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Difference between revisions of "Lie algebra, supersolvable"

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''triangular Lie algebra''
 
''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]]).
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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 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.
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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 <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]].
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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]].

Latest revision as of 01:00, 24 December 2018

triangular Lie algebra

A finite-dimensional 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).

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, 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 has many of the properties of solvable Lie algebras (cf. 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) 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.

How to Cite This Entry:
Lie algebra, supersolvable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_supersolvable&oldid=17913
This article was adapted from an original article by V.V. Gorbatsevich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article