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 | + | 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 $ | |
− | The Weyl group | + | 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]]. | 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 | + | 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 16:55, 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 ![]() |
[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
Weyl group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weyl_group&oldid=21960