View source for Lie group, solvable

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

A nilpotent, in particular an Abelian, Lie group is solvable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l0586901.png" /> is a complete [[Flag|flag]] in a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l0586902.png" /> (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l0586903.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l0586904.png" />), then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l0586905.png" /></td> </tr></table>

is a solvable algebraic subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l0586906.png" /> and, in particular, a solvable Lie group. If one chooses a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l0586907.png" /> compatible with the flag <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l0586908.png" />, then in it the elements of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l0586909.png" /> are represented by non-singular upper triangular matrices; the resulting solvable matrix Lie group is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869011.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869012.png" />.

The [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869013.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869014.png" /> is solvable if and only if the connected component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869015.png" /> of the identity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869016.png" /> is solvable. The Lie algebras of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869018.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869020.png" />, respectively (see [[Lie algebra, solvable|Lie algebra, solvable]]). By virtue of the correspondence between subalgebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869021.png" /> and connected Lie subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869022.png" />, all properties of solvable Lie algebras carry over to solvable Lie groups (see [[#References|[1]]], [[#References|[3]]]).

An analogue of Lie's theorem on solvable Lie algebras is true for solvable Lie groups: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869023.png" /> is a finite-dimensional complex representation of a solvable Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869024.png" />, then there is a complete flag <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869025.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869027.png" />. In particular, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869028.png" /> there is a common eigen vector for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869030.png" />.

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 [[#References|[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|Levi–Mal'tsev decomposition]]), solvable Lie groups play an important role in the study of arbitrary Lie groups. In an arbitrary connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869031.png" /> one also considers maximal solvable subgroups. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869032.png" />, they are Borel subgroups (cf. also [[Borel subgroup|Borel subgroup]]) and are conjugate in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869033.png" />. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869034.png" /> is a Borel subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869035.png" />.

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 [[#References|[6]]]. The [[Exponential mapping|exponential mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869036.png" /> need not be injective or surjective, even for a simply-connected solvable Lie group. Solvable Lie groups for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869037.png" /> is a diffeomorphism are said to be exponential (see [[Lie group, exponential|Lie group, exponential]]). A simply-connected solvable Lie group is diffeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869038.png" />, and an arbitrary connected solvable Lie group is diffeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869040.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869041.png" />-dimensional torus.

A connected linear solvable Lie group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869042.png" /> can be represented as a semi-direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869043.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869044.png" /> is a compact Abelian subgroup and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869045.png" /> 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 [[#References|[3]]]. For connected solvable Lie groups one can define [[#References|[4]]] an analogue of the Mal'tsev decomposition.

If the Lie algebra of a connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869046.png" /> is triangular (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869047.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869048.png" /> is said to be triangular (cf. also [[Lie algebra, supersolvable|Lie algebra, supersolvable]]). The analogue of Lie's theorem on solvable algebras is true for triangular Lie groups (cf. [[Lie theorem|Lie theorem]]). Maximal connected triangular subgroups of an arbitrary connected Lie group are conjugate [[#References|[5]]]. A connected triangular Lie group is isomorphic to a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058690/l05869049.png" /> and is an exponential group if it is simply connected.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I. Kaplansky,   "An introduction to differential algebra" , Hermann  (1957)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Chevalley,   "Théorie des groupes de Lie" , '''3''' , Hermann  (1955)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Auslander,   "An exposition of the structure of solvmanifolds. I Algebraic theory"  ''Bull. Amer. Math. Soc.'' , '''79'''  (1973)  pp. 227–261</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.I. Mal'tsev,   "On the theory of Lie groups in the large"  ''Mat. Sb.'' , '''16 (58)'''  (1945)  pp. 163–190  (In Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.J. Olver,   "Applications of Lie groups to differential equations" , Springer  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.S. Varadarajan,   "Lie groups, Lie algebras, and their representations" , Prentice-Hall  (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T.A. Springer,   "Linear algebraic groups" , Birkhäuser  (1981)</TD></TR></table>