View source for Lie algebra, supersolvable

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''triangular Lie algebra''

A finite-dimensional [[Lie algebra|Lie algebra]] $\mathfrak g$ over a field $k$ for which the eigen values of the operators $\ad X$ of the adjoint representation belong to $k$ for all $X\in\mathfrak g$ (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]).

A supersolvable Lie algebra is solvable. The class of supersolvable Lie algebras contains the class of nilpotent Lie algebras and is contained in the class of exponential Lie algebras (cf. [[Lie algebra, nilpotent|Lie algebra, nilpotent]]; [[Lie algebra, exponential|Lie algebra, exponential]]). It is closed with respect to transition to subalgebras, quotient algebras and finite direct sums, but it is not closed with respect to extensions.

A supersolvable Lie algebra over a [[Perfect field|perfect field]] has many of the properties of solvable Lie algebras (cf. [[Lie algebra, solvable|Lie algebra, solvable]]) over an algebraically closed field (Lie's theorem, the presence of a chain of ideals $\mathfrak g=\mathfrak g_0\supset\mathfrak g_1\supset\dots\supset\mathfrak g_n=\{0\}$ for which $\dim\mathfrak g_i=\dim\mathfrak g-i$, and others). In an arbitrary finite-dimensional Lie algebra $\mathfrak g$ there are maximal supersolvable subalgebras and they contain the nil radical. If $k=\mathbf R$ or $\mathbf C$, or if $k$ is perfect and $\mathfrak g$ is an algebraic linear Lie algebra, then all supersolvable subalgebras are conjugate. The Lie algebra $\mathfrak g$ over $k$ corresponding to a $k$-split algebraic group (cf. [[Split group|Split group]]) over a perfect field $k$ is a supersolvable Lie algebra.

Any supersolvable Lie algebra over a field of characteristic 0 can be isomorphically imbedded in the Lie algebra of upper-triangular matrices with coefficients from $k$ (which is itself supersolvable). The simplest example of a supersolvable Lie algebra that is not nilpotent is a two-dimensional Lie algebra with basis $X,Y$ and defining relation $[X,Y]=X$. For references see [[Lie group, supersolvable|Lie group, supersolvable]].