Difference between revisions of "Weyl group"
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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 of a Tits system is a generalization of this situation (for its exact definition see [[Tits system|Tits system]]). | ||
− | 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 | + | 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 <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" />. | 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" />. | ||
Line 26: | Line 26: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1a]</TD> <TD valign="top"> | + | <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> |
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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 <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]]). | ||
''A.S. Fedenko'' | ''A.S. Fedenko'' |
Revision as of 14:52, 24 March 2012
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 be a connected affine algebraic group defined over an algebraically closed field
. The Weyl group of
with respect to a torus
is the quotient group
![]() |
considered as a group of automorphisms of induced by the conjugations of
by elements of
. Here
is the normalizer (cf. Normalizer of a subset) and
is the centralizer of
in
. The group
is finite. If
is a maximal torus,
is said to be the Weyl group
of the algebraic group
. This definition does not depend on the choice of a maximal torus
(up to isomorphism). The action by conjugation of
on the set
of Borel subgroups (cf. Borel subgroup) in
containing
induces a simply transitive action of
on
. The action by conjugation of
on
induces an adjoint action of
on the Lie algebra
of
. Let
be the set of non-zero weights of the weight decomposition of
with respect to this action, which means that
is the root system of
with respect to
(cf. Weight of a representation of a Lie algebra).
is a subset of the group
of rational characters of the torus
, and
is invariant with respect to the action of
on
.
Let be a reductive group, let
be the connected component of the identity of its centre and let
be a maximal torus of
. The vector space
![]() |
is canonically identified with a subspace of the vector space
![]() |
As a subset of , the set
is a reduced root system in
, and the natural action of
on
defines an isomorphism between
and the Weyl group of the root system
. Thus,
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 of a finite-dimensional reductive Lie algebra
over an algebraically closed field of characteristic zero is defined as the Weyl group of its adjoint group. The adjoint action of
in the Cartan subalgebra
of
is a faithful representation of
. The group
is often identified with the image of this representation, being regarded as the corresponding linear group in
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 defined over an algebraically non-closed field. If
is a maximal
-split torus of
, then the quotient group
(the normalizer of
over its centralizer in
), regarded as the group of automorphisms of
induced by the conjugations of
by elements of
, is said to be the relative Weyl group of
.
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 is the quotient group
, where
is the normalizer in
of a maximal torus
of
. This Weyl group is isomorphic to a finite group of linear transformations of the Lie algebra
of
(the isomorphism is realized by the adjoint representation of
in
), and may be characterized with the aid of the root system
of the Lie algebra
of
(with respect to
), as follows: If
is a system of simple roots of the algebra, which are linear forms on the real vector space
, the Weyl group is generated by the reflections in the hyperplanes
. Thus,
is the Weyl group of the system
(as a linear group in
).
has a simple transitive action on the set of all chambers (cf. Chamber) of
(which, in this case, are referred to as Weyl chambers). It should be noted that, in general,
is not the semi-direct product of
and
; all the cases in which it is have been studied. The Weyl group of
is isomorphic to the Weyl group of the corresponding complex semi-simple algebraic group
(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