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tautological equivariant line bundle

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Bundles

Representation theory

Contents

Idea

A tautological equivariant line bundle is an equivariant tautological line bundle over a projective G-space.

Definition

Let GG be a finite group (or maybe a compact Lie group) and let VV be a GG-linear representation over some topological ground field kk, with kP(V)k P(V) its projective G-space.

Then the corresponding tautological equivariant line bundle V\mathcal{L}_V is the k *k^\ast-fiber bundle which is associated to the canonical k ×k^\times-principal bundle over projective G-space:

V (V{0})×k *k × [v,z]([v],vz) V{0}k ××V id×ptk × kP(V) = (V{0})×*k × \array{ \mathcal{L}_V & \coloneqq & \frac{ (V \setminus \{0\}) \times k^\ast }{ k^\times } & \overset{ [v,z] \mapsto \big( [v], v \cdot z \big) }{\hookrightarrow} & \frac{ V \setminus \{0\} }{ k^\times } \times V \\ \big\downarrow && \big\downarrow {}^{\mathrlap{ \frac{id \times pt}{ k^\times } }} \\ k P(V) &=& \frac{ (V \setminus \{0\}) \times \ast }{ k^\times } }

and equipped with the induced GG-action through VV (which passes to the quotient spaces since the kk-multiplication action commutes with it, by linearity).

Here

  • k ×k{0}k^\times \,\coloneqq\, k \setminus \{0\} is the group of units of kk;

  • ()×()k ×\frac{(-) \times (-)}{k^\times} denotes the quotient space of a product space by the diagonal action;

  • and k *k^\ast is kk equipped with the dual k ×k^\times-action

    k ××k * k * (g,z) z/g \array{ k^\times \times k^\ast &\longrightarrow& k^\ast \\ (g,z) &\mapsto& z/g }
  • so that, for z0z \neq 0,

    [v,z]=[v,z1]=[vz,1](V{0})×k *k × [v,z] \;=\; [v, z \cdot 1] \;=\; [v \cdot z, 1] \;\;\in\;\; \frac{ (V \setminus \{0\}) \times k^\ast }{ k^\times }

Last revised on November 22, 2020 at 13:21:46. See the history of this page for a list of all contributions to it.