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A prescription for constructing anti-Hermitian representations of a symmetric [[Lie algebra|Lie algebra]] (over the real numbers) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101101.png" /> from representations of an Inönü–Wigner contraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101102.png" />. One assumes that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101103.png" /> is a direct sum of vector spaces and
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<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/g/g110/g110110/g1101104.png" /></td> </tr></table>
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Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101105.png" /> and there exists an isomorphism of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101106.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g1101109.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011011.png" />. In addition, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011012.png" />.
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A prescription for constructing anti-Hermitian representations of a symmetric [[Lie algebra|Lie algebra]] (over the real numbers)  $  \mathfrak g $
 +
from representations of an Inönü–Wigner contraction  $  {\overline{\mathfrak g}\; } $.  
 +
One assumes that $  \mathfrak g = \mathfrak k + \mathfrak p $
 +
is a direct sum of vector spaces and
  
The best studied examples concern the (pseudo-) orthogonal algebras, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011013.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011015.png" /> [[#References|[a1]]], [[#References|[a2]]]. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011016.png" /> is an inhomogeneous Lie algebra with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011017.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011018.png" /> be the quadratic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011019.png" />-invariant element from the symmetric algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011021.png" /> is an anti-Hermitian representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011023.png" /> is a multiple of the unit operator, then the formula for the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011025.png" /> reads: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011026.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011027.png" />, and, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011028.png" />,
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$$
 +
[ \mathfrak k, \mathfrak k ] \subset  \mathfrak k, [ \mathfrak k, \mathfrak p ] \subset  \mathfrak p, [ \mathfrak p, \mathfrak p ] \subset  \mathfrak k.
 +
$$
  
<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/g/g110/g110110/g11011029.png" /></td> </tr></table>
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Then  $  {\overline{\mathfrak g}\; } = {\overline{\mathfrak k}\; } + {\overline{\mathfrak p}\; } $
 +
and there exists an isomorphism of vector spaces  $  \pi : \mathfrak g \rightarrow { {\overline{\mathfrak g}\; } } $
 +
such that  $  \pi ( \mathfrak k ) = {\overline{\mathfrak k}\; } $,
 +
$  \pi ( \mathfrak p ) = {\overline{\mathfrak p}\; } $
 +
and  $  [ \pi ( X ) , \pi ( Y ) ] = \pi ( [ X,Y ] ) $
 +
for all  $  X \in \mathfrak k $,
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$  Y \in \mathfrak g $.
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In addition, one has  $  [ {\overline{\mathfrak p}\; } , {\overline{\mathfrak p}\; } ] = 0 $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011030.png" /> is the second-degree Casimir element from the [[Universal enveloping algebra|universal enveloping algebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011031.png" /> while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011033.png" /> are parameters. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011034.png" /> is real and arbitrary and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011035.png" /> is pure imaginary and depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g110/g110110/g11011036.png" />.
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The best studied examples concern the (pseudo-) orthogonal algebras, when  $  \mathfrak g = \mathfrak s \mathfrak o ( m + 1,n ) $
 +
or  $  \mathfrak g = \mathfrak s \mathfrak o ( m,n + 1 ) $
 +
and  $  \mathfrak k = \mathfrak s \mathfrak o ( m,n ) $[[#References|[a1]]], [[#References|[a2]]]. Then  $  {\overline{\mathfrak g}\; } = \mathfrak i \mathfrak s \mathfrak o ( m,n ) $
 +
is an inhomogeneous Lie algebra with  $  {\overline{\mathfrak p}\; } = \mathbf R ^ {m + n } $.  
 +
Let  $  M  ^ {2} $
 +
be the quadratic  $  {\overline{\mathfrak k}\; } $-
 +
invariant element from the symmetric algebra of  $  {\overline{\mathfrak p}\; } $.  
 +
If  $  {\overline \rho \; } $
 +
is an anti-Hermitian representation of  $  {\overline{\mathfrak g}\; } $
 +
such that  $  {\overline \rho \; } ( M  ^ {2} ) $
 +
is a multiple of the unit operator, then the formula for the representation  $  \rho $
 +
of  $  \mathfrak g $
 +
reads:  $  \rho ( X ) = {\overline \rho \; } ( \pi ( X ) ) $
 +
for all  $  X \in \mathfrak k $,
 +
and, for all  $  Y \in \mathfrak p $,
 +
 
 +
$$
 +
\rho ( Y ) = \lambda  {\overline \rho \; } ( \pi ( Y ) ) + a [ {\overline \rho \; } ( \Delta ) , {\overline \rho \; } ( \pi ( Y ) ) ] ,
 +
$$
 +
 
