Solvable Lie algebra

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In mathematics, a Lie algebra is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra is the subalgebra of , denoted

that consists of all Lie brackets of pairs of elements of . The derived series is the sequence of subalgebras

If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is solvable.[1] The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory.

Any nilpotent Lie algebra is solvable, a fortiori, but the converse is not true. The solvable Lie algebras and the semisimple Lie algebras form two large and generally complementary classes, as is shown by the Levi decomposition.

A maximal solvable subalgebra is called a Borel subalgebra. The largest solvable ideal of a Lie algebra is called the radical.

Characterizations[edit]

Let be a finite-dimensional Lie algebra over a field of characteristic 0. The following are equivalent.

  • (i) is solvable.
  • (ii) , the adjoint representation of , is solvable.
  • (iii) There is a finite sequence of ideals of }:
  • (iv) is nilpotent.[2]
  • (v) For -dimensional, there is a finite sequence of subalgebras of :
with each an ideal in .[3] A sequence of this type is called an elementary sequence.
  • (vi) There is a finite sequence of subalgebras of ,
such that is an ideal in and is abelian.[4]
  • (vii) is solvable if and only if its Killing form satisfies for all X in and Y in .[5] This is Cartan's criterion for solvability.

Properties[edit]

Lie's Theorem states that if is a finite-dimensional vector space over an algebraically closed field of characteristic zero, and is a solvable linear Lie algebra over a subfield of , and if is a representation of over , then there exists a simultaneous eigenvector of the matrices for all elements . More generally, the result holds if all eigenvalues of lie in for all .[6]

  • Every Lie subalgebra, quotient and extension[clarification needed] of a solvable Lie algebra is solvable.
  • A solvable nonzero Lie algebra has a nonzero abelian ideal, the last nonzero term in the derived series.[7]
  • A homomorphic image of a solvable Lie algebra is solvable.[7]
  • If is a solvable ideal in and is solvable, then is solvable.[7]
  • If is finite-dimensional, then there is a unique solvable ideal containing all solvable ideals in . This ideal is the radical of , denoted .[7]
  • If are solvable ideals, then so is .[1]
  • A solvable Lie algebra has a unique largest nilpotent ideal , the set of all such that is nilpotent. If D is any derivation of , then .[8]

Completely solvable Lie algebras[edit]

A Lie algebra is called completely solvable or split solvable if it has an elementary sequence[when defined as?] of ideals in from to . A finite-dimensional nilpotent Lie algebra is completely solvable, and a completely solvable Lie algebra is solvable. Over an algebraically closed field a solvable Lie algebra is completely solvable, but the -dimensional real Lie algebra of the group of Euclidean isometries of the plane is solvable but not completely solvable.

A solvable Lie algebra is split solvable if and only if the eigenvalues of are in for all in .[7]

Examples[edit]

  • A semisimple Lie algebra is never solvable.[1]
  • Every abelian Lie algebra is solvable.
  • Every nilpotent Lie algebra is solvable.
  • Let be the subalgebra of consisting of upper triangular matrices. Then is solvable.
  • Let be the set of matrices on the form
Then is solvable, but not split solvable.[7] It is isomorphic with the Lie algebra of the group of translations and rotations in the plane.

Solvable Lie groups[edit]

Because the term "solvable" is also used for solvable groups in group theory, there are several possible definitions of solvable Lie group. For a Lie group , there is

  • termination of the usual derived series of the group (as an abstract group);
  • termination of the closures of the derived series;
  • having a solvable Lie al

See also[edit]

External links[edit]

Notes[edit]

  1. ^ a b c Humphreys 1972
  2. ^ Knapp 2002 Proposition 1.39.
  3. ^ Knapp 2002 Proposition 1.23.
  4. ^ Fulton & Harris 1991
  5. ^ Knapp 2002 Proposition 1.46.
  6. ^ Knapp 2002 Theorem 1.25.
  7. ^ a b c d e f Knapp 2002
  8. ^ Knapp 2002 Proposition 1.40.

References[edit]

  • Fulton, W.; Harris, J. (1991). Representation theory. A first course. Graduate Texts in Mathematics. 129. New York: Springer-Verlag. ISBN 978-0-387-97527-6. MR 1153249. 
  • Humphreys, James E. (1972). Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics. 9. New York: Springer-Verlag. ISBN 0-387-90053-5. 
  • Knapp, A. W. (2002). Lie groups beyond an introduction. Progress in Mathematics. 120 (2nd ed.). Boston·Basel·Berlin: Birkhäuser. ISBN 0-8176-4259-5. .