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Difference between revisions of "Weyl group"

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The Weyl group of symmetries of a [[Root system|root system]]. Depending on the actual realization of the root system, different Weyl groups are considered: Weyl groups of a semi-simple splittable Lie algebra, of a symmetric space, of an algebraic group, etc.
 
The Weyl group of symmetries of a [[Root system|root system]]. Depending on the actual realization of the root system, different Weyl groups are considered: Weyl groups of a semi-simple splittable Lie algebra, of a symmetric space, of an algebraic group, etc.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w0977101.png" /> be a connected affine [[Algebraic group|algebraic group]] defined over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w0977102.png" />. The Weyl group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w0977103.png" /> with respect to a torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w0977104.png" /> is the quotient group
+
Let $  G $
 +
be a connected affine [[Algebraic group|algebraic group]] defined over an algebraically closed field $  k $ .
 +
The Weyl group of  $  G $
 +
with respect to a torus  $  T \subset G $
 +
is the quotient group $$
 +
W(T,\  G)  =   N _{G} (T) / Z _{G} (T),
 +
$$
 +
considered as a group of automorphisms of  $  T $
 +
induced by the conjugations of  $  T $
 +
by elements of  $  N _{G} (T) $ .  
 +
Here  $  N _{G} (T) $
 +
is the normalizer (cf. [[Normalizer of a subset|Normalizer of a subset]]) and  $  Z _{G} (T) $
 +
is the [[Centralizer|centralizer]] of  $  T $
 +
in  $  G $ .  
 +
The group  $  W(T,\  G) $
 +
is finite. If  $  T _{0} $
 +
is a maximal torus,  $  W( T _{0} ,\  G) $
 +
is said to be the Weyl group $  W(G) $
 +
of the algebraic group  $  G $ .
 +
This definition does not depend on the choice of a maximal torus  $  T _{0} $ (
 +
up to isomorphism). The action by conjugation of  $  N _{G} ( T _{0} ) $
 +
on the set  $  B ^ {T _{0}} $
 +
of Borel subgroups (cf. [[Borel subgroup|Borel subgroup]]) in  $  G $
 +
containing  $  T _{0} $
 +
induces a simply transitive action of  $  W( T _{0} ,\  G) $
 +
on  $  B ^ {T _{0}} $ .  
 +
The action by conjugation of  $  T $
 +
on  $  G $
 +
induces an adjoint action of  $  T $
 +
on the Lie algebra  $  \mathfrak g $
 +
of  $  G $ .  
 +
Let  $  \Phi (T,\  G) $
 +
be the set of non-zero weights of the weight decomposition of  $  \mathfrak g $
 +
with respect to this action, which means that  $  \Phi (T,\  G) $
 +
is the root system of  $  \mathfrak g $
 +
with respect to $  T $ (
 +
cf. [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]]). $  \Phi (T,\  G) $
 +
is a subset of the group $  X(T) $
 +
of rational characters of the torus  $  T $ ,
 +
and  $  \Phi (T,\  G) $
 +
is invariant with respect to the action of  $  W(T,\  G) $
 +
on  $  X(T) $ .
  
<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/w097710/w0977105.png" /></td> </tr></table>
 
