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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.

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) $ .


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]]).

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

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> H. Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch linearen Transformationen I" ''Math. Z.'' , '''23''' (1925) pp. 271–309 {{MR|1544744}} {{ZBL|}} </TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> H. Weyl, "Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch linearen Transformationen II" ''Math. Z.'' , '''24''' (1925) pp. 328–395 {{MR|1544744}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979)) {{MR|0148716}} {{MR|0143793}} {{ZBL|0121.27504}} {{ZBL|0109.26201}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> N. Bourbaki, "Lie groups and Lie algebras" , ''Elements of mathematics'' , Hermann (1975) (Translated from French) {{MR|2109105}} {{MR|1890629}} {{MR|1728312}} {{MR|0979493}} {{MR|0682756}} {{MR|0524568}} {{ZBL|0319.17002}} </TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top"> A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. I.H.E.S.'' , '''27''' (1965) pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top"> A. Borel, J. Tits, "Complément à l'article "Groupes réductifs" " ''Publ. Math. I.H.E.S.'' , '''41''' (1972) pp. 253–276 {{MR|0315007}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> 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 {{MR|0230838}} {{ZBL|0263.14016}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) {{MR|0145455}} {{ZBL|0111.18101}} </TD></TR></table>



====Comments====
The affine Weyl group is the Weyl group of an affine [[Kac–Moody algebra|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|Coxeter group]].

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

====References====
<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  $  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''