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A [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l0585201.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l0585202.png" /> satisfying one of the following equivalent conditions:
+
{{TEX|done}}
 +
A [[Lie algebra|Lie algebra]] $  \mathfrak g $
 +
over a field $  K $
 +
satisfying one of the following equivalent conditions:
  
1) the terms of the derived series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l0585203.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l0585204.png" /> are equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l0585205.png" /> for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l0585206.png" />;
+
1) the terms of the derived series $  D ^{k} \mathfrak g $
 +
of $  \mathfrak g $
 +
are equal to $  \{ 0 \} $
 +
for sufficiently large $  k $ ;
  
2) there is a finite decreasing chain of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l0585207.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l0585208.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l0585209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852011.png" /> (that is, the Lie algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852012.png" /> are Abelian for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852013.png" />);
 
  
3) there is a finite decreasing chain of subalgebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852014.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852017.png" /> is an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852018.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852019.png" /> is a one-dimensional (Abelian) Lie algebra for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852020.png" />.
+
2) there is a finite decreasing chain of ideals $  \{ \mathfrak g _{i} \} _ {0 \leq i \leq n} $
 +
of $  \mathfrak g $
 +
such that $  \mathfrak g _{0} = \mathfrak g $ ,  
 +
$  \mathfrak g _{n} = \{ 0 \} $
 +
and $  [ \mathfrak g _{i} ,\  \mathfrak g _{i} ] \subset \mathfrak g _{i+1} $ (
 +
that is, the Lie algebras $  \mathfrak g _{i} / \mathfrak g _{i+1} $
 +
are Abelian for all $  0 \leq i < n $ );
  
A nilpotent Lie algebra (cf. [[Lie algebra, nilpotent|Lie algebra, nilpotent]]) is solvable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852021.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/l058520/l05852022.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852023.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/l058520/l05852024.png" /></td> </tr></table>
+
3) there is a finite decreasing chain of subalgebras $  \{ \mathfrak g _{i} ^ \prime  \} _ {0 \leq i \leq m} $
 +
such that $  \mathfrak g _{0} ^ \prime  = \mathfrak g $ ,
 +
$  \mathfrak g _{m} ^ \prime  = \{ 0 \} $ ,
 +
$  \mathfrak g _{i+1} ^ \prime  $
 +
is an ideal of $  \mathfrak g _{i} ^ \prime  $ ,
 +
and $  \mathfrak g _{i} ^ \prime  / \mathfrak g _{i+1} ^ \prime  $
 +
is a one-dimensional (Abelian) Lie algebra for $  0 \leq i < m $ .
  
is a solvable subalgebra of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852025.png" /> of all linear transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852026.png" />. If one chooses a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852027.png" /> compatible with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852028.png" />, then with respect to that basis, elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852029.png" /> are represented by upper triangular matrices; the resulting solvable linear Lie algebra is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852031.png" />.
 
  
The class of solvable Lie algebras is closed with respect to transition to a subalgebra, a quotient algebra or an extension. In particular, any subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852032.png" /> is solvable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852034.png" /> is an algebraically closed field, then any finite-dimensional solvable Lie algebra is isomorphic to a subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852035.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852036.png" />. One of the main properties of solvable Lie algebras is expressed in Lie's theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852037.png" /> be a solvable Lie algebra over an algebraically closed field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852038.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852039.png" /> be a finite-dimensional linear representation of it. Then in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852040.png" /> there is a complete flag <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852041.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852042.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852043.png" /> is irreducible, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852044.png" />. Ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852045.png" /> can be chosen so as to form a complete flag, that is, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852046.png" />.
+
A nilpotent Lie algebra (cf. [[Lie algebra, nilpotent|Lie algebra, nilpotent]]) is solvable. If $  F = \{ V _{i} \} $
 +
is a complete [[Flag|flag]] in a finite-dimensional vector space $  V $
 +
over $  K $ ,
 +
then$$
 +
\mathfrak b ( F \  )  =   \{ {x \in \mathfrak g \mathfrak l (V)} : {x V _{i} \subset V
 +
_{i}  \textrm{ for   all }  i} \}
 +
$$
 +
is a solvable subalgebra of the Lie algebra $  \mathfrak g \mathfrak l (V) $
 +
of all linear transformations of $  V $ .  
 +
If one chooses a basis in $  V $
 +
compatible with $  F $ ,  
 +
then with respect to that basis, elements of $  \mathfrak b (F \  ) $
 +
are represented by upper triangular matrices; the resulting solvable linear Lie algebra is denoted by $  \mathfrak t ( n ,\  K ) $ ,  
 +
where $  n = \mathop{\rm dim}\nolimits \  V $ .
  
A finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852047.png" /> over a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852048.png" /> is solvable if and only if the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852049.png" /> is nilpotent. Another criterion for solvability (Cartan's criterion) is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852050.png" /> is solvable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852051.png" /> is orthogonal to the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852052.png" /> with respect to the [[Killing form|Killing form]] (or any bilinear form associated with a faithful finite-dimensional representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852053.png" />).
 
  
Solvable Lie algebras were first considered by S. Lie in connection with the study of solvable Lie transformation groups. The study of solvable Lie algebras acquired great significance after the introduction of the concept of the radical (that is, the largest solvable ideal) of an arbitrary finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852054.png" />, and it was proved that in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852055.png" /> the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852056.png" /> is the semi-direct sum of its radical and a maximal semi-simple subalgebra (see [[Levi–Mal'tsev decomposition|Levi–Mal'tsev decomposition]]). This made it possible to reduce the problem of classifying arbitrary Lie algebras to the enumeration of semi-simple (which for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852057.png" /> had already been done by W. Killing) and solvable Lie algebras. The classification of solvable Lie algebras (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852058.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852059.png" />) has been carried out only in dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852060.png" />.
+
The class of solvable Lie algebras is closed with respect to transition to a subalgebra, a quotient algebra or an extension. In particular, any subalgebra of $  \mathfrak t ( n ,\  K ) $
 +
is solvable. If $  \mathop{\rm char}\nolimits \  K = 0 $
 +
and $  K $
 +
is an algebraically closed field, then any finite-dimensional solvable Lie algebra is isomorphic to a subalgebra of $  \mathfrak t ( n ,\  K ) $
 +
for some $  n $ .  
 +
One of the main properties of solvable Lie algebras is expressed in Lie's theorem: Let $  \mathfrak g $
 +
be a solvable Lie algebra over an algebraically closed field of characteristic $  0 $
 +
and let $  \rho : \  \mathfrak g \rightarrow \mathfrak g \mathfrak l (V) $
 +
be a finite-dimensional linear representation of it. Then in $  V $
 +
there is a complete flag $  F $
 +
such that $  \rho ( \mathfrak g ) \subset \mathfrak b (F \  ) $ .  
 +
In particular, if $  \rho $
 +
is irreducible, then $  \mathop{\rm dim}\nolimits \  V = 1 $ .  
 +
Ideals of $  \mathfrak g $
 +
can be chosen so as to form a complete flag, that is, so that $  \mathop{\rm dim}\nolimits \  \mathfrak g _{i} = \mathop{\rm dim}\nolimits \  \mathfrak g - i $ .
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852061.png" /> is a solvable algebraic subalgebra (cf. [[Algebraic algebra|Algebraic algebra]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852062.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852063.png" /> is a finite-dimensional space over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852064.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852065.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852066.png" /> splits into the semi-direct product of the nilpotent ideal formed by all nilpotent transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852067.png" /> and the Abelian subalgebra consisting of the semi-simple transformations [[#References|[6]]]. In general, any split solvable Lie algebra, that is, a finite-dimensional solvable Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852068.png" /> every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852069.png" /> of which splits into a sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852070.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852073.png" /> is semi-simple, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852074.png" /> is nilpotent, has a similar structure [[#References|[8]]]. To every finite-dimensional solvable Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852075.png" /> there uniquely corresponds a minimal split solvable Lie algebra containing it (the Mal'tsev decomposition). The problem of classifying solvable Lie algebras that have a given Mal'tsev decomposition has been solved [[#References|[8]]]. Thus, the problem of classifying solvable Lie algebras reduces, in a certain sense, to the study of nilpotent Lie algebras.
 