  
considered as a group of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w0977106.png" /> induced by the conjugations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w0977107.png" /> by elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w0977108.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w0977109.png" /> is the normalizer (cf. [[Normalizer of a subset|Normalizer of a subset]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771010.png" /> is the [[Centralizer|centralizer]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771012.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771013.png" /> is finite. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771014.png" /> is a maximal torus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771015.png" /> is said to be the Weyl group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771016.png" /> of the algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771017.png" />. This definition does not depend on the choice of a maximal torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771018.png" /> (up to isomorphism). The action by conjugation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771019.png" /> on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771020.png" /> of Borel subgroups (cf. [[Borel subgroup|Borel subgroup]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771021.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771022.png" /> induces a simply transitive action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771024.png" />. The action by conjugation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771025.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771026.png" /> induces an adjoint action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771027.png" /> on the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771028.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771029.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771030.png" /> be the set of non-zero weights of the weight decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771031.png" /> with respect to this action, which means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771032.png" /> is the root system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771033.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771034.png" /> (cf. [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]]). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771035.png" /> is a subset of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771036.png" /> of rational characters of the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771037.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771038.png" /> is invariant with respect to the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771040.png" />.
+
Let  $  G $
 +
be a [[Reductive group|reductive group]], let  $  Z(G) ^{0} $
 +
be the connected component of the identity of its centre and let  $  T _{0} $
 +
be a maximal torus of  $  G $ .  
 +
The vector space $$
 +
X(T _{0} /Z(G) ^{0} ) _ {\mathbf Q}
 +
  =   X(T _{0} /Z(G) ^{0} ) \otimes _ {\mathbf Z} \mathbf Q
 +
$$
 +
is canonically identified with a subspace of the vector space $$
 +
X(T _{0} ) _ {\mathbf Q}
 +
  =   X(T _{0} ) \otimes _ {\mathbf Z} \mathbf Q .
 +
$$
 +
As a subset of $  X {( T _{0} )} _ {\mathbf Q} $ ,
 +
the set $  \Phi ( T _{0} ,\  G) $
 +
is a reduced root system in $  X( T _{0} /Z(G) ^{0} ) _ {\mathbf Q} $ ,
 +
and the natural action of $  W( T _{0} ,\  G) $
 +
on $  {X( T _{0} )} _ {\mathbf Q} $
 +
defines an isomorphism between  $  W( T _{0} ,\  G) $
 +
and the Weyl group of the root system  $  \Phi (T _{0} ,\  G) $ .  
 +
Thus,  $  W(T _{0} ,\  G) $
 +
displays all the properties of a Weyl group of a reduced root system; e.g. it is generated by reflections (cf. [[Reflection|Reflection]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771041.png" /> be a [[Reductive group|reductive group]], let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771042.png" /> be the connected component of the identity of its centre and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771043.png" /> be a maximal torus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771044.png" />. The vector space
+
The Weyl group of a Tits system is a generalization of this situation (for its exact definition see [[Tits system|Tits system]]).
 
 
<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/w097710/w09771045.png" /></td> </tr></table>
 
 
 
is canonically identified with a subspace of the vector space
 
 
 
<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/w097710/w09771046.png" /></td> </tr></table>
 
  
As a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771047.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771048.png" /> is a reduced root system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771049.png" />, and the natural action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771050.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771051.png" /> defines an isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771052.png" /> and the Weyl group of the root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771053.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771054.png" /> displays all the properties of a Weyl group of a reduced root system; e.g. it is generated by reflections (cf. [[Reflection|Reflection]]).
+
The Weyl group  $  W $
 
+
of a finite-dimensional reductive Lie algebra  $  \mathfrak g $
The Weyl group of a Tits system is a generalization of this situation (for its exact definition see [[Tits system|Tits system]]).
+
over an algebraically closed field of characteristic zero is defined as the Weyl group of its adjoint group. The adjoint action of  $  W $
 +
in the Cartan subalgebra  $  \mathfrak p $
 +
of $  \mathfrak g $
 +
is a faithful representation of $  W $ .  
 +
The group  $  W $
 +
is often identified with the image of this representation, being regarded as the corresponding linear group in  $  \mathfrak p $
 +
generated by the reflections. The concept of a "Weyl group" was first used by H. Weyl
  
The Weyl group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771055.png" /> of a finite-dimensional reductive Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771056.png" /> over an algebraically closed field of characteristic zero is defined as the Weyl group of its adjoint group. The adjoint action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771057.png" /> in the Cartan subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771058.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771059.png" /> is a faithful representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771060.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771061.png" /> is often identified with the image of this representation, being regarded as the corresponding linear group in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771062.png" /> generated by the reflections. The concept of a "Weyl group" was first used by H. Weyl
+
in the special case of finite-dimensional semi-simple Lie algebras over the field of complex numbers. A Weyl group may also be defined for an arbitrary splittable semi-simple finite-dimensional Lie algebra, as the Weyl group of its root system. A relative Weyl group may be defined for an affine algebraic group  $  G $
 +
defined over an algebraically non-closed field. If  $  T $
 +
is a maximal  $  k $ -
 +
split torus of $  G $ ,
 +
then the quotient group $  N _{G} (T)/ Z _{G} (T) $ (
 +
the normalizer of $  T $
 +
over its centralizer in  $  G $ ),  
 +
regarded as the group of automorphisms of  $  T $
 +
induced by the conjugations of  $  T $
 +
by elements of $  N _{G} (T) $ ,
 +
is said to be the relative Weyl group of  $  G $ .
  
in the special case of finite-dimensional semi-simple Lie algebras over the field of complex numbers. A Weyl group may also be defined for an arbitrary splittable semi-simple finite-dimensional Lie algebra, as the Weyl group of its root system. A relative Weyl group may be defined for an affine algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771063.png" /> defined over an algebraically non-closed field. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771064.png" /> is a maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771065.png" />-split torus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771066.png" />, then the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771067.png" /> (the normalizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771068.png" /> over its centralizer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771069.png" />), regarded as the group of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771070.png" /> induced by the conjugations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771071.png" /> by elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771072.png" />, is said to be the relative Weyl group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771073.png" />.
 
