product $ g = k an $ of elements $ k,\ a,\ n $ of analytic subgroups $ K,\ A,\ N $ , respectively, where $ K $ , $ A $ and $ N $ are defined as follows. Let $ \mathfrak g = \mathfrak k + \mathfrak P $ be a Cartan decomposition of the Lie algebra $ \mathfrak g $ of $ G $ ; let $ \mathfrak a $ be the maximal commutative subspace of the space $ \mathfrak P $ , and let $ \mathfrak N $ be a nilpotent Lie subalgebra of $ \mathfrak g $ such that $ \mathfrak N $ is the linear hull of the root vectors in some system of positive roots with respect to $ \mathfrak a $ . The decomposition of the Lie algebra as the direct sum of the subalgebras $ \mathfrak k $ , $ \mathfrak a $ and $ \mathfrak N $ is called the Iwasawa decomposition [1] of the semi-simple real Lie algebra $ \mathfrak g $ . The groups $ K $ , $ A $ and $ N $ are defined to be the analytic subgroups of $ G $ corresponding to the subalgebras $ \mathfrak k $ , $ \mathfrak a $ and $ \mathfrak N $ , respectively. The groups $ K $ , $ A $ and $ N $ are closed; $ A $ and $ N $ are simply-connected; $ K $ contains the centre of $ G $ , and the image of $ K $ under the adjoint representation of $ G $ is a maximal compact subgroup of the adjoint group of $ G $ . The mapping $ (k,\ a,\ n) \rightarrow kan $ is an analytic diffeomorphism of the manifold $ K \times A \times N $ onto the Lie group $ G $ . The Iwasawa decomposition plays a fundamental part in the representation theory of semi-simple Lie groups. The Iwasawa decomposition can be defined also for connected semi-simple algebraic groups over a $ p $ - adic field (or, more generally, for groups of $ p $ - adic type) (see [4], [5]).

References

[1] K. Iwasawa, "On some types of topological groups" Ann. of Math. , 50 (1949) pp. 507–558 MR0029911 Zbl 0034.01803
[2] M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[3] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) MR0514561 Zbl 0451.53038
[4] F. Bruhat, "Sur une classe de sous-groupes compacts maximaux des groupes de Chevalley sur un corps -adique" Publ. Math. IHES , 23 (1964) pp. 45–74 MR179298
[5] N. Iwahori, H. Matsumoto, "On some Bruhat decomposition and the structure of the Hecke rings of -adic Chevalley groups" Publ. Math. IHES , 25 (1965) pp. 5–48 MR185016


Comments

An example of an Iwasawa decomposition is $ \mathop{\rm SL}\nolimits _{n} ( \mathbf R ) = K A N $ with $ K = \mathop{\rm SO}\nolimits _{n} ( \mathbf R ) $ , $ A $ the subgroup of diagonal matrices of $ \mathop{\rm SL}\nolimits _{n} ( \mathbf R ) $ and $ N $ a lower triangular matrix with $ 1 $ ' s on the diagonal. So, in particular, every element of $ \mathop{\rm SL}\nolimits _{n} ( \mathbf R ) $ gets written as a product of a special orthogonal matrix and a lower triangular matrix.

References

[a1] S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4 MR0754767 Zbl 0543.58001
How to Cite This Entry:
Iwasawa decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Iwasawa_decomposition&oldid=44276
This article was adapted from an original article by A.S. FedenkoA.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article