A method of construction of a categorical quotient

1975 ◽  
Vol 15 (4) ◽  
pp. 525-529
Author(s):  
Yu. E. Borovskii
Keyword(s):  
Categorical Quotient

scholarly journals Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?

2011 ◽  
Vol 147 (2) ◽  
pp. 428-466 ◽  
Author(s):  
Jean-Louis Colliot-Thélène ◽  
Boris Kunyavskiĭ ◽  
Vladimir L. Popov ◽  
Zinovy Reichstein
Keyword(s):  
Lie Algebra ◽  
Rational Functions ◽  
Adjoint Action ◽  
Affirmative Answer ◽  
Simple Groups ◽  
Brauer Group ◽  
Reductive Algebraic Group ◽  
Categorical Quotient ◽  
Field Extensions ◽  
Reductive Lie Algebra

AbstractLet k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let 𝔤 be its Lie algebra. Let k(G), respectively, k(𝔤), be the field of k-rational functions on G, respectively, 𝔤. The conjugation action of G on itself induces the adjoint action of G on 𝔤. We investigate the question whether or not the field extensions k(G)/k(G)G and k(𝔤)/k(𝔤)G are purely transcendental. We show that the answer is the same for k(G)/k(G)G and k(𝔤)/k(𝔤)G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type An or Cn, and negative for groups of other types, except possibly G2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

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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