 +
where  $  \Delta $
 +
is the second-degree Casimir element from the [[Universal enveloping algebra|universal enveloping algebra]] of $  {\overline{\mathfrak k}\; } $
 +
while $  \lambda $
 +
and $  a $
 +
are parameters. Here, $  \lambda $
 +
is real and arbitrary and $  a $
 +
is pure imaginary and depends on $  {\overline \rho \; } ( M  ^ {2} ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Weimar,  "The range of validity of the Gell-Mann formula"  ''Nuovo Cim. Lett.'' , '''4'''  (1972)  pp. 43–50</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Hermann,  "Lie groups for physicists" , Benjamin  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Weimar,  "The range of validity of the Gell-Mann formula"  ''Nuovo Cim. Lett.'' , '''4'''  (1972)  pp. 43–50</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Hermann,  "Lie groups for physicists" , Benjamin  (1966)</TD></TR></table>

Latest revision as of 19:41, 5 June 2020


A prescription for constructing anti-Hermitian representations of a symmetric Lie algebra (over the real numbers) $ \mathfrak g $ from representations of an Inönü–Wigner contraction $ {\overline{\mathfrak g}\; } $. One assumes that $ \mathfrak g = \mathfrak k + \mathfrak p $ is a direct sum of vector spaces and

$$ [ \mathfrak k, \mathfrak k ] \subset \mathfrak k, [ \mathfrak k, \mathfrak p ] \subset \mathfrak p, [ \mathfrak p, \mathfrak p ] \subset \mathfrak k. $$

Then $ {\overline{\mathfrak g}\; } = {\overline{\mathfrak k}\; } + {\overline{\mathfrak p}\; } $ and there exists an isomorphism of vector spaces $ \pi : \mathfrak g \rightarrow { {\overline{\mathfrak g}\; } } $ such that $ \pi ( \mathfrak k ) = {\overline{\mathfrak k}\; } $, $ \pi ( \mathfrak p ) = {\overline{\mathfrak p}\; } $ and $ [ \pi ( X ) , \pi ( Y ) ] = \pi ( [ X,Y ] ) $ for all $ X \in \mathfrak k $, $ Y \in \mathfrak g $. In addition, one has $ [ {\overline{\mathfrak p}\; } , {\overline{\mathfrak p}\; } ] = 0 $.

The best studied examples concern the (pseudo-) orthogonal algebras, when $ \mathfrak g = \mathfrak s \mathfrak o ( m + 1,n ) $ or $ \mathfrak g = \mathfrak s \mathfrak o ( m,n + 1 ) $ and $ \mathfrak k = \mathfrak s \mathfrak o ( m,n ) $[a1], [a2]. Then $ {\overline{\mathfrak g}\; } = \mathfrak i \mathfrak s \mathfrak o ( m,n ) $ is an inhomogeneous Lie algebra with $ {\overline{\mathfrak p}\; } = \mathbf R ^ {m + n } $. Let $ M ^ {2} $ be the quadratic $ {\overline{\mathfrak k}\; } $- invariant element from the symmetric algebra of $ {\overline{\mathfrak p}\; } $. If $ {\overline \rho \; } $ is an anti-Hermitian representation of $ {\overline{\mathfrak g}\; } $ such that $ {\overline \rho \; } ( M ^ {2} ) $ is a multiple of the unit operator, then the formula for the representation $ \rho $ of $ \mathfrak g $ reads: $ \rho ( X ) = {\overline \rho \; } ( \pi ( X ) ) $ for all $ X \in \mathfrak k $, and, for all $ Y \in \mathfrak p $,

$$ \rho ( Y ) = \lambda {\overline \rho \; } ( \pi ( Y ) ) + a [ {\overline \rho \; } ( \Delta ) , {\overline \rho \; } ( \pi ( Y ) ) ] , $$

where $ \Delta $ is the second-degree Casimir element from the universal enveloping algebra of $ {\overline{\mathfrak k}\; } $ while $ \lambda $ and $ a $ are parameters. Here, $ \lambda $ is real and arbitrary and $ a $ is pure imaginary and depends on $ {\overline \rho \; } ( M ^ {2} ) $.

References

[a1] E. Weimar, "The range of validity of the Gell-Mann formula" Nuovo Cim. Lett. , 4 (1972) pp. 43–50
[a2] R. Hermann, "Lie groups for physicists" , Benjamin (1966)
How to Cite This Entry:
Gell-Mann formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gell-Mann_formula&oldid=47062
This article was adapted from an original article by P. Stovicek (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article