Namespaces
Variants
Actions

Difference between revisions of "Weight of a representation of a Lie algebra"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
''in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w0975401.png" />''
+
<!--
 +
w0975401.png
 +
$#A+1 = 95 n = 0
 +
$#C+1 = 95 : ~/encyclopedia/old_files/data/W097/W.0907540 Weight of a representation of a Lie algebra
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
A linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w0975402.png" /> from the [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w0975403.png" /> into its field of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w0975404.png" /> for which there exists a non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w0975405.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w0975406.png" /> such that for the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w0975407.png" /> one has
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/w/w097/w097540/w0975408.png" /></td> </tr></table>
+
''in a vector space  $  V $''
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w0975409.png" /> and some integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754010.png" /> (which in general depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754012.png" />). Here 1 denotes the identity transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754013.png" />. One also says in such a case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754014.png" /> is a weight of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754016.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754017.png" /> defined by the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754018.png" />. The set of all vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754019.png" /> which satisfy this condition, together with zero, forms a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754020.png" />, which is known as the weight subspace of the weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754021.png" /> (or corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754022.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754024.png" /> is said to be a weight space or weight module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754027.png" /> of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754028.png" />.
+
A linear mapping  $  \alpha $
 +
from the [[Lie algebra|Lie algebra]]  $  L $
 +
into its field of definition  $  k $
 +
for which there exists a non-zero vector  $  x $
 +
of $  V $
 +
such that for the representation $  \rho $
 +
one has
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754029.png" /> is a finite-dimensional module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754030.png" /> of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754031.png" />, its contragredient module (cf. [[Contragredient representation|Contragredient representation]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754032.png" /> is a weight module of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754033.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754035.png" /> are weight modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754036.png" /> of weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754038.png" />, respectively, then their tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754039.png" /> is a weight module of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754040.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754041.png" /> is a nilpotent Lie algebra, a weight subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754042.png" /> of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754043.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754044.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754045.png" />-submodule of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754046.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754047.png" />. If, in addition,
+
$$
 +
( \rho ( h) - \alpha ( h) 1)  ^ {n} _ {x,h} ( x) = 0
 +
$$
  
<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/w/w097/w097540/w09754048.png" /></td> </tr></table>
+
for all  $  h \in L $
 +
and some integer  $  n _ {x,h} > 0 $(
 +
which in general depends on  $  x $
 +
and  $  h $).
 +
Here 1 denotes the identity transformation of  $  V $.
 +
One also says in such a case that  $  \alpha $
 +
is a weight of the  $  L $-
 +
module  $  V $
 +
defined by the representation  $  \rho $.  
 +
The set of all vectors  $  x \in V $
 +
which satisfy this condition, together with zero, forms a subspace  $  V _  \alpha  $,
 +
which is known as the weight subspace of the weight  $  \alpha $(
 +
or corresponding to  $  \alpha $).  
 +
If  $  V = V _  \alpha  $,
 +
then  $  V $
 +
is said to be a weight space or weight module over  $  L $
 +
of weight  $  \alpha $.
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754049.png" /> is a splitting Lie algebra of linear transformations of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754050.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754051.png" /> can be decomposed into a direct sum of a finite number of weight subspaces of different weights:
+
If  $  V $
 +
is a finite-dimensional module over  $  L $
 +
of weight  $  \alpha $,
 +
its contragredient module (cf. [[Contragredient representation|Contragredient representation]])  $  V  ^ {*} $
 +
is a weight module of weight  $  - \alpha $;
 +
if  $  V $
 +
and  $  W $
 +
are weight modules over  $  L $
 +
of weights  $  \alpha $
 +
and  $  \beta $,
 +
respectively, then their tensor product  $  V \otimes W $
 +
is a weight module of weight  $  \alpha + \beta $.  
 +
If  $  L $
 +
is a nilpotent Lie algebra, a weight subspace  $  V _  \alpha  $
 +
of weight $  \alpha $
 +
in  $  V $
 +
is an  $  L $-
 +
submodule of the  $  L $-
 +
module  $  V $.
 +
If, in addition,
  
<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/w/w097/w097540/w09754052.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm dim} _ {k}  V  < \infty
 +
$$
  
(the weight decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754053.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754054.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754055.png" /> is a nilpotent subalgebra of a finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754056.png" />, considered as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754057.png" />-module with respect to the adjoint representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754058.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754059.png" /> (cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754060.png" /> is a splitting Lie algebra of linear transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754061.png" />, then the corresponding weight decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754062.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754063.png" />,
+
and  $  \rho ( L) $
 +
is a splitting Lie algebra of linear transformations of the module  $  V $,  
 +
then $  V $
 +
can be decomposed into a direct sum of a finite number of weight subspaces of different weights:
  
