nLab
Weyl group

Contents

Context

Group Theory

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

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          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Lie theory

          ∞-Lie theory (higher geometry)

          Background

          Smooth structure

          Higher groupoids

          Lie theory

          ∞-Lie groupoids

          ∞-Lie algebroids

          Formal Lie groupoids

          Cohomology

          Homotopy

          Examples

          \infty-Lie groupoids

          \infty-Lie groups

          \infty-Lie algebroids

          \infty-Lie algebras

          Contents

          Idea

          In Lie theory

          In Lie theory, a Weyl group is a group associated with a compact Lie group that can either be abstractly defined in terms of a root system or in terms of a maximal torus. More generally there are Weyl groups associated with symmetric spaces.

          The Weyl group of a compact Lie group GG is equivalently the quotient group of the normalizer of any maximal torus TT by that torus.

          WN GT/T. W \simeq N_G T / T \,.

          In equivariant homotopy theory

          In equivariant homotopy theory one uses the term Weyl group more general for the quotient group

          W GH=(N GH)/H W_G H = (N_G H) / H

          of the normalizer of any subgroup HGH \hookrightarrow G by that subgroup (e.g. May 96, p. 13).

          The relevance of the Weyl group in this sense is that it is the maximal group which canonically acts on HH-fixed points of a topological G-space. (See for instance at tom Dieck splitting.)

          Notice that W GG=1W_G G = 1 and W G1=GW_G 1 = G.

          On the other hand, if H=NGH = N \subset G is a normal subgroup, then its normalizer is GG itself, in which case the Weyl group is just the plain quotient group

          W GNG/N. W_G N \;\simeq\; G/N \,.

          Definition

          Given a compact Lie group GG with chosen maximal torus TT, its Weyl group W(G)=W(G,T)W(G)=W(G,T) is the group of automorphisms of TT which are restrictions of inner automorphisms of GG.

          This is the quotient group of the normalizer subgroup of TGT \subset G by TT

          WN G(T)/T. W \simeq N_G(T)/T \,.

          Properties

          References

          • eom: Weyl group; wikipedia Weyl group

          • N. Chriss, V. Ginzburg, Representation theory and complex geometry, Birkhäuser 1997. x+495 pp.

          • Walter Borho, Robert MacPherson, Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 15, 707–710 MR82f:14002

          • Peter May, Equivariant homotopy and cohomology theory CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf)

          Last revised on September 18, 2018 at 04:46:14. See the history of this page for a list of all contributions to it.