View source for Lie algebra, supersolvable

''triangular Lie algebra''

A finite-dimensional [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l0585301.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l0585302.png" /> for which the eigen values of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l0585303.png" /> of the adjoint representation belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l0585304.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l0585305.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l0585306.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l0585307.png" />, and others). In an arbitrary finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l0585308.png" /> there are maximal supersolvable subalgebras and they contain the nil radical. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l0585309.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l05853010.png" />, or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l05853011.png" /> is perfect and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l05853012.png" /> is an algebraic linear Lie algebra, then all supersolvable subalgebras are conjugate. The Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l05853013.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l05853014.png" /> corresponding to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l05853015.png" />-split algebraic group (cf. [[Split group|Split group]]) over a perfect field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l05853016.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l05853017.png" /> (which is itself supersolvable). The simplest example of a supersolvable Lie algebra that is not nilpotent is a two-dimensional Lie algebra with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l05853018.png" /> and defining relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058530/l05853019.png" />. For references see [[Lie group, supersolvable|Lie group, supersolvable]].