An Explicit Cell Decomposition of the Wonderful Compactification of a Semisimple Algebraic Group

2003 ◽  
Vol 46 (1) ◽  
pp. 140-148 ◽  
Author(s):  
Lex E. Renner
Keyword(s):  
Algebraic Group ◽  
Cell Decomposition ◽  
Bruhat Decomposition ◽  
Semisimple Algebraic Group ◽  
Reductive Monoid

AbstractWe determine an explicit cell decomposition of the wonderful compactification of a semisimple algebraic group. To do this we first identify the B × B-orbits using the generalized Bruhat decomposition of a reductive monoid. From there we show how each cell is made up from B × B orbits.

scholarly journals On the cohomology of loop spaces of compact Lie groups

1985 ◽  
Vol 26 (1) ◽  
pp. 91-99 ◽  
Author(s):  
Howard Hiller
Keyword(s):  
Homogeneous Spaces ◽  
Cell Decomposition ◽  
Point Of View ◽  
Loop Spaces ◽  
Bruhat Decomposition ◽  
Infinite Dimensional ◽  
Dimensional Group ◽  
Analogous Question ◽  
A Cell ◽  
Simply Connected

Let G be a compact, simply-connected Lie group. The cohomology of the loop space ΏG has been described by Bott, both in terms of a cell decomposition [1] and certain homogeneous spaces called generating varieties [2]. It is possible to view ΏG as an infinite dimensional “Grassmannian” associated to an appropriate infinite dimensional group, cf. [3], [7]. From this point of view the above cell-decomposition of Bott arises from a Bruhat decomposition of the associated group. We choose a generator H ∈ H2(ΏG, ℤ) and call it the hyperplane class. For a finite-dimensional Grassmannian the highest power of H carries geometric information about the variety, namely, its degree. An analogous question for ΏG is: What is the largest integer Nk = Nk(G) which divides Hk ∈ H2k(ΏG, ℤ)?Of course, if G = SU(2) = S3, one knows Nk = h!. In general, the deviation of Nk from k! measures the failure of H to generate a divided polynomial algebra in H*(ΏG, ℤ).

scholarly journals Hopf-theoretic approach to motives of twisted flag varieties

2021 ◽  
Vol 157 (5) ◽  
pp. 963-996
Author(s):  
Victor Petrov ◽  
Nikita Semenov
Keyword(s):  
Hopf Algebra ◽  
Algebraic Group ◽  
Cohomology Theory ◽  
Algebra Structure ◽  
Flag Varieties ◽  
Theoretic Approach ◽  
Hopf Algebra Structure ◽  
Uniform Approach ◽  
Semisimple Algebraic Group ◽  
Oriented Cohomology

Let $G$ be a split semisimple algebraic group over a field and let $A^*$ be an oriented cohomology theory in the Levine–Morel sense. We provide a uniform approach to the $A^*$ -motives of geometrically cellular smooth projective $G$ -varieties based on the Hopf algebra structure of $A^*(G)$ . Using this approach, we provide various applications to the structure of motives of twisted flag varieties.

Smoothness of Quotients Associated With a Pair of Commuting Involutions

2004 ◽  
Vol 56 (5) ◽  
pp. 945-962 ◽  
Author(s):  
Aloysius G. Helminck ◽  
Gerald W. Schwarz
Keyword(s):  
Fixed Point ◽  
Algebraic Group ◽  
Symmetric Case ◽  
Closed Field ◽  
Categorical Quotient ◽  
Algebraically Closed Field ◽  
Semisimple Algebraic Group ◽  
Point Groups

AbstractLet σ, θ be commuting involutions of the connected semisimple algebraic group G where σ, θ and G are defined over an algebraically closed field , char = 0. Let H := Gσ and K := Gθ be the fixed point groups. We have an action (H × K) × G → G, where ((h, k), g) ⟼ hgk–1, h ∈ H, k ∈ K, g ∈ G. Let G//(H × K) denote the categorical quotient Spec (G)H×K. We determine when this quotient is smooth. Our results are a generalization of those of Steinberg [Ste75], Pittie [Pit72] and Richardson [Ric82] in the symmetric case where σ = θ and H = K.

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