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 $G$ .

Some Results in the Connective K-Theory of Lie Groups

1988 ◽  
Vol 31 (2) ◽  
pp. 194-199
Author(s):  
L. Magalhães
Keyword(s):  
Hopf Algebra ◽  
Lie Groups ◽  
Lie Group ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
K Theory ◽  
Torsion Free ◽  
Connected Lie Group

AbstractIn this paper we give a description of:(1) the Hopf algebra structure of k*(G; L) when G is a compact, connected Lie group and L is a ring of type Q(P) so that H*(G; L) is torsion free;(2) the algebra structure of k*(G2; L) for L = Z2 or Z.

The mod 2 homology of the space of loops on the exceptional Lie group

1989 ◽  
Vol 112 (3-4) ◽  
pp. 187-202 ◽  
Author(s):  
Akira Kono ◽  
Kazumuto Kozima
Keyword(s):  
Hopf Algebra ◽  
Lie Groups ◽  
Lie Group ◽  
Steenrod Algebra ◽  
Algebra Structure ◽  
Spectral Sequences ◽  
Hopf Algebra Structure ◽  
Simple Lie Groups ◽  
Simply Connected

SynopsisThe Hopf algebra structure of H*(ΩG, F2) and the action of the dual Steenrod algebra are completely and explicitly determined when G isone of the connected, simply connected, exceptional, simple Lie groups. The approach is homological, using connected coverings and spectral sequences.

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 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.

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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