Twisted Weyl Groups of Compact Lie Groups and Nonabelian Cohomology

Ming Liu,Xia Zhang

doi:10.3390/math8010021

Abstract

Given a compact connected Lie group G with an S 1 -module structure and a maximal compact torus T of G S 1 , we define twisted Weyl group W ( G , S 1 , T ) of G associated to S 1 -module and show that two elements of T are δ -conjugate if and only if they are in one W ( G , S 1 , T ) -orbit. Based on this, we prove that the natural map W ( G , S 1 , T ) \ H 1 ( S 1 , T ) → H 1 ( S 1 , G ) is bijective, which reduces the calculation for the nonabelian cohomology H 1 ( S 1 , G ) .

Highlights

Let G be a compact connected Lie group and σ be an automorphism on G [1]
A maximal compact torus of G0Zn, An J. [3] defined the twisted Weyl group W ( G, Zn, T ) of G associated to Zn -module and reduced the calculation of H 1 (Zn, G ) to the action of W ( G, Zn, T ) on T, where
Picking a topological generator δ of S1, which can be regarded as an automorphism of G [4], we first define the twisted Weyl group W ( G, δ, T )

Summary

Introduction

Let G be a compact connected Lie group and σ be an automorphism on G [1]. The twisted conjugate action τ σ [2] of G on itself associated to σ is defined as τ σ : G × G → G, ( g, h) 7→ ghσ ( g)−1 , τgσ (h). [3] defined the twisted Weyl group W ( G, Zn , T ) of G associated to Zn -module and reduced the calculation of H 1 (Zn , G ) to the action of W ( G, Zn , T ) on T, where. Of G associated to δ and define the twisted Weyl group W ( G, S1 , T ) associated to S1 -module, where T is a maximal compact torus in G S. Based on the underlying properties of W ( G, S1 , T ), we study the action of W ( G, S1 , T ) on the first cohomology H 1 (S1 , T ) of the compact Lie group S1 with coefficients in T [5], and prove that the natural map. For basic knowledge on compact Lie groups and twisted conjugate actions, one can refer [2,13,14]; for the nonabelian cohomology of Lie groups, one can refer [5,15,16,17]

Twisted Weyl Groups of Compact Lie Groups

Twisted Weyl Groups of Compact Lie Groups and Nonabelian Cohomology

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