Lie group, solvable

A Lie group that is solvable as an abstract group (cf. Solvable group). In what follows real or complex solvable Lie groups are considered.

A nilpotent, in particular an Abelian, Lie group is solvable. If $ F = \{ V _{i} \} $ is a complete flag in a finite-dimensional vector space $ V $ (over $ \mathbf R $ or $ \mathbf C $ ), then $$ B (F \ ) = \{ {g \in \mathop{\rm GL}\nolimits (V)} : { g V _{i} \subset V _{i} \textrm{ for all } i} \} $$ is a solvable algebraic subgroup of $ \mathop{\rm GL}\nolimits (V) $ and, in particular, a solvable Lie group. If one chooses a basis in $ V $ compatible with the flag $ F $ , then in it the elements of the group $ B (F \ ) $ are represented by non-singular upper triangular matrices; the resulting solvable matrix Lie group is denoted by $ T ( n ,\ K ) $ , where $ K = \mathbf R $ or $ \mathbf C $ .

The Lie algebra $ \mathfrak g $ of the group $ G $ is solvable if and only if the connected component $ (G) _{0} $ of the identity of $ G $ is solvable. The Lie algebras of the groups $ B (F \ ) $ and $ T ( n ,\ K ) $ are $ \mathfrak t (F \ ) $ and $ \mathfrak t ( n ,\ K ) $ , respectively (see Lie algebra, solvable). By virtue of the correspondence between subalgebras of $ \mathfrak g $ and connected Lie subgroups of $ G $ , all properties of solvable Lie algebras carry over to solvable Lie groups (see [1], [3]).

An analogue of Lie's theorem on solvable Lie algebras is true for solvable Lie groups: If $ \rho : \ G \rightarrow \mathop{\rm GL}\nolimits (V) $ is a finite-dimensional complex representation of a solvable Lie group $ G $ , then there is a complete flag $ F $ in $ V $ such that $ \rho (G) \subset B (F \ ) $ . In particular, in $ V $ there is a common eigen vector for all $ \rho (g) $ , $ g \in G $ .

Solvable Lie groups were first considered by S. Lie, who conjectured that continuous groups could play the same role in the theory of integration of differential equations by quadratures as Galois groups do in the theory of algebraic equations. However, generally speaking, the group of automorphisms of a differential equation is trivial, and so meaningful results in this direction have been obtained only for linear and some other equations. Thus, for these equations the fact that the solutions can be expressed by quadratures and exponentials of them is actually equivalent to the fact that the corresponding (matrix) Galois group is solvable [2]. If this group is nilpotent, then exponentials of quadratures do not occur in the solution.

By the Levi–Mal'tsev theorem on the decomposition of an arbitrary simply-connected Lie group into a semi-direct product (cf. Levi–Mal'tsev decomposition), solvable Lie groups play an important role in the study of arbitrary Lie groups. In an arbitrary connected Lie group $ G $ one also considers maximal solvable subgroups. If $ K = \mathbf C $ , they are Borel subgroups (cf. also Borel subgroup) and are conjugate in $ G $ . For example, $ B (F \ ) $ is a Borel subgroup of $ \mathop{\rm GL}\nolimits (V) $ .

A simply-connected solvable Lie group always has a faithful finite-dimensional representation, but for non-simply-connected solvable Lie groups this is not always so. An arbitrary connected subgroup of a simply-connected solvable Lie group is closed and simply connected [6]. The exponential mapping $ \mathop{\rm exp}\nolimits : \ \mathfrak g \rightarrow G $ need not be injective or surjective, even for a simply-connected solvable Lie group. Solvable Lie groups for which $ \mathop{\rm exp}\nolimits $ is a diffeomorphism are said to be exponential (see Lie group, exponential). A simply-connected solvable Lie group is diffeomorphic to $ \mathbf R ^{N} $ , and an arbitrary connected solvable Lie group is diffeomorphic to $ \mathbf R ^{n} \times T ^{m} $ , where $ T ^{m} $ is an $ m $ -dimensional torus.

A connected linear solvable Lie group over $ \mathbf R $ can be represented as a semi-direct product $ K \cdot S $ , where $ K $ is a compact Abelian subgroup and $ S $ is a simply-connected normal subgroup. An algebraic connected solvable group over any field of characteristic 0 splits into the semi-direct product of the normal subgroup consisting of unipotent elements and the Abelian subgroup consisting of semi-simple elements [3]. For connected solvable Lie groups one can define [4] an analogue of the Mal'tsev decomposition.

If the Lie algebra of a connected Lie group $ G $ is triangular (over $ \mathbf R $ ), then $ G $ is said to be triangular (cf. also Lie algebra, supersolvable). The analogue of Lie's theorem on solvable algebras is true for triangular Lie groups (cf. Lie theorem). Maximal connected triangular subgroups of an arbitrary connected Lie group are conjugate [5]. A connected triangular Lie group is isomorphic to a subgroup of $ T ( n ,\ K ) $ and is an exponential group if it is simply connected.

References

[1]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
[2]	I. Kaplansky, "An introduction to differential algebra" , Hermann (1957) MR0093654 Zbl 0083.03301
[3]	C. Chevalley, "Théorie des groupes de Lie" , 3 , Hermann (1955) MR0068552 Zbl 0186.33104 Zbl 0054.01303 Zbl 0063.00843
[4]	L. Auslander, "An exposition of the structure of solvmanifolds. I Algebraic theory" Bull. Amer. Math. Soc. , 79 (1973) pp. 227–261 MR0486307 MR0486308 Zbl 0265.22016
[5]	E.B. Vinberg, "The Mozorov–Borel theorem for real Lie groups" Soviet Math. Dokl. , 2 : 6 (1961) pp. 1416–1419 Dokl. Akad. Nauk SSSR , 141 (1961) pp. 270–273
[6]	A.I. Mal'tsev, "On the theory of Lie groups in the large" Mat. Sb. , 16 (58) (1945) pp. 163–190 (In Russian) Zbl 0061.04602

References

[a1]	P.J. Olver, "Applications of Lie groups to differential equations" , Springer (1986) MR0836734 Zbl 0588.22001
[a2]	V.S. Varadarajan, "Lie groups, Lie algebras, and their representations" , Prentice-Hall (1974) MR0376938 Zbl 0371.22001
[a3]	T.A. Springer, "Linear algebraic groups" , Birkhäuser (1981) MR0632835 Zbl 0453.14022