  
Apart from the radical, in an arbitrary finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852076.png" /> one can distinguish maximal solvable subalgebras. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852077.png" /> is an algebraically closed field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852078.png" />, then all such subalgebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852079.png" /> (they are called Borel subalgebras) are conjugate. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852080.png" /> is a Borel subalgebra of the Lie algebra of all matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852081.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852082.png" /> is not algebraically closed or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852083.png" /> is finite, then Lie's theorem is false, in general. However, it can be extended to the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852084.png" /> is perfect and contains the characteristic roots of all the characteristic polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852085.png" /> of the adjoint transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852087.png" />. If this condition is satisfied for the adjoint representation of a solvable Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852088.png" /> (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852089.png" /> is said to be triangular. Many properties of solvable Lie algebras over an algebraically closed field carry over to triangular Lie algebras. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058520/l05852090.png" />, then all maximal triangular subalgebras of an arbitrary finite-dimensional Lie algebra are conjugate (see [[#References|[1]]], [[#References|[7]]]). Maximal triangular subalgebras are used in the study of semi-simple Lie algebras over an algebraically non-closed field as a good analogue of Borel subalgebras. They also play a fundamental role in the description of the connected uniform subgroups (cf. [[Uniform subgroup|Uniform subgroup]]) of Lie groups [[#References|[9]]].
+
A finite-dimensional Lie algebra $  \mathfrak g $
 +
over a field of characteristic 0 $
 +
is solvable if and only if the algebra $  D ^{2} \mathfrak g = [ \mathfrak g ,\  \mathfrak g ] $
 +
is nilpotent. Another criterion for solvability (Cartan's criterion) is: $  \mathfrak g $
 +
is solvable if and only if $  D ^{2} \mathfrak g $
 +
is orthogonal to the whole of $  \mathfrak g $
 +
with respect to the [[Killing form|Killing form]] (or any bilinear form associated with a faithful finite-dimensional representation of $  \mathfrak g $ ).
  
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'' , '''27''' (1965) pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) {{MR|0682756}} {{ZBL|0319.17002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) {{MR|0148716}} {{MR|0143793}} {{ZBL|0121.27504}} {{ZBL|0109.26201}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) {{MR|0218496}} {{ZBL|0132.27803}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> C. Chevalley, "Théorie des groupes de Lie" , '''3''' , Hermann (1955) {{MR|0068552}} {{ZBL|0186.33104}} {{ZBL|0054.01303}} {{ZBL|0063.00843}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> E.B. Vinberg, "The Morozov–Borel theorem for real Lie groups" ''Soviet Math. Dokl.'' , '''2''' (1961) pp. 1416–1419 ''Dokl. Akad. Nauk SSSR'' , '''141''' (1961) pp. 270–273 {{MR|0142683}} {{ZBL|0112.02505}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.I. Mal'tsev, "Solvable Lie algebras" ''Izv. Akad. Nauk SSSR'' , '''9''' (1945) pp. 329–352 (In Russian) {{MR|}} {{ZBL|0061.05303}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> A.L. Onishchik, "On Lie groups transitive on compact manifolds II" ''Math. USSR Sb.'' , '''3''' (1967) pp. 373–388 ''Mat. Sb.'' , '''74''' (1967) pp. 398–416 {{MR|}} {{ZBL|0198.28903}} </TD></TR></table>
 
  
 +
Solvable Lie algebras were first considered by S. Lie in connection with the study of solvable Lie transformation groups. The study of solvable Lie algebras acquired great significance after the introduction of the concept of the radical (that is, the largest solvable ideal) of an arbitrary finite-dimensional Lie algebra $  \mathfrak g $ ,
 +
and it was proved that in the case $  \mathop{\rm char}\nolimits \  K = 0 $
 +
the algebra $  \mathfrak g $
 +
is the semi-direct sum of its radical and a maximal semi-simple subalgebra (see [[Levi–Mal'tsev decomposition|Levi–Mal'tsev decomposition]]). This made it possible to reduce the problem of classifying arbitrary Lie algebras to the enumeration of semi-simple (which for $  K = \mathbf C $
 +
had already been done by W. Killing) and solvable Lie algebras. The classification of solvable Lie algebras (for $  K = \mathbf C $
 +
or $  \mathbf R $ )
 +
has been carried out only in dimensions $  \leq 6 $ .
  
