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''of a topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g0434301.png" />''
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A representation of an everywhere-dense subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g0434302.png" /> in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g0434303.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g0434304.png" /> is an Abelian subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g0434305.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g0434306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g0434307.png" /> are nilpotent groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g0434308.png" />, normalized by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g0434309.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343010.png" /> is the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343011.png" /> of non-singular real matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343013.png" /> is the subgroup of diagonal matrices, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343014.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343015.png" />) is the subgroup of lower-triangular (upper-triangular) matrices with unit elements on the principal diagonal, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343016.png" /> is the subset of matrices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343017.png" /> whose principal minors are non-zero, then the decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343018.png" /> is known as the Gauss decomposition of the general linear group and is directly connected with the [[Gauss method|Gauss method]] for the solution of systems of linear equations: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343022.png" />, is the non-singular coefficient matrix of the system of linear equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343023.png" />, then it may be converted by the Gauss method into the triangular form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343024.png" /> by multiplying it from the left by the lower-triangular matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343026.png" />. A rigorous definition of the Gauss decomposition necessitates the introduction of the following terms. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343027.png" /> be a [[Topological group|topological group]], let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343028.png" /> be a subgroup of it, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343030.png" /> be nilpotent subgroups in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343031.png" />, normalized by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343032.png" />. The subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343033.png" /> is called a triangular truncation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343034.png" /> if: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343037.png" /> is the commutator subgroup of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343040.png" /> are connected solvable subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343041.png" />; and 2) the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343042.png" /> is everywhere dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343043.png" />, and the decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343044.png" /> is unique. The decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343045.png" /> is called a triangular decomposition in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343046.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343047.png" /> is an [[Abelian group|Abelian group]], this decomposition is called a completely-triangular decomposition or a Gauss decomposition. In such a case the subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343049.png" /> are solvable. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343050.png" /> be an irreducible (continuous) representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343051.png" /> in a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343052.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343053.png" /> be the subspace of all vectors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343054.png" /> which are fixed with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343055.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343056.png" /> will then be invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343057.png" />, while the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343058.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343059.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343060.png" /> will be irreducible. The representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343061.png" /> unambiguously defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343062.png" />, up to an equivalence. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343063.png" /> also denote the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343064.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343065.png" /> restricting to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343066.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343067.png" /> and being trivial on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343068.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343069.png" /> denote the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343070.png" /> on the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343071.png" /> induced by this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343072.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343073.png" /> is contained (as an invariant part) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343074.png" />, and the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343075.png" /> is one-dimensional. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343076.png" /> is an Abelian subgroup, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343077.png" /> is one-dimensional and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343078.png" /> is a character of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343079.png" />. The following examples of triangular decompositions of Lie groups are known. 1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343080.png" /> be a reductive connected complex [[Lie group|Lie group]] with [[Cartan subalgebra|Cartan subalgebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343081.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343082.png" /> be a reductive connected subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343083.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343084.png" />. The subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343085.png" /> is then a triangular truncation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343086.png" />. 2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343087.png" /> be a reductive connected linear Lie group; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343088.png" /> will then contain a triangular truncation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343089.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343090.png" /> is a simply-connected Abelian subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343091.png" /> (generated by the non-compact roots in the Lie algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343092.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343093.png" /> is the centralizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343094.png" /> in the maximal compact subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343095.png" />. 3) In particular, any reductive connected complex Lie group permits a Gauss decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343096.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343097.png" /> is the Cartan subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g04343099.png" /> (respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430100.png" />) is an analytic subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430101.png" /> whose Lie algebra is spanned by all root vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430103.png" /> (respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430104.png" />), with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430105.png" /> denoting the roots with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430106.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430108.png" /> are opposite Borel subgroups (cf. [[Borel subgroup|Borel subgroup]]). In examples 1)–3) the subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430110.png" /> are simply connected, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430111.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430112.png" /> in the Zariski topology, while the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430113.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430114.png" />, is an isomorphism of algebraic varieties (and, in particular, a homeomorphism). This implies that the algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043430/g043430115.png" /> is rational.
