Lie algebra, solvable
A Lie algebra over a field
satisfying one of the following equivalent conditions:
1) the terms of the derived series of
are equal to
for sufficiently large
;
2) there is a finite decreasing chain of ideals of
such that
,
and
(that is, the Lie algebras
are Abelian for all
);
3) there is a finite decreasing chain of subalgebras such that
,
,
is an ideal of
, and
is a one-dimensional (Abelian) Lie algebra for
.
A nilpotent Lie algebra (cf. Lie algebra, nilpotent) is solvable. If is a complete flag in a finite-dimensional vector space
over
, then
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is a solvable subalgebra of the Lie algebra of all linear transformations of
. If one chooses a basis in
compatible with
, then with respect to that basis, elements of
are represented by upper triangular matrices; the resulting solvable linear Lie algebra is denoted by
, where
.
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 is solvable. If
and
is an algebraically closed field, then any finite-dimensional solvable Lie algebra is isomorphic to a subalgebra of
for some
. One of the main properties of solvable Lie algebras is expressed in Lie's theorem: Let
be a solvable Lie algebra over an algebraically closed field of characteristic
and let
be a finite-dimensional linear representation of it. Then in
there is a complete flag
such that
. In particular, if
is irreducible, then
. Ideals of
can be chosen so as to form a complete flag, that is, so that
.
A finite-dimensional Lie algebra over a field of characteristic
is solvable if and only if the algebra
is nilpotent. Another criterion for solvability (Cartan's criterion) is:
is solvable if and only if
is orthogonal to the whole of
with respect to the Killing form (or any bilinear form associated with a faithful finite-dimensional representation of
).
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 , and it was proved that in the case
the algebra
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
had already been done by W. Killing) and solvable Lie algebras. The classification of solvable Lie algebras (for
or
) has been carried out only in dimensions
.
If is a solvable algebraic subalgebra (cf. Algebraic algebra) of
, where
is a finite-dimensional space over a field
of characteristic
, then
splits into the semi-direct product of the nilpotent ideal formed by all nilpotent transformations of
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
every element
of which splits into a sum
, where
,
,
is semi-simple, and
is nilpotent, has a similar structure [8]. To every finite-dimensional solvable Lie algebra over
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 one can distinguish maximal solvable subalgebras. If
is an algebraically closed field of characteristic
, then all such subalgebras of
(they are called Borel subalgebras) are conjugate. For example,
is a Borel subalgebra of the Lie algebra of all matrices of order
. If
is not algebraically closed or if
is finite, then Lie's theorem is false, in general. However, it can be extended to the case when
is perfect and contains the characteristic roots of all the characteristic polynomials
of the adjoint transformations
,
. If this condition is satisfied for the adjoint representation of a solvable Lie algebra
(cf. Adjoint representation of a Lie group), then
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
, 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) |
[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 |
Comments
References
[a1] | J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) MR0323842 Zbl 0254.17004 |
Lie algebra, solvable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_solvable&oldid=17061