  
====Comments====
+
If $  \mathfrak g $
 +
is a solvable algebraic subalgebra (cf. [[Algebraic algebra|Algebraic algebra]]) of $  \mathfrak g \mathfrak l (V) $ ,
 +
where $  V $
 +
is a finite-dimensional space over a field $  K $
 +
of characteristic $  0 $ ,
 +
then $  \mathfrak g $
 +
splits into the semi-direct product of the nilpotent ideal formed by all nilpotent transformations of $  \mathfrak g $
 +
and the Abelian subalgebra consisting of the semi-simple transformations [[#References|[6]]]. In general, any split solvable Lie algebra, that is, a finite-dimensional solvable Lie algebra over $  K $
 +
every element $  x $
 +
of which splits into a sum $  x = s + n $ ,
 +
where $  s ,\  n \in \mathfrak g $ ,
 +
$  [ s ,\  n ] = 0 $ ,
 +
$  s $
 +
is semi-simple, and $  n $
 +
is nilpotent, has a similar structure [[#References|[8]]]. To every finite-dimensional solvable Lie algebra over $  K $
 +
there uniquely corresponds a minimal split solvable Lie algebra containing it (the Mal'tsev decomposition). The problem of classifying solvable Lie algebras that have a given Mal'tsev decomposition has been solved [[#References|[8]]]. Thus, the problem of classifying solvable Lie algebras reduces, in a certain sense, to the study of nilpotent Lie algebras.
  
 +
Apart from the radical, in an arbitrary finite-dimensional Lie algebra $  \mathfrak g $
 +
one can distinguish maximal solvable subalgebras. If $  K $
 +
is an algebraically closed field of characteristic $  0 $ ,
 +
then all such subalgebras of $  \mathfrak g $ (
 +
they are called Borel subalgebras) are conjugate. For example, $  \mathfrak t ( n ,\  K ) $
 +
is a Borel subalgebra of the Lie algebra of all matrices of order $  n $ .
 +
If $  K $
 +
is not algebraically closed or if $  \mathop{\rm char}\nolimits \  K $
 +
is finite, then Lie's theorem is false, in general. However, it can be extended to the case when $  K $
 +
is perfect and contains the characteristic roots of all the characteristic polynomials $  \rho (x) $
 +
of the adjoint transformations $  \mathop{\rm ad}\nolimits \  x $ ,
 +
$  x \in \mathfrak g $ .
 +
If this condition is satisfied for the adjoint representation of a solvable Lie algebra $  \mathfrak g $ (
 +
cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]), then $  \mathfrak g $
 +
is said to be triangular. Many properties of solvable Lie algebras over an algebraically closed field carry over to triangular Lie algebras. In particular, if $  \mathop{\rm char}\nolimits \  K = 0 $ ,
 +
then all maximal triangular subalgebras of an arbitrary finite-dimensional Lie algebra are conjugate (see [[#References|[1]]], [[#References|[7]]]). Maximal triangular subalgebras are used in the study of semi-simple Lie algebras over an algebraically non-closed field as a good analogue of Borel subalgebras. They also play a fundamental role in the description of the connected uniform subgroups (cf. [[Uniform subgroup|Uniform subgroup]]) of Lie groups [[#References|[9]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) {{MR|0323842}} {{ZBL|0254.17004}} </TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'' , '''27''' (1965) pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) {{MR|0682756}} {{ZBL|0319.17002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) {{MR|0148716}} {{MR|0143793}} {{ZBL|0121.27504}} {{ZBL|0109.26201}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) {{MR|0218496}} {{ZBL|0132.27803}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> C. Chevalley, "Théorie des groupes de Lie" , '''3''' , Hermann (1955) {{MR|0068552}} {{ZBL|0186.33104}} {{ZBL|0054.01303}} {{ZBL|0063.00843}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> E.B. Vinberg, "The Morozov–Borel theorem for real Lie groups" ''Soviet Math. Dokl.'' , '''2''' (1961) pp. 1416–1419 ''Dokl. Akad. Nauk SSSR'' , '''141''' (1961) pp. 270–273 {{MR|0142683}} {{ZBL|0112.02505}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.I. Mal'tsev, "Solvable Lie algebras" ''Izv. Akad. Nauk SSSR'' , '''9''' (1945) pp. 329–352 (In Russian) {{MR|}} {{ZBL|0061.05303}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> A.L. Onishchik, "On Lie groups transitive on compact manifolds II" ''Math. USSR Sb.'' , '''3''' (1967) pp. 373–388 ''Mat. Sb.'' , '''74''' (1967) pp. 398–416 {{MR|}} {{ZBL|0198.28903}} </TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) {{MR|0323842}} {{ZBL|0254.17004}} </TD></TR></table>
 +
[[Category:Nonassociative rings and algebras]]

Latest revision as of 08:49, 8 April 2023

A Lie algebra $ \mathfrak g $ over a field $ K $ satisfying one of the following equivalent conditions:

1) the terms of the derived series $ D ^{k} \mathfrak g $ of $ \mathfrak g $ are equal to $ \{ 0 \} $ for sufficiently large $ k $ ;


