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

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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 [2], [4], [5]). 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 [3]).

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 [4]. 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 [1]).

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 [5].

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.

References

[1]	G. Birkhoff, "Representability of Lie algebras and Lie groups by matrices" Ann. of Math. (2) , 38 (1937) pp. 526–532 MR1503351 Zbl 0016.24402 Zbl 63.0090.01
[2]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
[3]	J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
[4]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) MR0514561 Zbl 0451.53038
[5]	C. Chevalley, "Théorie des groupes de Lie" , 3 , Hermann (1955) MR0068552 Zbl 0186.33104 Zbl 0054.01303 Zbl 0063.00843

Comments

The theory of unitary representations of nilpotent Lie groups is well understood, and goes back to the fundamental paper [a1] of A.A. Kirillov. This theory, which is usually called the "orbit method" , has extensions to the case of solvable Lie groups, although the results are not as complete as in the nilpotent case. See also [a3].

References

[a1]	A.A. Kirillov, "Unitary representations of nilpotent Lie groups" Russian Math. Surveys , 17 : 4 (1962) pp. 53–104 Uspekhi Mat. Nauk , 17 : 4 (1962) pp. 57–110 MR0142001 Zbl 0106.25001
[a2]	M.S. Raghunathan, "Discrete subgroups of Lie groups" , Springer (1972) MR0507234 MR0507236 Zbl 0254.22005
[a3]	L. Pukanszky, "Leçons sur les représentations des groupes" , Dunod (1967) MR0217220 Zbl 0152.01201