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 +
''of a topological group  $  G $''
 +
 
 +
A representation of an everywhere-dense subset $  G _ {0} \subset  G $
 +
in the form $  G _ {0} = NH N  ^ {*} $,  
 +
where $  H $
 +
is an Abelian subgroup of $  G $,  
 +
and $  N $
 +
and $  N  ^ {*} $
 +
are nilpotent groups of $  G $,  
 +
normalized by $  H $.  
 +
If $  G $
 +
is the group $  \mathop{\rm GL} ( m, \mathbf R ) $
 +
of non-singular real matrices of order $  m $,  
 +
$  H $
 +
is the subgroup of diagonal matrices, $  N $(
 +
respectively, $  N  ^ {*} $)  
 +
is the subgroup of lower-triangular (upper-triangular) matrices with unit elements on the principal diagonal, and $  G _ {0} $
 +
is the subset of matrices in $  G $
 +
whose principal minors are non-zero, then the decomposition $  G _ {0} = NH N  ^ {*} $
 +
is known as the Gauss decomposition of the general linear group and is directly connected with the [[Gauss method|Gauss method]] for the solution of systems of linear equations: If $  g _ {0} = nh n  ^ {*} $,  
 +
where $  n \in N $,  
 +
$  h \in H $,  
 +
$  n  ^ {*} \in N  ^ {*} $,  
 +
is the non-singular coefficient matrix of the system of linear equations $  g _ {0} x = b $,  
 +
then it may be converted by the Gauss method into the triangular form $  h n  ^ {*} $
 +
by multiplying it from the left by the lower-triangular matrix $  n  ^ {-} 1 $,  
 +
$  n \in N $.  
 +
A rigorous definition of the Gauss decomposition necessitates the introduction of the following terms. Let $  G $
 +
be a [[Topological group|topological group]], let $  H $
 +
be a subgroup of it, and let $  N $
 +
and $  N  ^ {*} $
 +
be nilpotent subgroups in $  G $,  
 +
normalized by $  H $.  
 +
The subgroup $  H $
 +
is called a triangular truncation of $  G $
 +
if: 1) $  N \in D( R) $,  
 +
$  N  ^ {*} \subset  D( R  ^ {*} ) $,  
 +
where $  D( X) $
 +
is the commutator subgroup of the group $  X $
 +
and $  R $
 +
and $  R  ^ {*} $
 +
are connected solvable subgroups of $  G $;  
 +
and 2) the set $  G _ {0} = NH N  ^ {*} $
 +
is everywhere dense in $  G $,  
 +
and the decomposition $  NH N  ^ {*} $
 +
is unique. The decomposition $  G _ {0} = NH N  ^ {*} $
 +
is called a triangular decomposition in $  G $.  
 +
If $  H $
 +
is an [[Abelian group|Abelian group]], this decomposition is called a completely-triangular decomposition or a Gauss decomposition. In such a case the subgroups $  B = NH = HN $,  
 +
$  B  ^ {*} = N  ^ {*} H = H N  ^ {*} $
 +
are solvable. Let $  \pi $
 +
be an irreducible (continuous) representation of $  G $
 +
in a finite-dimensional vector space $  V $,  
 +
and let $  V _ {0} $
 +
be the subspace of all vectors in $  V $
 +
which are fixed with respect to $  N  ^ {*} $;  
 +
$  V _ {0} $
 +
will then be invariant with respect to $  H $,  
 +
while the representation $  \alpha $
 +
of $  H $
 +
on $  V _ {0} $
 +
will be irreducible. The representation $  \alpha $
 +
unambiguously defines $  \pi $,  
 +
up to an equivalence. Let $  \alpha $
 +
also denote the representation of $  B $
 +
on $  V _ {0} $
 +
restricting to $  \alpha $
 +
on $  H $
 +
and being trivial on $  N $.  
 +
Let $  e ( \alpha ) $
 +
denote the representation of $  G $
 +
on the vector space $  C ( G , V _ {0} ) $
 +
induced by this $  \alpha $.  