2) there is a finite decreasing chain of ideals $ \{ \mathfrak g _{i} \} _ {0 \leq i \leq n} $ of $ \mathfrak g $ such that $ \mathfrak g _{0} = \mathfrak g $ , $ \mathfrak g _{n} = \{ 0 \} $ and $ [ \mathfrak g _{i} ,\ \mathfrak g _{i} ] \subset \mathfrak g _{i+1} $ ( that is, the Lie algebras $ \mathfrak g _{i} / \mathfrak g _{i+1} $ are Abelian for all $ 0 \leq i < n $ );


3) there is a finite decreasing chain of subalgebras $ \{ \mathfrak g _{i} ^ \prime \} _ {0 \leq i \leq m} $ such that $ \mathfrak g _{0} ^ \prime = \mathfrak g $ , $ \mathfrak g _{m} ^ \prime = \{ 0 \} $ , $ \mathfrak g _{i+1} ^ \prime $ is an ideal of $ \mathfrak g _{i} ^ \prime $ , and $ \mathfrak g _{i} ^ \prime / \mathfrak g _{i+1} ^ \prime $ is a one-dimensional (Abelian) Lie algebra for $ 0 \leq i < m $ .


A nilpotent Lie algebra (cf. Lie algebra, nilpotent) is solvable. If $ F = \{ V _{i} \} $ is a complete flag in a finite-dimensional vector space $ V $ over $ K $ , then$$ \mathfrak b ( F \ ) = \{ {x \in \mathfrak g \mathfrak l (V)} : {x V _{i} \subset V _{i} \textrm{ for all } i} \} $$ is a solvable subalgebra of the Lie algebra $ \mathfrak g \mathfrak l (V) $ of all linear transformations of $ V $ . If one chooses a basis in $ V $ compatible with $ F $ , then with respect to that basis, elements of $ \mathfrak b (F \ ) $ are represented by upper triangular matrices; the resulting solvable linear Lie algebra is denoted by $ \mathfrak t ( n ,\ K ) $ , where $ n = \mathop{\rm dim}\nolimits \ V $ .


The class of solvable Lie algebras is closed with respect to transition to a subalgebra, a quotient algebra or an extension. In particular, any subalgebra of $ \mathfrak t ( n ,\ K ) $ is solvable. If $ \mathop{\rm char}\nolimits \ K = 0 $ and $ K $ is an algebraically closed field, then any finite-dimensional solvable Lie algebra is isomorphic to a subalgebra of $ \mathfrak t ( n ,\ K ) $ for some $ n $ . One of the main properties of solvable Lie algebras is expressed in Lie's theorem: Let $ \mathfrak g $ be a solvable Lie algebra over an algebraically closed field of characteristic $ 0 $ and let $ \rho : \ \mathfrak g \rightarrow \mathfrak g \mathfrak l (V) $ be a finite-dimensional linear representation of it. Then in $ V $ there is a complete flag $ F $ such that $ \rho ( \mathfrak g ) \subset \mathfrak b (F \ ) $ . In particular, if $ \rho $ is irreducible, then $ \mathop{\rm dim}\nolimits \ V = 1 $ . Ideals of $ \mathfrak g $ can be chosen so as to form a complete flag, that is, so that $ \mathop{\rm dim}\nolimits \ \mathfrak g _{i} = \mathop{\rm dim}\nolimits \ \mathfrak g - i $ .


A finite-dimensional Lie algebra $ \mathfrak g $ over a field of characteristic $ 0 $ is solvable if and only if the algebra $ D ^{2} \mathfrak g = [ \mathfrak g ,\ \mathfrak g ] $ is nilpotent. Another criterion for solvability (Cartan's criterion) is: $ \mathfrak g $ is solvable if and only if $ D ^{2} \mathfrak g $ is orthogonal to the whole of $ \mathfrak g $ with respect to the Killing form (or any bilinear form associated with a faithful finite-dimensional representation of $ \mathfrak g $ ).


Solvable Lie algebras were first considered by S. Lie in connection with the study of solvable Lie transformation groups. The study of solvable Lie algebras acquired great significance after the introduction of the concept of the radical (that is, the largest solvable ideal) of an arbitrary finite-dimensional Lie algebra $ \mathfrak g $ , and it was proved that in the case $ \mathop{\rm char}\nolimits \ K = 0 $ the algebra $ \mathfrak g $ is the semi-direct sum of its radical and a maximal semi-simple subalgebra (see Levi–Mal'tsev decomposition). This made it possible to reduce the problem of classifying arbitrary Lie algebras to the enumeration of semi-simple (which for $ K = \mathbf C $ had already been done by W. Killing) and solvable Lie algebras. The classification of solvable Lie algebras (for $ K = \mathbf C $ or $ \mathbf R $ ) has been carried out only in dimensions $ \leq 6 $ .


