# Gauss decomposition

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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 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.