Namespaces
Variants
Actions

Difference between revisions of "Casimir element"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
c0206901.png
 +
$#A+1 = 43 n = 0
 +
$#C+1 = 43 : ~/encyclopedia/old_files/data/C020/C.0200690 Casimir element,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''Casimir operator''
 
''Casimir operator''
  
 
A central element of special form in the universal enveloping algebra of a semi-simple Lie algebra. Such operators were first introduced, for a particular case, by H. Casimir [[#References|[1]]].
 
A central element of special form in the universal enveloping algebra of a semi-simple Lie algebra. Such operators were first introduced, for a particular case, by H. Casimir [[#References|[1]]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c0206901.png" /> be a semi-simple finite-dimensional Lie algebra over a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c0206902.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c0206903.png" /> be an invariant symmetric bilinear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c0206904.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c0206905.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c0206906.png" />) which is non-degenerate on a Cartan subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c0206907.png" />. Then a Casimir element of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c0206908.png" /> with respect to the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c0206909.png" /> is an element of the universal enveloping algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069010.png" /> that is representable in the form
+
Let $  \mathfrak g $
 +
be a semi-simple finite-dimensional Lie algebra over a field of characteristic 0 $,  
 +
and let $  B $
 +
be an invariant symmetric bilinear form on $  \mathfrak g $(
 +
that is, $  B ( [ x , y ] , z ) =B ( x , [ y , z ] ) $
 +
for all $  x , y , z \in \mathfrak g $)  
 +
which is non-degenerate on a Cartan subalgebra $  \mathfrak g _ {0} \subset  \mathfrak g $.  
 +
Then a Casimir element of the Lie algebra $  \mathfrak g $
 +
with respect to the form $  B $
 +
is an element of the universal enveloping algebra $  U ( \mathfrak g ) $
 +
that is representable in the form
  
<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/c/c020/c020690/c02069011.png" /></td> </tr></table>
+
$$
 +
= \sum _ { i=1 } ^ { k }
 +
e _ {i} f _ {i} .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069013.png" /> are dual bases of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069014.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069015.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069018.png" /> is the Kronecker symbol and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069019.png" />. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069020.png" /> does not depend on the choice of the dual bases in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069021.png" /> and belongs to the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069022.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069023.png" /> is a simple algebra, then a Casimir element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069024.png" /> defined by the [[Killing form|Killing form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069025.png" /> is the unique (up to a scalar multiplier) central element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069026.png" /> that is representable as a homogeneous quadratic polynomial in the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069027.png" />.
+
Here $  \{ e _ {i} \} $,  
 +
$  \{ f _ {i} \} $
 +
are dual bases of $  \mathfrak g _ {0} $
 +
with respect to $  B $,  
 +
that is, $  B ( e _ {i} , f _ {i} ) = \delta _ {ij} $,  
 +
$  i = 1 \dots k $,  
 +
where $  \delta _ {ij} $
 +
is the Kronecker symbol and $  k = \mathop{\rm dim}  \mathfrak g _ {0} $.  
 +
The element $  b $
 +
does not depend on the choice of the dual bases in $  \mathfrak g _ {0} $
 +
and belongs to the centre of $  U ( \mathfrak g _ {0} ) $.  
 +
If $  \mathfrak g $
 +
is a simple algebra, then a Casimir element of $  \mathfrak g $
 +
defined by the [[Killing form|Killing form]] $  B $
 +
is the unique (up to a scalar multiplier) central element in $  U ( \mathfrak g ) $
 +
that is representable as a homogeneous quadratic polynomial in the elements of $  \mathfrak g $.
  
Every linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069028.png" /> of a semi-simple algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069029.png" /> in a finite-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069030.png" /> defines an invariant symmetric bilinear form
+
Every linear representation $  \phi $
 +
of a semi-simple algebra $  \mathfrak g $
 +
in a finite-dimensional space $  V $
 +
defines an invariant symmetric bilinear form
  
<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/c/c020/c020690/c02069031.png" /></td> </tr></table>
+
$$
 +
B _  \phi  ( x , y )  = \
 +
\mathop{\rm Tr} ( \phi (x) \phi (y) )
 +
$$
  
