Lie algebras MOC

Solvable Lie algebra

A Lie algebra is solvable iff its derived series

terminates in the zero subalgebra, #m/def/lie i.e. for some .1 Clearly this is a special case of a Nilpotent Lie algebra.

Properties

  1. If is solvable, then so too are all subalgebras and homomorphic images.
  2. If is a solvable ideal such that the quotient is solvable, then is solvable.
  3. If are solvable ideals, then so to is .
Proof of 1–3

Clearly if , then for , so if the latter terminates so to does the former. Similarly given a epimorphism we have , and given

proving ^P1 by induction.

Let be the projection, and say . Then so . But then applying ^P1 the derived series of must terminate, and thus the derived series of terminates, proving ^P2.

By the second isomorphism theorem we have the isomorphism

Since the latter is the homomorphic image of , by ^P1 it is solvable, and thus is solvable by ^P2, proving ^P3.

See also

#state/tidy | #lang/en | #SemBr

Footnotes

  1. 1972. Introduction to Lie Algebras and Representation Theory, §3,1, p. 10 ↩