  
 
For the Weyl group of a symmetric space, see [[Symmetric space|Symmetric space]]. The Weyl group of a real connected non-compact semi-simple algebraic group is identical with the Weyl group of the corresponding symmetric space. For the affine Weyl group see [[Root system|Root system]].
 
For the Weyl group of a symmetric space, see [[Symmetric space|Symmetric space]]. The Weyl group of a real connected non-compact semi-simple algebraic group is identical with the Weyl group of the corresponding symmetric space. For the affine Weyl group see [[Root system|Root system]].
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Tits, "Reductive groups over local fields" A. Borel (ed.) W. Casselman (ed.) , ''Automorphic forms, representations and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771074.png" />-functions'' , ''Proc. Symp. Pure Math.'' , '''33:1''' , Amer. Math. Soc. (1979) pp. 29–69 {{MR|0546588}} {{ZBL|0415.20035}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.E. Humphreys, "Reflection groups and Coxeter groups" , Cambridge Univ. Press (1991) {{MR|1066460}} {{ZBL|0768.20016}} {{ZBL|0725.20028}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Tits, "Reductive groups over local fields" A. Borel (ed.) W. Casselman (ed.) , ''Automorphic forms, representations and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771074.png" />-functions'' , ''Proc. Symp. Pure Math.'' , '''33:1''' , Amer. Math. Soc. (1979) pp. 29–69 {{MR|0546588}} {{ZBL|0415.20035}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.E. Humphreys, "Reflection groups and Coxeter groups" , Cambridge Univ. Press (1991) {{MR|1066460}} {{ZBL|0768.20016}} {{ZBL|0725.20028}} </TD></TR></table>
  
The Weyl group of a connected compact Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771075.png" /> is the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771076.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771077.png" /> is the normalizer in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771078.png" /> of a maximal torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771079.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771080.png" />. This Weyl group is isomorphic to a finite group of linear transformations of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771081.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771082.png" /> (the isomorphism is realized by the adjoint representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771083.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771084.png" />), and may be characterized with the aid of the root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771085.png" /> of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771086.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771087.png" /> (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771088.png" />), as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771089.png" /> is a system of simple roots of the algebra, which are linear forms on the real vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771090.png" />, the Weyl group is generated by the reflections in the hyperplanes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771091.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771092.png" /> is the Weyl group of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771093.png" /> (as a linear group in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771094.png" />). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771095.png" /> has a simple transitive action on the set of all chambers (cf. [[Chamber|Chamber]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771096.png" /> (which, in this case, are referred to as Weyl chambers). It should be noted that, in general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771097.png" /> is not the semi-direct product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w09771099.png" />; all the cases in which it is have been studied. The Weyl group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w097710100.png" /> is isomorphic to the Weyl group of the corresponding complex semi-simple algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097710/w097710101.png" /> (cf. [[Complexification of a Lie group|Complexification of a Lie group]]).
+
The Weyl group of a connected compact Lie group $  G $
 +
is the quotient group $  W = N/T $ ,  
 +
where $  N $
 +
is the normalizer in $  G $
 +
of a maximal torus $  T $
 +
of $  G $ .  
 +
This Weyl group is isomorphic to a finite group of linear transformations of the Lie algebra $  \mathfrak t $
 +
of $  T $ (
 +
the isomorphism is realized by the adjoint representation of $  N $
 +
in $  \mathfrak t $ ),  
 +
and may be characterized with the aid of the root system $  \Delta $
 +
of the Lie algebra $  \mathfrak g $
 +
of $  G $ (
 +
with respect to $  \mathfrak t $ ),  
 +
as follows: If $  \alpha _{1} \dots \alpha _{r} $
 +
is a system of simple roots of the algebra, which are linear forms on the real vector space $  \mathfrak t $ ,  
 +
the Weyl group is generated by the reflections in the hyperplanes $  \alpha _{i} (x) = 0 $ .  
 +
Thus, $  W $
 +
is the Weyl group of the system $  \Delta $ (
 +
as a linear group in $  \mathfrak t $ ).  
 +
$  W $
 +
has a simple transitive action on the set of all chambers (cf. [[Chamber|Chamber]]) of $  \Delta $ (
 +
which, in this case, are referred to as Weyl chambers). It should be noted that, in general, $  N $
 +
is not the semi-direct product of $  W $
 +
and $  T $ ;  
 +
all the cases in which it is have been studied. The Weyl group of $  G $
 +
is isomorphic to the Weyl group of the corresponding complex semi-simple algebraic group $  G _{\mathbf C} $ (
 +
cf. [[Complexification of a Lie group|Complexification of a Lie group]]).
  
