Difference between revisions of "Lie algebra, exponential"
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| − | + | ''Lie algebra of type $ ( E) $'' | |
| − | + | A finite-dimensional real [[Lie algebra|Lie algebra]] $ \mathfrak g $ | |
| + | for any element $ X $ | ||
| + | of which the operator of adjoint representation (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]) $ \mathop{\rm ad} X $ | ||
| + | does not have purely imaginary eigen values. The [[Exponential mapping|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|Lie group, exponential]]). | ||
| − | + | 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 $ | |
| + | 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. | ||
| − | + | 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. | is a solvable, but not an exponential Lie algebra. | ||
| − | A Lie algebra | + | A Lie algebra $ \mathfrak g $ |
| + | is exponential if and only if all roots of $ \mathfrak g $( | ||
| + | cf. [[Root system|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|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.
Lie algebra, exponential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_algebra,_exponential&oldid=13258