User:Maximilian Janisch/latexlist/Algebraic Groups/Weyl group

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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 $k$ be a connected affine algebraic group defined over an algebraically closed field $k$. The Weyl group of $k$ with respect to a torus $T \subset G$ is the quotient group

\begin{equation} W ( T , G ) = N _ { G } ( T ) / Z _ { G } ( T ) \end{equation}

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 $Z _ { \zeta } ( T )$ is the centralizer of $T$ in $k$. 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 $k$. 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 $k$ containing $T _ { 0 }$ induces a simply transitive action of $W ( T _ { 0 } , G )$ on $B ^ { T } 0$. The action by conjugation of $T$ on $k$ induces an adjoint action of $T$ on the Lie algebra $8$ of $k$. Let $\Phi ( T , G )$ be the set of non-zero weights of the weight decomposition of $8$ with respect to this action, which means that $\Phi ( T , G )$ is the root system of $8$ 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 $k$ be a reductive group, let $Z ( G )$ be the connected component of the identity of its centre and let $T _ { 0 }$ be a maximal torus of $k$. The vector space

\begin{equation} X ( T _ { 0 } / Z ( G ) ^ { 0 } ) _ { Q } = X ( T _ { 0 } / Z ( G ) ^ { 0 } ) \bigotimes _ { Z } Q \end{equation}

is canonically identified with a subspace of the vector space

\begin{equation} X ( T _ { 0 } ) _ { Q } = X ( T _ { 0 } ) \bigotimes _ { Z } Q \end{equation}

As a subset of $X ( T _ { 0 } ) _ { Q }$, the set $\Phi ( T _ { 0 } , G )$ is a reduced root system in $X ( T _ { 0 } / Z ( G ) ^ { 0 } ) _ { Q }$, and the natural action of $W ( T _ { 0 } , G )$ on $X ( T _ { 0 } ) _ { Q }$ 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 $v$ of a finite-dimensional reductive Lie algebra $8$ over an algebraically closed field of characteristic zero is defined as the Weyl group of its adjoint group. The adjoint action of $v$ in the Cartan subalgebra $t$ of $8$ is a faithful representation of $v$. The group $v$ is often identified with the image of this representation, being regarded as the corresponding linear group in $t$ 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 $k$ defined over an algebraically non-closed field. If $T$ is a maximal $k$-split torus of $k$, then the quotient group $N _ { G } ( T ) / Z _ { G } ( T )$ (the normalizer of $T$ over its centralizer in $k$), 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 $k$.

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 $L$-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 $k$ is the quotient group $W = N / T$, where $M$ is the normalizer in $k$ of a maximal torus $T$ of $k$. This Weyl group is isomorphic to a finite group of linear transformations of the Lie algebra $1$ of $T$ (the isomorphism is realized by the adjoint representation of $M$ in $1$), and may be characterized with the aid of the root system $\Delta$ of the Lie algebra $8$ of $k$ (with respect to $1$), as follows: If $\alpha 1 , \ldots , \alpha _ { \gamma }$ is a system of simple roots of the algebra, which are linear forms on the real vector space $1$, the Weyl group is generated by the reflections in the hyperplanes $\alpha _ { i } ( x ) = 0$. Thus, $v$ is the Weyl group of the system $\Delta$ (as a linear group in $1$). $v$ 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, $M$ is not the semi-direct product of $v$ and $T$; all the cases in which it is have been studied. The Weyl group of $k$ is isomorphic to the Weyl group of the corresponding complex semi-simple algebraic group $G _ { C }$ (cf. Complexification of a Lie group).

A.S. Fedenko