scholarly journals Invariant analytic hypersurfaces in complex Lie groups

2004 ◽  
Vol 70 (2) ◽  
pp. 343-349
Author(s):  
Bruce Gilligan
Keyword(s):  
Lie Groups ◽  
Parabolic Subgroup ◽  
Lie Group ◽  
Commutator Subgroup ◽  
Holomorphic Functions ◽  
Related Question ◽  
Right Action ◽  
Analytic Hypersurface ◽  
Holomorphic Reduction ◽  
The Right

Suppose G is a complex Lie group and H is a closed complex subgroup of G. Let G′ denote the commutator subgroup of G. If there are no nonconstant holomorphic functions on G/H and H is not contained in any proper parabolic subgroup of G, then Akhiezer [2] asked whether every analytic hypersurface in G which is invariant under the right action of H is also invariant under the right action of G′. In this paper we answer a related question in two settings. Under the assumptions stated above we show that the orbits of the radical of G in G/H cannot be Cousin groups, provided G/H is Kähler. We also introduce an intermediate fibration of G/H induced by the holomorphic reduction of the radical orbits and resolve the related question in a situation arising from this fibration.

scholarly journals BIQUOTIENT ACTIONS ON UNIPOTENT LIE GROUPS

2008 ◽  
Vol 19 (08) ◽  
pp. 997-1010
Author(s):  
ANNETT PÜTTMANN
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Nilpotent Lie Group ◽  
The Right

We consider pairs (V, H) of subgroups of a connected unipotent complex Lie group G for which the induced V × H-action on G by multiplication from the left and from the right is free. We prove that this action is proper if the Lie algebra 𝔤 of G is 3-step nilpotent. If 𝔤 is 2-step nilpotent, then there is a global slice of the action that is isomorphic to ℂn. Furthermore, a global slice isomorphic to ℂn exists if dim V = 1 = dim H or dim V = 1 and 𝔤 is 3-step nilpotent. We give an explicit example of a 3-step nilpotent Lie group and a pair of 2-dimensional subgroups such that the induced action is proper but the corresponding geometric quotient is not affine.

scholarly journals Levi’s Problem for Pseudoconvex Homogeneous Manifolds

2017 ◽  
Vol 60 (4) ◽  
pp. 736-746
Author(s):  
Bruce Gilligan
Keyword(s):  
Lie Group ◽  
Holomorphic Functions ◽  
Homogeneous Manifolds ◽  
Exhaustion Function ◽  
Holomorphic Reduction

AbstractSuppose G is a connected complex Lie group and H is a closed complex subgroup. Then there exists a closed complex subgroup J of G containing H such that the fibration π:G/H ⟶ G/J is the holomorphic reduction of G/H; i.e., G/J is holomorphically separable and O(G/H)≅ π*O(G/J). In this paper we prove that if G/H is pseudoconvex, i.e., if G/H admits a continuous plurisubharmonic exhaustion function, then G/J is Stein and J/H has no non-constant holomorphic functions.

scholarly journals On holomorphic reflexivity conditions for complex Lie groups

2021 ◽  
pp. 1-22
Author(s):  
O. Yu. Aristov
Keyword(s):  
Hopf Algebra ◽  
Banach Algebra ◽  
Lie Groups ◽  
Lie Group ◽  
Holomorphic Functions ◽  
Analytic Functional ◽  
Pontryagin Duality

Abstract We consider Akbarov's holomorphic version of the non-commutative Pontryagin duality for a complex Lie group. We prove, under the assumption that $G$ is a Stein group with finitely many components, that (1) the topological Hopf algebra of holomorphic functions on $G$ is holomorphically reflexive if and only if $G$ is linear; (2) the dual cocommutative topological Hopf algebra of exponential analytic functional on $G$ is holomorphically reflexive. We give a counterexample, which shows that the first criterion cannot be extended to the case of infinitely many components. Nevertheless, we conjecture that, in general, the question can be solved in terms of the Banach-algebra linearity of $G$ .