 
''A.S. Fedenko''
 
''A.S. Fedenko''

Latest revision as of 17:55, 17 December 2019

The Weyl group of symmetries of a root system. Depending on the actual realization of the root system, different Weyl groups are considered: Weyl groups of a semi-simple splittable Lie algebra, of a symmetric space, of an algebraic group, etc.

Let $ G $ be a connected affine algebraic group defined over an algebraically closed field $ k $ . The Weyl group of $ G $ with respect to a torus $ T \subset G $ is the quotient group $$ W(T,\ G) = N _{G} (T) / Z _{G} (T), $$ considered as a group of automorphisms of $ T $ induced by the conjugations of $ T $ by elements of $ N _{G} (T) $ . Here $ N _{G} (T) $ is the normalizer (cf. Normalizer of a subset) and $ Z _{G} (T) $ is the centralizer of $ T $ in $ G $ . The group $ W(T,\ G) $ is finite. If $ T _{0} $ is a maximal torus, $ W( T _{0} ,\ G) $ is said to be the Weyl group $ W(G) $ of the algebraic group $ G $ . This definition does not depend on the choice of a maximal torus $ T _{0} $ ( up to isomorphism). The action by conjugation of $ N _{G} ( T _{0} ) $ on the set $ B ^ {T _{0}} $ of Borel subgroups (cf. Borel subgroup) in $ G $ containing $ T _{0} $ induces a simply transitive action of $ W( T _{0} ,\ G) $ on $ B ^ {T _{0}} $ . The action by conjugation of $ T $ on $ G $ induces an adjoint action of $ T $ on the Lie algebra $ \mathfrak g $ of $ G $ . Let $ \Phi (T,\ G) $ be the set of non-zero weights of the weight decomposition of $ \mathfrak g $ with respect to this action, which means that $ \Phi (T,\ G) $ is the root system of $ \mathfrak g $ with respect to $ T $ ( cf. Weight of a representation of a Lie algebra). $ \Phi (T,\ G) $ is a subset of the group $ X(T) $ of rational characters of the torus $ T $ , and $ \Phi (T,\ G) $ is invariant with respect to the action of $ W(T,\ G) $ on $ X(T) $ .


Let $ G $ be a reductive group, let $ Z(G) ^{0} $ be the connected component of the identity of its centre and let $ T _{0} $ be a maximal torus of $ G $ . The vector space $$ X(T _{0} /Z(G) ^{0} ) _ {\mathbf Q} = X(T _{0} /Z(G) ^{0} ) \otimes _ {\mathbf Z} \mathbf Q $$ is canonically identified with a subspace of the vector space $$ X(T _{0} ) _ {\mathbf Q} = X(T _{0} ) \otimes _ {\mathbf Z} \mathbf Q . $$ As a subset of $ X {( T _{0} )} _ {\mathbf Q} $ , the set $ \Phi ( T _{0} ,\ G) $ is a reduced root system in $ X( T _{0} /Z(G) ^{0} ) _ {\mathbf Q} $ , and the natural action of $ W( T _{0} ,\ G) $ on $ {X( T _{0} )} _ {\mathbf Q} $ defines an isomorphism between $ W( T _{0} ,\ G) $ and the Weyl group of the root system $ \Phi (T _{0} ,\ G) $ . Thus, $ W(T _{0} ,\ G) $ displays all the properties of a Weyl group of a reduced root system; e.g. it is generated by reflections (cf. Reflection).

The Weyl group of a Tits system is a generalization of this situation (for its exact definition see Tits system).