 +
Then $  \pi $
 +
is contained (as an invariant part) in $  e ( \alpha ) $,  
 +
and the space $  \mathop{\rm Hom} _ {G} ( \pi , e( \alpha )) $
 +
is one-dimensional. If $  H $
 +
is an Abelian subgroup, then $  V _ {0} $
 +
is one-dimensional and $  \alpha $
 +
is a character of the group $  H $.  
 +
The following examples of triangular decompositions of Lie groups are known. 1) Let $  G $
 +
be a reductive connected complex [[Lie group|Lie group]] with [[Cartan subalgebra|Cartan subalgebra]] $  H _ {0} $
 +
and let $  H $
 +
be a reductive connected subgroup in $  G $
 +
containing $  H _ {0} $.  
 +
The subgroup $  H $
 +
is then a triangular truncation of $  G $.  
 +
2) Let $  G $
 +
be a reductive connected linear Lie group; $  G $
 +
will then contain a triangular truncation $  H = MA $,  
 +
where $  A $
 +
is a simply-connected Abelian subgroup in $  G $(
 +
generated by the non-compact roots in the Lie algebra of $  G $),  
 +
and $  M $
 +
is the centralizer of $  A $
 +
in the maximal compact subgroup $  K \subset  G $.  
 +
3) In particular, any reductive connected complex Lie group permits a Gauss decomposition $  G _ {0} = NH N  ^ {*} $,  
 +
where $  H $
 +
is the Cartan subgroup of $  G $
 +
and $  N $(
 +
respectively $  N  ^ {*} $)  
 +
is an analytic subgroup in $  G $
 +
whose Lie algebra is spanned by all root vectors $  e _  \alpha  $,  
 +
$  \alpha < 0 $(
 +
respectively $  \alpha > 0 $),  
 +
with $  \alpha $
 +
denoting the roots with respect to $  H $,  
 +
i.e. $  HN $
 +
and $  H N  ^ {*} $
 +
are opposite Borel subgroups (cf. [[Borel subgroup|Borel subgroup]]). In examples 1)–3) the subgroups $  N $
 +
and $  N  ^ {*} $
 +
are simply connected, $  G _ {0} $
 +
is open in $  G $
 +
in the Zariski topology, while the mapping $  N \times H \times N  ^ {*} $,
 +
$  ( n, h, n  ^ {*} ) \rightarrow nh n  ^ {*} $,  
 +
is an isomorphism of algebraic varieties (and, in particular, a homeomorphism). This implies that the algebraic variety $  G $
 +
is rational.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.P. Zhelobenko,  "Compact Lie groups and their representations" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.P. Zhelobenko,  "Compact Lie groups and their representations" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR></table>

Latest revision as of 19:41, 5 June 2020


of a topological group $ G $

A representation of an everywhere-dense subset $ G _ {0} \subset G $ in the form $ G _ {0} = NH N ^ {*} $, where $ H $ is an Abelian subgroup of $ G $, and $ N $ and $ N ^ {*} $ are nilpotent groups of $ G $, normalized by $ H $. If $ G $ is the group $ \mathop{\rm GL} ( m, \mathbf R ) $ of non-singular real matrices of order $ m $, $ H $ is the subgroup of diagonal matrices, $ N $( respectively, $ N ^ {*} $) is the subgroup of lower-triangular (upper-triangular) matrices with unit elements on the principal diagonal, and $ G _ {0} $ is the subset of matrices in $ G $ whose principal minors are non-zero, then the decomposition $ G _ {0} = NH N ^ {*} $ is known as the Gauss decomposition of the general linear group and is directly connected with the Gauss method for the solution of systems of linear equations: If $ g _ {0} = nh n ^ {*} $, where $ n \in N $, $ h \in H $, $ n ^ {*} \in N ^ {*} $, is the non-singular coefficient matrix of the system of linear equations $ g _ {0} x = b $, then it may be converted by the Gauss method into the triangular form $ h n ^ {*} $ by multiplying it from the left by the lower-triangular matrix $ n ^ {-} 1 $, $ n \in N $. A rigorous definition of the Gauss decomposition necessitates the