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

scholarly journals The Higher Order Riesz Transform andBMOType Space Associated with Schrödinger Operators on Stratified Lie Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Yu Liu ◽  
Jianfeng Dong
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Riesz Transform ◽  
Integral Operators ◽  
Higher Order ◽  
Reverse Hölder Class ◽  
Stratified Lie Group ◽  
Homogeneous Dimension ◽  
Stratified Lie Groups ◽  
Reverse Hölder

Assume thatGis a stratified Lie group andQis the homogeneous dimension ofG. Let-Δbe the sub-Laplacian onGandW≢0a nonnegative potential belonging to certain reverse Hölder classBsfors≥Q/2. LetL=-Δ+Wbe a Schrödinger operator on the stratified Lie groupG. In this paper, we prove the boundedness of some integral operators related toL, such asL-1∇2,L-1W, andL-1(-Δ) on the spaceBMOL(G).

ON THE MODEL THEORY OF THE LOGARITHMIC FUNCTION IN COMPACT LIE GROUPS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350055
Author(s):  
SONIA L'INNOCENTE ◽  
FRANÇOISE POINT ◽  
CARLO TOFFALORI
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Model Theory ◽  
Effective Model ◽  
Logarithmic Function ◽  
Compact Lie Groups ◽  
Model Completeness ◽  
Schanuel's Conjecture ◽  
Logarithm Function ◽  
Natural Expansion

Given a compact linear Lie group G, we form a natural expansion of the theory of the reals where G and the graph of a logarithm function on G live. We prove its effective model-completeness and decidability modulo a suitable variant of Schanuel's Conjecture.

THE FERMI–WALKER DERIVATIVE IN LIE GROUPS

2013 ◽  
Vol 10 (07) ◽  
pp. 1320011 ◽  
Author(s):  
FATMA KARAKUŞ ◽  
YUSUF YAYLI
Keyword(s):  
Vector Field ◽  
Lie Groups ◽  
Lie Group ◽  
Necessary Conditions ◽  
Rotating Frame ◽  
Darboux Vector

In this study, Fermi–Walker derivative, Fermi–Walker parallelism, non-rotating frame, Fermi–Walker termed Darboux vector concepts are given for Lie groups in E4. First, we get any Frénet curve and any vector field along the Frénet curve in a Lie group. Then, the Fermi–Walker derivative is defined for the Lie group. Fermi–Walker derivative and Fermi–Walker parallelism are analyzed in Lie groups. Finally, the necessary conditions for Fermi–Walker parallelism are explained.

Hegel on Sophocles’ Oedipus the King and the Moral Accountability of Ancient Tragic Heroes

2018 ◽  
Vol 41 (2) ◽  
pp. 159-176 ◽  
Author(s):  
Rachel Falkenstern
Keyword(s):  
Historical Account ◽  
Right Action ◽  
Tragic Hero ◽  
Moral Accountability ◽  
The Right

AbstractThis paper argues that Hegel’s account of subjectivity and agency as historically coined is essential to an accurate understanding of his theory of tragedy. Focusing on Sophocles’ Oedipus the King, I argue that Hegel’s historical account of agency is necessary for understanding his theory of the ancient tragic hero. Although Hegel’s theory of ancient tragedy is often described in terms of a conflict between ethical spheres embodied in two individuals, the conflict in Oedipus is between Oedipus’ deeds and his later knowledge of what has actually occurred. I show how this seemingly subjective conflict is in keeping with Hegel’s theory. Further, while Hegel sees Oedipus as wrong to take full moral accountability for the consequences of his deeds, at the same time, for Hegel, this is the right action for a tragic hero, and the very thing that renders Oedipus timelessly and tragically heroic, rather than a mere victim of fate.

CRITERION OF PROPER ACTIONS FOR 3-STEP NILPOTENT LIE GROUPS

2007 ◽  
Vol 18 (07) ◽  
pp. 783-795 ◽  
Author(s):  
TARO YOSHINO
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Lie Group ◽  
Closed Subgroup ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Affirmative Solution

For a nilpotent Lie group G and its closed subgroup L, Lipsman [13] conjectured that the L-action on some homogeneous space of G is proper in the sense of Palais if and only if the action is free. Nasrin [14] proved this conjecture assuming that G is a 2-step nilpotent Lie group. However, Lipsman's conjecture fails for the 4-step nilpotent case. This paper gives an affirmative solution to Lipsman's conjecture for the 3-step nilpotent case.

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