<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/w/w097/w097540/w09754064.png" /></td> </tr></table>
+
$$
 +
= V _  \sigma  \oplus V _  \delta  \oplus \dots \oplus V _  \tau  $$
  
is called the Fitting decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754065.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754066.png" />, the weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754067.png" /> are called the roots, while the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754068.png" /> are called the root subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754069.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754070.png" />. If, in addition, one specifies the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754071.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754072.png" /> in a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754073.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754074.png" /> is a splitting Lie algebra of linear transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754075.png" />, and
+
(the weight decomposition of $  V $
 +
with respect to $  L $).  
 +
If  $  L $
 +
is a nilpotent subalgebra of a finite-dimensional Lie algebra  $  M $,  
 +
considered as an  $  L $-
 +
module with respect to the adjoint representation $  { \mathop{\rm ad} } _ {M} $
 +
of  $  M $(
 +
cf. [[Adjoint representation of a Lie group|Adjoint representation of a Lie group]]), and  $  { \mathop{\rm ad} } _ {M}  L $
 +
is a splitting Lie algebra of linear transformations of $  M $,
 +
then the corresponding weight decomposition of  $  M $
 +
with respect to  $  L $,
  
<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/w/w097/w097540/w09754076.png" /></td> </tr></table>
+
$$
 +
= M _  \alpha  \oplus M _  \beta  \oplus \dots \oplus M _  \gamma  $$
  
is the corresponding weight decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754077.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754078.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754079.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754080.png" /> is a weight of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754081.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754082.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754083.png" /> otherwise. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754084.png" /> is a root, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754085.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754086.png" /> otherwise. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754087.png" /> is a field of characteristic zero, the weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754088.png" /> and the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754089.png" /> are linear functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754090.png" /> which vanish on the commutator subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754091.png" />.
+
is called the Fitting decomposition of  $  M $
 +
with respect to  $  L $,
 +
the weights  $  \alpha , \beta \dots \gamma $
 +
are called the roots, while the spaces  $  M _  \alpha  , M _  \beta  \dots M _  \gamma  $
 +
are called the root subspaces of  $  M $
 +
with respect to  $  L $.
 +
If, in addition, one specifies the representation  $  \rho $
 +
of the algebra  $  M $
 +
in a finite-dimensional vector space  $  V $
 +
for which  $  \rho ( L) $
 +
is a splitting Lie algebra of linear transformations of  $  V $,
 +
and
 +
 
 +
$$
 +
V  =  V _  \sigma  \oplus V _  \delta  \oplus \dots \oplus V _  \tau  $$
 +
 
 +
is the corresponding weight decomposition of $  V $
 +
with respect to $  L $,  
 +
then $  \rho ( M _  \alpha  )( V _  \sigma  ) \subseteq V _ {\alpha + \sigma }  $
 +
if $  \alpha + \sigma $
 +
is a weight of $  V $
 +
with respect to $  L $,  
 +
and $  \rho ( M _  \alpha  )( V _  \sigma  ) = 0 $
 +
otherwise. In particular, if $  \alpha + \beta $
 +
is a root, then $  [ M _  \alpha  , M _  \beta  ] \subseteq M _ {\alpha + \beta }  $,  
 +
and $  [ M _  \alpha  , M _  \beta  ] = 0 $
 +
otherwise. If $  k $
 +
is a field of characteristic zero, the weights $  \sigma , \delta \dots \tau $
 +
and the roots $  \alpha , \beta \dots \gamma $
 +
are linear functions on $  L $
 +
which vanish on the commutator subalgebra of $  L $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.P. Zhelobenko,  "Compact Lie groups and their representations" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Jacobson,  "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.P. Zhelobenko,  "Compact Lie groups and their representations" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A set (algebra, Lie algebra, etc.) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754092.png" /> of linear transformations of a vector space over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754093.png" /> is called split or splitting if the characteristic polynomial of each of the transformations has all its roots in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754094.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754095.png" /> contains a splitting field (cf. [[Splitting field of a polynomial|Splitting field of a polynomial]]) of the characteristic polynomial of each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754096.png" />.
+
A set (algebra, Lie algebra, etc.) $  L $
 +
of linear transformations of a vector space over a field $  k $
 +
is called split or splitting if the characteristic polynomial of each of the transformations has all its roots in $  k $,  
 +
i.e. if $  k $
 +
contains a splitting field (cf. [[Splitting field of a polynomial|Splitting field of a polynomial]]) of the characteristic polynomial of each $  h \in L $.
  
A representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754097.png" /> of Lie algebras is split if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097540/w09754098.png" /> is a split Lie algebra of linear transformations.
+
A representation $  \rho : L \rightarrow  \mathop{\rm End} ( V) $
 +
of Lie algebras is split if $  \rho ( L) $
 +
is a split Lie algebra of linear transformations.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Groupes et algèbres de Lie" , Hermann  (1975)  pp. Chapts. VII-VIII</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Groupes et algèbres de Lie" , Hermann  (1975)  pp. Chapts. VII-VIII</TD></TR></table>

Latest revision as of 08:29, 6 June 2020


in a vector space $ V $

A linear mapping $ \alpha $ from the Lie algebra $ L $ into its field of definition $ k $ for which there exists a non-zero vector $ x $ of $ V $ such that for the representation $ \rho $ one has

$$ ( \rho ( h) - \alpha ( h) 1) ^ {n} _ {x,h} ( x) = 0 $$

for all $ h \in L $ and some integer $ n _ {x,h} > 0 $( which in general depends on $ x $ and $ h $). Here 1 denotes the identity transformation of $ V $. One also says in such a case that $ \alpha $ is a weight of the $ L $- module $ V $ defined by the representation $ \rho $. The set of all vectors $ x \in V $ which satisfy this condition, together with zero, forms a subspace $ V _ \alpha $, which is known as the weight subspace of the weight $ \alpha $( or corresponding to $ \alpha $). If $ V = V _ \alpha $, then $ V $ is said to be a weight space or weight module over $ L $ of weight $ \alpha $.

If $ V $ is a finite-dimensional module over $ L $ of weight $ \alpha $, its contragredient module (cf. Contragredient representation) $ V ^ {*} $ is a weight module of weight $ - \alpha $; if $ V $ and $ W $ are weight modules over $ L $ of weights $ \alpha $ and $ \beta $, respectively, then their tensor product $ V \otimes W $ is a weight module of weight $ \alpha + \beta $. If $ L $ is a nilpotent Lie algebra, a weight subspace $ V _ \alpha $ of weight $ \alpha $ in $ V $ is an $ L $- submodule of the $ L $- module $ V $. If, in addition,

$$ \mathop{\rm dim} _ {k} V < \infty $$

and $ \rho ( L) $ is a splitting Lie algebra of linear transformations of the module $ V $, then $ V $ can be decomposed into a direct sum of a finite number of weight subspaces of different weights:

$$ V = V _ \sigma \oplus V _ \delta \oplus \dots \oplus V _ \tau $$

(the weight decomposition of $ V $ with respect to $ L $). If $ L $ is a nilpotent subalgebra of a finite-dimensional Lie algebra $ M $, considered as an $ L $- module with respect to the adjoint representation $ { \mathop{\rm ad} } _ {M} $ of $ M $( cf. Adjoint representation of a Lie group), and $ { \mathop{\rm ad} } _ {M} L $ is a splitting Lie algebra of linear transformations of $ M $, then the corresponding weight decomposition of $ M $ with respect to $ L $,

$$ M = M _ \alpha \oplus M _ \beta \oplus \dots \oplus M _ \gamma $$

is called the Fitting decomposition of $ M $ with respect to $ L $, the weights $ \alpha , \beta \dots \gamma $ are called the roots, while the spaces $ M _ \alpha , M _ \beta \dots M _ \gamma $ are called the root subspaces of $ M $ with respect to $ L $. If, in addition, one specifies the representation $ \rho $ of the algebra $ M $ in a finite-dimensional vector space $ V $ for which $ \rho ( L) $ is a splitting Lie algebra of linear transformations of $ V $, and

$$ V = V _ \sigma \oplus V _ \delta \oplus \dots \oplus V _ \tau $$

is the corresponding weight decomposition of $ V $ with respect to $ L $, then $ \rho ( M _ \alpha )( V _ \sigma ) \subseteq V _ {\alpha + \sigma } $ if $ \alpha + \sigma $ is a weight of $ V $ with respect to $ L $, and $ \rho ( M _ \alpha )( V _ \sigma ) = 0 $ otherwise. In particular, if $ \alpha + \beta $ is a root, then $ [ M _ \alpha , M _ \beta ] \subseteq M _ {\alpha + \beta } $, and $ [ M _ \alpha , M _ \beta ] = 0 $ otherwise. If $ k $ is a field of characteristic zero, the weights $ \sigma , \delta \dots \tau $ and the roots $ \alpha , \beta \dots \gamma $ are linear functions on $ L $ which vanish on the commutator subalgebra of $ L $.

References

[1] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[2] D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)

Comments

A set (algebra, Lie algebra, etc.) $ L $ of linear transformations of a vector space over a field $ k $ is called split or splitting if the characteristic polynomial of each of the transformations has all its roots in $ k $, i.e. if $ k $ contains a splitting field (cf. Splitting field of a polynomial) of the characteristic polynomial of each $ h \in L $.

A representation $ \rho : L \rightarrow \mathop{\rm End} ( V) $ of Lie algebras is split if $ \rho ( L) $ is a split Lie algebra of linear transformations.

References

[a1] N. Bourbaki, "Groupes et algèbres de Lie" , Hermann (1975) pp. Chapts. VII-VIII
How to Cite This Entry:
Weight of a representation of a Lie algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weight_of_a_representation_of_a_Lie_algebra&oldid=49194
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article