introduction of the following terms. Let $ G $ be a topological group, let $ H $ be a subgroup of it, and let $ N $ and $ N ^ {*} $ be nilpotent subgroups in $ G $, normalized by $ H $. The subgroup $ H $ is called a triangular truncation of $ G $ if: 1) $ N \in D( R) $, $ N ^ {*} \subset D( R ^ {*} ) $, where $ D( X) $ is the commutator subgroup of the group $ X $ and $ R $ and $ R ^ {*} $ are connected solvable subgroups of $ G $; and 2) the set $ G _ {0} = NH N ^ {*} $ is everywhere dense in $ G $, and the decomposition $ NH N ^ {*} $ is unique. The decomposition $ G _ {0} = NH N ^ {*} $ is called a triangular decomposition in $ G $. If $ H $ is an Abelian group, this decomposition is called a completely-triangular decomposition or a Gauss decomposition. In such a case the subgroups $ B = NH = HN $, $ B ^ {*} = N ^ {*} H = H N ^ {*} $ are solvable. Let $ \pi $ be an irreducible (continuous) representation of $ G $ in a finite-dimensional vector space $ V $, and let $ V _ {0} $ be the subspace of all vectors in $ V $ which are fixed with respect to $ N ^ {*} $; $ V _ {0} $ will then be invariant with respect to $ H $, while the representation $ \alpha $ of $ H $ on $ V _ {0} $ will be irreducible. The representation $ \alpha $ unambiguously defines $ \pi $, up to an equivalence. Let $ \alpha $ also denote the representation of $ B $ on $ V _ {0} $ restricting to $ \alpha $ on $ H $ and being trivial on $ N $. Let $ e ( \alpha ) $ denote the representation of $ G $ on the vector space $ C ( G , V _ {0} ) $ induced by this $ \alpha $. Then $ \pi $ is contained (as an invariant part) in $ e ( \alpha ) $, and the space $ \mathop{\rm Hom} _ {G} ( \pi , e( \alpha )) $ is one-dimensional. If $ H $ is an Abelian subgroup, then $ V _ {0} $ is one-dimensional and $ \alpha $ is a character of the group $ H $. The following examples of triangular decompositions of Lie groups are known. 1) Let $ G $ be a reductive connected complex Lie group with Cartan subalgebra $ H _ {0} $ and let $ H $ be a reductive connected subgroup in $ G $ containing $ H _ {0} $. The subgroup $ H $ is then a triangular truncation of $ G $. 2) Let $ G $ be a reductive connected linear Lie group; $ G $ will then contain a triangular truncation $ H = MA $, where $ A $ is a simply-connected Abelian subgroup in $ G $( generated by the non-compact roots in the Lie algebra of $ G $), and $ M $ is the centralizer of $ A $ in the maximal compact subgroup $ K \subset G $. 3) In particular, any reductive connected complex Lie group permits a Gauss decomposition $ G _ {0} = NH N ^ {*} $, where $ H $ is the Cartan subgroup of $ G $ and $ N $( respectively $ N ^ {*} $) is an analytic subgroup in $ G $ whose Lie algebra is spanned by all root vectors $ e _ \alpha $, $ \alpha < 0 $( respectively $ \alpha > 0 $), with $ \alpha $ denoting the roots with respect to $ H $, i.e. $ HN $ and $ H N ^ {*} $ are opposite Borel subgroups (cf. Borel subgroup). In examples 1)–3) the subgroups $ N $ and $ N ^ {*} $ are simply connected, $ G _ {0} $ is open in $ G $ in the Zariski topology, while the mapping $ N \times H \times N ^ {*} $, $ ( n, h, n ^ {*} ) \rightarrow nh n ^ {*} $, is an isomorphism of algebraic varieties (and, in particular, a homeomorphism). This implies that the algebraic variety $ G $ is rational.

References

[1] D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)
How to Cite This Entry:
Gauss decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_decomposition&oldid=47046
This article was adapted from an original article by D.P. Zhelobenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article