# User:Maximilian Janisch/latexlist/Algebraic Groups/Lie group, nilpotent

A Lie group that is nilpotent as an abstract group (cf. Nilpotent group). An Abelian Lie group is nilpotent. If $F = \{ V _ { i } \}$ is a flag in a finite-dimensional vector space $V$ over a field $K$, then

\begin{equation} N ( F ) = \{ g \in GL ( V ) : g v \equiv v \operatorname { mod } V _ { i } \text { for all } v \in V _ { i } , i \geq 1 \} \end{equation}

is a nilpotent algebraic group over $K$; in a basis compatible with $H ^ { \prime }$ its elements are represented by triangular matrices with ones on the main diagonal. If $H ^ { \prime }$ is a complete flag (that is, if $\operatorname { lim } V _ { k } = k$), then the matrix nilpotent Lie group $N ( n , k )$ corresponding to $N ( F )$ consists of all matrices of order $n = \operatorname { dim } V$ of the form mentioned above.

If $K$ is a complete normed field, then $N ( F )$ is a nilpotent Lie group over $K$. Its Lie algebra is $n ( F )$ (see Lie algebra, nilpotent). More generally, the Lie algebra of a Lie group $k$ over a field $K$ of characteristic 0 is nilpotent if and only if the connected component $G$ of the identity of $k$ is nilpotent. This makes it possible to carry over to nilpotent Lie groups the properties of nilpotent Lie algebras (see , , ). The group version of Engel's theorem admits the following strengthening (Kolchin's theorem): If $k$ is a subgroup of $GL ( V )$, where $V$ is a finite-dimensional vector space over an arbitrary field $K$, and if every $g \in G$ is unipotent, then there is a complete flag $H ^ { \prime }$ in $V$ such that $G \subset N ( F )$ (and $k$ automatically turns out to be nilpotent) (see ).

Nilpotent Lie groups are solvable, so the properties of solvable Lie groups carry over them, and often in a strengthened from, since every nilpotent Lie group is triangular. A connected Lie group $k$ is nilpotent if and only if in canonical coordinates (see Lie group) the group operation in $k$ is written polynomially . Every simply-connected real nilpotent Lie group $k$ is isomorphic to an algebraic group, and moreover, to an algebraic subgroup of $N ( n , R )$.

A faithful representation of $k$ in $N ( n , R )$ can be chosen so that the automorphism group $G$ can be imbedded in $GL ( n , R )$ as the normalizer of the image of $k$ (see ).

If $k$ is a connected matrix real nilpotent Lie group, then it splits into the direct product of a compact Abelian Lie group and a simply-connected Lie group. A connected linear algebraic group $k$ over a field of characteristic 0 splits into the direct product of an Abelian normal subgroup consisting of the semi-simple elements and a normal subgroup consisting of the unipotent elements .

Nilpotent Lie groups were formerly called special Lie groups or Lie groups of rank 0. In the representation theory of semi-simple Lie groups, when studying discrete subgroups of such groups, substantial use was made of horospherical Lie groups that are nilpotent Lie groups.

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Maximilian Janisch/latexlist/Algebraic Groups/Lie group, nilpotent. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximilian_Janisch/latexlist/Algebraic_Groups/Lie_group,_nilpotent&oldid=44030