## A homotopy construction of the adjoint representation for Lie groups (2000)

### BibTeX

@MISC{Castellana00ahomotopy,

author = {Natàlia Castellana},

title = {A homotopy construction of the adjoint representation for Lie groups},

year = {2000}

}

### OpenURL

### Abstract

Let G be a compact, simply-connected, simple Lie group and T ⊂ G a maximal torus. The purpose of this paper is to study the connection between various fibrations over BG (where G is a compact, simply-connected, simple Lie group) associated to the adjoint representation and homotopy colimits over poset categories �, hocolim�BGI where GI are certain connected maximal rank subgroups of G. 1.

### Citations

20 | Homotopy fixed point methods for Lie groups and finite loop spaces
- Dwyer, Wilkerson
- 1994
(Show Context)
Citation Context ...orthogonal, unitary or symplectic groups. The program for understanding the homotopy properties of compact Lie groups led to the concept of a p-compact group introduced by Dwyer and Wilkerson in 1994 =-=[3]-=-. p-Compact groups are p-local versions of finite loop spaces. Namely, a p-compact group is a triple X =(X, BX, e) where BX is a p-complete pointed space, such that H∗ (X; Fp) is finite and e: ΩBX → X... |

16 |
Homotopy limits, completions and localisations
- Bousfield, Kan
- 1975
(Show Context)
Citation Context ...BX, e) where BX is a p-complete pointed space, such that H∗ (X; Fp) is finite and e: ΩBX → X is a homotopy equivalence. The obvious examples of p-compact groups are the p completions (in the sense of =-=[2]-=-) of compact connected Lie groups and their classifying spaces. Many properties of compact Lie group theory can be reinterpreted as homotopy theoretic properties of the classifying spaces (see [4]) in... |

11 |
Quillen’s theorem on buildings and the loops on a symmetric space, L’Enseignement Mathematique 34
- Mitchell
- 1988
(Show Context)
Citation Context ...cit simplicial model of Bousfield–Kan for the homotopy colimit described in [2]. It is worth pointing out that the third homeomorphism is implicit in the work of Mitchell. Our methods are inspired by =-=[7]-=-. For a G-space X, let EG ×G X denote the Borel construction for X. On taking Borel constructions on either side of the equalities in the above theorem, we obtain: Corollary 1·2. There exist subgroups... |

6 |
The adjoint action of a Lie group on the space of loops
- Kozima
- 1993
(Show Context)
Citation Context ... on H ∗ (Λ(BG2), F2), it remains to calculate Sq 4 x5. We leave this for later. We now proceed to solve the algebraic extension problems. These extensions are nontrivial and they were first solved in =-=[6]-=-. Our homotopy decomposition simplifies these computations considerably. Consider the homotopy decomposition given by Theorem 1·2. The Bousfield–Kan spectral sequence for this homotopy decomposition p... |

5 | The elementary geometric structure of compact Lie groups
- Dwyer, Wilkerson
- 1998
(Show Context)
Citation Context ... of [2]) of compact connected Lie groups and their classifying spaces. Many properties of compact Lie group theory can be reinterpreted as homotopy theoretic properties of the classifying spaces (see =-=[4]-=-) in such a way that the concepts extend to the category of p-compact groups. For example, they admit the concept of a maximal torus and Weyl group. For these reasons, p-compact groups can be consider... |

4 |
On the characteristic zero cohomology of the free loop
- Smith
- 1981
(Show Context)
Citation Context ... EG ×G S LG (Φ3)hG : hocolim�(G/GI)hG −→ EG ×G Gc. The spaces (G/GI)hG are homeomorphic to EG/GI and they have the homotopy type of BGI. Notice that the space EG ×G Gc is homotopy equivalent to Λ(BG) =-=[8]-=-, where Λ(BG) denotes the space of free loops on the space BG. The category �0 can be identified with the subcategory of � consisting of all proper subsets of {0, 1,... ,l} containing 0. Under this id... |

3 | Homotopy classification of spaces with interesting cohomology and a conjecture of Cooke - Aguadé, Broto, et al. - 1994 |

1 |
Homotopy localization and v1-periodic spaces
- Farjoun
- 1992
(Show Context)
Citation Context ... hocolim�BGI (Φ3)hG ✲ EG ×G Gc T C ❄ hocolim� ˜ BGI Φ ❄ ✲ T (Ad) 4. Farjoun localization of homotopy colimits and applications Let LBZ/p denote the Farjoun localization with respect to the map BZ/p →∗=-=[5]-=-. In this section we will show that LBZ/p applied to the p-completion of the spaces EG ×G S LG and EG ×G Gc yields a point up to homotopy. Recall some of the basic facts about LBZ/p. Let BG denote the... |