solvable Lie algebra

Let 𝔤 be a Lie algebraMathworldPlanetmath. The lower central seriesPlanetmathPlanetmath of 𝔤 is the filtrationPlanetmathPlanetmath of subalgebrasMathworldPlanetmathPlanetmath


of 𝔤, inductively defined for every natural numberMathworldPlanetmath k as follows:

𝒟1𝔤 := [𝔤,𝔤]
𝒟k𝔤 := [𝔤,𝒟k-1𝔤]

The upper central series of 𝔤 is the filtration


defined inductively by

𝒟1𝔤 := [𝔤,𝔤]
𝒟k𝔤 := [𝒟k-1𝔤,𝒟k-1𝔤]

In fact both 𝒟k𝔤 and 𝒟k𝔤 are ideals of 𝔤, and 𝒟k𝔤𝒟k𝔤 for all k. The Lie algebra 𝔤 is defined to be nilpotent if 𝒟k𝔤=0 for some k, and solvable if 𝒟k𝔤=0 for some k.

A subalgebra 𝔥 of 𝔤 is said to be nilpotent or solvable if 𝔥 is nilpotent or solvable when considered as a Lie algebra in its own right. The terms may also be applied to ideals of 𝔤, since every ideal of 𝔤 is also a subalgebra.

Title solvable Lie algebra
Canonical name SolvableLieAlgebra
Date of creation 2013-03-22 12:41:06
Last modified on 2013-03-22 12:41:06
Owner djao (24)
Last modified by djao (24)
Numerical id 4
Author djao (24)
Entry type Definition
Classification msc 17B30
Defines nilpotent Lie algebra
Defines solvable
Defines nilpotent
Defines lower central series
Defines upper central series