on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069032.png" />, which is non-degenerate on the subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069033.png" /> complementary to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069034.png" />, and therefore also defines some Casimir element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069035.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069036.png" /> is an irreducible representation, then the extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069037.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069038.png" /> takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069039.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069040.png" />.
+
on $  \mathfrak g $,  
 +
which is non-degenerate on the subalgebra $  \mathfrak g _ {0} \subset  \mathfrak g $
 +
complementary to $  \mathop{\rm ker}  \phi $,  
 +
and therefore also defines some Casimir element $  b _  \phi  \in U ( \mathfrak g ) $.  
 +
If $  \phi $
 +
is an irreducible representation, then the extension of $  \phi $
 +
onto $  U ( \mathfrak g ) $
 +
takes $  b _  \phi  $
 +
into $  ( k / \mathop{\rm dim}  V ) E $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Casimir,  B.L. van der Waerden,  "Algebraischer Beweis der Vollständigen Reduzibilität der Darstellungen halbeinfacher Liescher Gruppen"  ''Math. Ann.'' , '''111'''  (1935)  pp. 1–2</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Jacobson,  "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.A. Naimark,  "Theory of group representations" , Springer  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. Dixmier,  "Enveloping algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Casimir,  B.L. van der Waerden,  "Algebraischer Beweis der Vollständigen Reduzibilität der Darstellungen halbeinfacher Liescher Gruppen"  ''Math. Ann.'' , '''111'''  (1935)  pp. 1–2</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Jacobson,  "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.A. Naimark,  "Theory of group representations" , Springer  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  J. Dixmier,  "Enveloping algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The Casimir element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069041.png" /> determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069042.png" /> is called the Casimir element of the linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020690/c02069043.png" />.
+
The Casimir element $  b _  \phi  $
 +
determined by $  \phi $
 +
is called the Casimir element of the linear representation $  \phi $.
  
 
An additional good reference is [[#References|[a1]]].
 
An additional good reference is [[#References|[a1]]].

Latest revision as of 10:08, 4 June 2020


Casimir operator

A central element of special form in the universal enveloping algebra of a semi-simple Lie algebra. Such operators were first introduced, for a particular case, by H. Casimir [1].

Let $ \mathfrak g $ be a semi-simple finite-dimensional Lie algebra over a field of characteristic $ 0 $, and let $ B $ be an invariant symmetric bilinear form on $ \mathfrak g $( that is, $ B ( [ x , y ] , z ) =B ( x , [ y , z ] ) $ for all $ x , y , z \in \mathfrak g $) which is non-degenerate on a Cartan subalgebra $ \mathfrak g _ {0} \subset \mathfrak g $. Then a Casimir element of the Lie algebra $ \mathfrak g $ with respect to the form $ B $ is an element of the universal enveloping algebra $ U ( \mathfrak g ) $ that is representable in the form

$$ b = \sum _ { i=1 } ^ { k } e _ {i} f _ {i} . $$

Here $ \{ e _ {i} \} $, $ \{ f _ {i} \} $ are dual bases of $ \mathfrak g _ {0} $ with respect to $ B $, that is, $ B ( e _ {i} , f _ {i} ) = \delta _ {ij} $, $ i = 1 \dots k $, where $ \delta _ {ij} $ is the Kronecker symbol and $ k = \mathop{\rm dim} \mathfrak g _ {0} $. The element $ b $ does not depend on the choice of the dual bases in $ \mathfrak g _ {0} $ and belongs to the centre of $ U ( \mathfrak g _ {0} ) $. If $ \mathfrak g $ is a simple algebra, then a Casimir element of $ \mathfrak g $ defined by the Killing form $ B $ is the unique (up to a scalar multiplier) central element in $ U ( \mathfrak g ) $ that is representable as a homogeneous quadratic polynomial in the elements of $ \mathfrak g $.

Every linear representation $ \phi $ of a semi-simple algebra $ \mathfrak g $ in a finite-dimensional space $ V $ defines an invariant symmetric bilinear form

$$ B _ \phi ( x , y ) = \ \mathop{\rm Tr} ( \phi (x) \phi (y) ) $$

on $ \mathfrak g $, which is non-degenerate on the subalgebra $ \mathfrak g _ {0} \subset \mathfrak g $ complementary to $ \mathop{\rm ker} \phi $, and therefore also defines some Casimir element $ b _ \phi \in U ( \mathfrak g ) $. If $ \phi $ is an irreducible representation, then the extension of $ \phi $ onto $ U ( \mathfrak g ) $ takes $ b _ \phi $ into $ ( k / \mathop{\rm dim} V ) E $.

References

[1] H. Casimir, B.L. van der Waerden, "Algebraischer Beweis der Vollständigen Reduzibilität der Darstellungen halbeinfacher Liescher Gruppen" Math. Ann. , 111 (1935) pp. 1–2
[2] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)
[3] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)
[4] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[5] M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian)
[6] J. Dixmier, "Enveloping algebras" , North-Holland (1977) (Translated from French)

Comments

The Casimir element $ b _ \phi $ determined by $ \phi $ is called the Casimir element of the linear representation $ \phi $.

An additional good reference is [a1].

References

[a1] J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972)
How to Cite This Entry:
Casimir element. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Casimir_element&oldid=11861
This article was adapted from an original article by D.P. Zhelobenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article