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

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.

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Maximilian Janisch/latexlist/Algebraic Groups/Weyl group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximilian_Janisch/latexlist/Algebraic_Groups/Weyl_group&oldid=44072