If $ \mathfrak g $ is a solvable algebraic subalgebra (cf. Algebraic algebra) of $ \mathfrak g \mathfrak l (V) $ , where $ V $ is a finite-dimensional space over a field $ K $ of characteristic $ 0 $ , then $ \mathfrak g $ splits into the semi-direct product of the nilpotent ideal formed by all nilpotent transformations of $ \mathfrak g $ and the Abelian subalgebra consisting of the semi-simple transformations [6]. In general, any split solvable Lie algebra, that is, a finite-dimensional solvable Lie algebra over $ K $ every element $ x $ of which splits into a sum $ x = s + n $ , where $ s ,\ n \in \mathfrak g $ , $ [ s ,\ n ] = 0 $ , $ s $ is semi-simple, and $ n $ is nilpotent, has a similar structure [8]. To every finite-dimensional solvable Lie algebra over $ K $ there uniquely corresponds a minimal split solvable Lie algebra containing it (the Mal'tsev decomposition). The problem of classifying solvable Lie algebras that have a given Mal'tsev decomposition has been solved [8]. Thus, the problem of classifying solvable Lie algebras reduces, in a certain sense, to the study of nilpotent Lie algebras.

Apart from the radical, in an arbitrary finite-dimensional Lie algebra $ \mathfrak g $ one can distinguish maximal solvable subalgebras. If $ K $ is an algebraically closed field of characteristic $ 0 $ , then all such subalgebras of $ \mathfrak g $ ( they are called Borel subalgebras) are conjugate. For example, $ \mathfrak t ( n ,\ K ) $ is a Borel subalgebra of the Lie algebra of all matrices of order $ n $ . If $ K $ is not algebraically closed or if $ \mathop{\rm char}\nolimits \ K $ is finite, then Lie's theorem is false, in general. However, it can be extended to the case when $ K $ is perfect and contains the characteristic roots of all the characteristic polynomials $ \rho (x) $ of the adjoint transformations $ \mathop{\rm ad}\nolimits \ x $ , $ x \in \mathfrak g $ . If this condition is satisfied for the adjoint representation of a solvable Lie algebra $ \mathfrak g $ ( cf. Adjoint representation of a Lie group), then $ \mathfrak g $ is said to be triangular. Many properties of solvable Lie algebras over an algebraically closed field carry over to triangular Lie algebras. In particular, if $ \mathop{\rm char}\nolimits \ K = 0 $ , then all maximal triangular subalgebras of an arbitrary finite-dimensional Lie algebra are conjugate (see [1], [7]). Maximal triangular subalgebras are used in the study of semi-simple Lie algebras over an algebraically non-closed field as a good analogue of Borel subalgebras. They also play a fundamental role in the description of the connected uniform subgroups (cf. Uniform subgroup) of Lie groups [9].

References

[1] A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[2] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
[3] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) MR0148716 MR0143793 Zbl 0121.27504 Zbl 0109.26201
[4] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[5] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
[6] C. Chevalley, "Théorie des groupes de Lie" , 3 , Hermann (1955) MR0068552 Zbl 0186.33104 Zbl 0054.01303 Zbl 0063.00843
[7] E.B. Vinberg, "The Morozov–Borel theorem for real Lie groups" Soviet Math. Dokl. , 2 (1961) pp. 1416–1419 Dokl. Akad. Nauk SSSR , 141 (1961) pp. 270–273 MR0142683 Zbl 0112.02505
[8] A.I. Mal'tsev, "Solvable Lie algebras" Izv. Akad. Nauk SSSR , 9 (1945) pp. 329–352 (In Russian) Zbl 0061.05303
[9] A.L. Onishchik, "On Lie groups transitive on compact manifolds II" Math. USSR Sb. , 3 (1967) pp. 373–388 Mat. Sb. , 74 (1967) pp. 398–416 Zbl 0198.28903
[a1] J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) MR0323842 Zbl 0254.17004
How to Cite This Entry:
Lie algebra, solvable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_solvable&oldid=21987
This article was adapted from an original article by V.V. Gorbatsevich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article