The Weyl group $ W $ of a finite-dimensional reductive Lie algebra $ \mathfrak g $ over an algebraically closed field of characteristic zero is defined as the Weyl group of its adjoint group. The adjoint action of $ W $ in the Cartan subalgebra $ \mathfrak p $ of $ \mathfrak g $ is a faithful representation of $ W $ . The group $ W $ is often identified with the image of this representation, being regarded as the corresponding linear group in $ \mathfrak p $ generated by the reflections. The concept of a "Weyl group" was first used by H. Weyl

in the special case of finite-dimensional semi-simple Lie algebras over the field of complex numbers. A Weyl group may also be defined for an arbitrary splittable semi-simple finite-dimensional Lie algebra, as the Weyl group of its root system. A relative Weyl group may be defined for an affine algebraic group $ G $ defined over an algebraically non-closed field. If $ T $ is a maximal $ k $ - split torus of $ G $ , then the quotient group $ N _{G} (T)/ Z _{G} (T) $ ( the normalizer of $ T $ over its centralizer in $ G $ ), regarded as the group of automorphisms of $ T $ induced by the conjugations of $ T $ by elements of $ N _{G} (T) $ , is said to be the relative Weyl group of $ G $ .


For the Weyl group of a symmetric space, see Symmetric space. The Weyl group of a real connected non-compact semi-simple algebraic group is identical with the Weyl group of the corresponding symmetric space. For the affine Weyl group see Root system.

References

[1a] H. Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch linearen Transformationen I" Math. Z. , 23 (1925) pp. 271–309 MR1544744
[1b] H. Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch linearen Transformationen II" Math. Z. , 24 (1925) pp. 328–395 MR1544744
[2] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[3] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) MR0148716 MR0143793 Zbl 0121.27504 Zbl 0109.26201
[4] N. Bourbaki, "Lie groups and Lie algebras" , Elements of mathematics , Hermann (1975) (Translated from French) MR2109105 MR1890629 MR1728312 MR0979493 MR0682756 MR0524568 Zbl 0319.17002
[5a] A. Borel, J. Tits, "Groupes réductifs" Publ. Math. I.H.E.S. , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[5b] A. Borel, J. Tits, "Complément à l'article "Groupes réductifs" " Publ. Math. I.H.E.S. , 41 (1972) pp. 253–276 MR0315007
[6] F. Bruhat, J. Tits, "Groupes algébriques simples sur un corps local" T.A. Springer (ed.) et al. (ed.) , Proc. Conf. local fields (Driebergen, 1966) , Springer (1967) pp. 23–36 MR0230838 Zbl 0263.14016
[7] S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) MR0145455 Zbl 0111.18101


Comments

The affine Weyl group is the Weyl group of an affine Kac–Moody algebra. One may define a Weyl group for an arbitrary Kac–Moody algebra.

The Weyl group as an abstract group is a Coxeter group.

Weyl groups play an important role in representation theory (see Character formula).

References

[a1] J. Tits, "Reductive groups over local fields" A. Borel (ed.) W. Casselman (ed.) , Automorphic forms, representations and -functions , Proc. Symp. Pure Math. , 33:1 , Amer. Math. Soc. (1979) pp. 29–69 MR0546588 Zbl 0415.20035
[a2] J.E. Humphreys, "Reflection groups and Coxeter groups" , Cambridge Univ. Press (1991) MR1066460 Zbl 0768.20016 Zbl 0725.20028

The Weyl group of a connected compact Lie group $ G $ is the quotient group $ W = N/T $ , where $ N $ is the normalizer in $ G $ of a maximal torus $ T $ of $ G $ . This Weyl group is isomorphic to a finite group of linear transformations of the Lie algebra $ \mathfrak t $ of $ T $ ( the isomorphism is realized by the adjoint representation of $ N $ in $ \mathfrak t $ ), and may be characterized with the aid of the root system $ \Delta $ of the Lie algebra $ \mathfrak g $ of $ G $ ( with respect to $ \mathfrak t $ ), as follows: If $ \alpha _{1} \dots \alpha _{r} $ is a system of simple roots of the algebra, which are linear forms on the real vector space $ \mathfrak t $ , the Weyl group is generated by the reflections in the hyperplanes $ \alpha _{i} (x) = 0 $ . Thus, $ W $ is the Weyl group of the system $ \Delta $ ( as a linear group in $ \mathfrak t $ ). $ W $ has a simple transitive action on the set of all chambers (cf. Chamber) of $ \Delta $ ( which, in this case, are referred to as Weyl chambers). It should be noted that, in general, $ N $ is not the semi-direct product of $ W $ and $ T $ ; all the cases in which it is have been studied. The Weyl group of $ G $ is isomorphic to the Weyl group of the corresponding complex semi-simple algebraic group $ G _{\mathbf C} $ ( cf. Complexification of a Lie group).

A.S. Fedenko

How to Cite This Entry:
Weyl group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Weyl_group&oldid=44291
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article