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''Lie algebra of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l0584002.png" />''
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A finite-dimensional real [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l0584003.png" /> for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l0584004.png" /> of which the operator of adjoint representation (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l0584005.png" /> does not have purely imaginary eigen values. The [[Exponential mapping|exponential mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l0584006.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l0584007.png" /> into the corresponding simply-connected Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l0584008.png" /> is a diffeomorphism, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l0584009.png" /> is an exponential Lie group (cf. [[Lie group, exponential|Lie group, exponential]]).
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Every exponential Lie algebra is solvable (cf. [[Lie algebra, solvable|Lie algebra, solvable]]). A nilpotent Lie algebra (cf. [[Lie algebra, nilpotent|Lie algebra, nilpotent]]) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840010.png" /> is an exponential Lie algebra. The class of exponential Lie algebras is intermediate between the classes of all solvable and all supersolvable Lie algebras (cf. [[Lie algebra, supersolvable|Lie algebra, supersolvable]]); it is closed with respect to transition to subalgebras, quotient algebras and finite direct sums, but it is not closed with respect to extensions.
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''Lie algebra of type  $  ( E) $''
  
The simplest example of an exponential Lie algebra that is not a supersolvable Lie algebra is the three-dimensional Lie algebra with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840011.png" /> and multiplication specified by the formulas
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A finite-dimensional real [[Lie algebra|Lie algebra]]  $  \mathfrak g $
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for any element  $  X $
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of which the operator of adjoint representation (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]])  $  \mathop{\rm ad}  X $
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does not have purely imaginary eigen values. The [[Exponential mapping|exponential mapping]]  $  \mathop{\rm exp} :  \mathfrak g \rightarrow G $
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of the algebra $  \mathfrak g $
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into the corresponding simply-connected Lie group  $  G $
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is a diffeomorphism, and  $  G $
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is an exponential Lie group (cf. [[Lie group, exponential|Lie group, exponential]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840012.png" /></td> </tr></table>
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Every exponential Lie algebra is solvable (cf. [[Lie algebra, solvable|Lie algebra, solvable]]). A nilpotent Lie algebra (cf. [[Lie algebra, nilpotent|Lie algebra, nilpotent]]) over  $  \mathbf R $
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is an exponential Lie algebra. The class of exponential Lie algebras is intermediate between the classes of all solvable and all supersolvable Lie algebras (cf. [[Lie algebra, supersolvable|Lie algebra, supersolvable]]); it is closed with respect to transition to subalgebras, quotient algebras and finite direct sums, but it is not closed with respect to extensions.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840013.png" /> is a real matrix that has complex but not purely imaginary eigen values. The three-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840014.png" /> with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840015.png" /> and defining relations
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The simplest example of an exponential Lie algebra that is not a supersolvable Lie algebra is the three-dimensional Lie algebra with basis $  X , Y , Z $
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and multiplication specified by the formulas
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840016.png" /></td> </tr></table>
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$$
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[ X , Y ]  = 0 ,\ \
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[ Z , X ]  = a _ {11} X + a _ {12} Y ,\ \
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[ Z , Y ]  = a _ {21} X + a _ {22} Y ,
 +
$$
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 +
where  $  [ a _ {ij} ] $
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is a real matrix that has complex but not purely imaginary eigen values. The three-dimensional Lie algebra  $  \mathfrak g _ {0} $
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with basis  $  X , Y , Z $
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and defining relations
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 +
$$
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[ X , Y ]  = 0 ,\ \
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[ Z , X ]  = Y ,\ \
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[ Z , Y ]  = - X
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$$
  
 
is a solvable, but not an exponential Lie algebra.
 
is a solvable, but not an exponential Lie algebra.
  
A Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840017.png" /> is exponential if and only if all roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840018.png" /> (cf. [[Root system|Root system]]) have the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840021.png" /> are real linear forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840023.png" /> is proportional to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840024.png" /> (see ), or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840025.png" /> has no quotient algebra containing a subalgebra isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058400/l05840026.png" />.
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A Lie algebra $  \mathfrak g $
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is exponential if and only if all roots of $  \mathfrak g $(
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cf. [[Root system|Root system]]) have the form $  \alpha + i \beta $,  
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where $  \alpha $
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and $  \beta $
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are real linear forms on $  \mathfrak g $
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and $  \beta $
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is proportional to $  \alpha $(
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see ), or if $  \mathfrak g $
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has no quotient algebra containing a subalgebra isomorphic to $  \mathfrak g _ {0} $.
  
 
For references see [[Lie group, exponential|Lie group, exponential]].
 
For references see [[Lie group, exponential|Lie group, exponential]].

Latest revision as of 22:16, 5 June 2020


Lie algebra of type $ ( E) $

A finite-dimensional real Lie algebra $ \mathfrak g $ for any element $ X $ of which the operator of adjoint representation (cf. Adjoint representation of a Lie group) $ \mathop{\rm ad} X $ does not have purely imaginary eigen values. The exponential mapping $ \mathop{\rm exp} : \mathfrak g \rightarrow G $ of the algebra $ \mathfrak g $ into the corresponding simply-connected Lie group $ G $ is a diffeomorphism, and $ G $ is an exponential Lie group (cf. Lie group, exponential).

Every exponential Lie algebra is solvable (cf. Lie algebra, solvable). A nilpotent Lie algebra (cf. Lie algebra, nilpotent) over $ \mathbf R $ is an exponential Lie algebra. The class of exponential Lie algebras is intermediate between the classes of all solvable and all supersolvable Lie algebras (cf. Lie algebra, supersolvable); it is closed with respect to transition to subalgebras, quotient algebras and finite direct sums, but it is not closed with respect to extensions.

The simplest example of an exponential Lie algebra that is not a supersolvable Lie algebra is the three-dimensional Lie algebra with basis $ X , Y , Z $ and multiplication specified by the formulas

$$ [ X , Y ] = 0 ,\ \ [ Z , X ] = a _ {11} X + a _ {12} Y ,\ \ [ Z , Y ] = a _ {21} X + a _ {22} Y , $$

where $ [ a _ {ij} ] $ is a real matrix that has complex but not purely imaginary eigen values. The three-dimensional Lie algebra $ \mathfrak g _ {0} $ with basis $ X , Y , Z $ and defining relations

$$ [ X , Y ] = 0 ,\ \ [ Z , X ] = Y ,\ \ [ Z , Y ] = - X $$

is a solvable, but not an exponential Lie algebra.

A Lie algebra $ \mathfrak g $ is exponential if and only if all roots of $ \mathfrak g $( cf. Root system) have the form $ \alpha + i \beta $, where $ \alpha $ and $ \beta $ are real linear forms on $ \mathfrak g $ and $ \beta $ is proportional to $ \alpha $( see ), or if $ \mathfrak g $ has no quotient algebra containing a subalgebra isomorphic to $ \mathfrak g _ {0} $.

For references see Lie group, exponential.

How to Cite This Entry:
Lie algebra, exponential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_exponential&oldid=47623
This article was adapted from an original article by V.V. Gorbatsevich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article