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

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