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 $$N (F \ ) = \{ {g \in \mathop{\rm GL}\nolimits (V)} : { g v \equiv v \mathop{\rm mod}\nolimits \ V _{i} \textrm{ for all } v \in V _{i} , i \geq 1} \}$$ is a nilpotent algebraic group over $K$ ; in a basis compatible with $F$ its elements are represented by triangular matrices with ones on the main diagonal. If $F$ is a complete flag (that is, if $\mathop{\rm dim}\nolimits \ V _{k} = k$ ), then the matrix nilpotent Lie group $N ( n , k )$ corresponding to $N (F \ )$ consists of all matrices of order $n = \mathop{\rm dim}\nolimits \ V$ 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 $\mathfrak n (F \ )$ (see Lie algebra, nilpotent). More generally, the Lie algebra of a Lie group $G$ over a field $K$ of characteristic 0 is nilpotent if and only if the connected component $G _{0}$ of the identity of $G$ 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 $G$ is a subgroup of $\mathop{\rm GL}\nolimits (V)$ , 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 $F$ in $V$ such that $G \subset N (F \ )$ (and $G$ 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 $G$ is nilpotent if and only if in canonical coordinates (see Lie group) the group operation in $G$ is written polynomially [4]. Every simply-connected real nilpotent Lie group $G$ is isomorphic to an algebraic group, and moreover, to an algebraic subgroup of $N (n ,\ \mathbf R )$ .

A faithful representation of $G$ in $N ( n ,\ \mathbf R )$ can be chosen so that the automorphism group $\mathop{\rm Aut}\nolimits \ G$ can be imbedded in $\mathop{\rm GL}\nolimits ( n ,\ \mathbf R )$ as the normalizer of the image of $G$ (see [1]).

If $G$ 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 $G$ 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