scholarly journals THE ASYMPTOTIC SEMIGROUP OF A SUBSEMIGROUP OF A NILPOTENT LIE GROUP

2021 ◽  
Author(s):  
HERBERT ABELS
Keyword(s):  
Lie Group ◽  
Nilpotent Lie Group ◽  
Simply Connected

AbstractLet S be a subsemigroup of a simply connected nilpotent Lie group G. We construct an asymptotic semigroup S0 in the associated graded Lie group G0 of G. We can compute the image of S0 in the abelianization $$ {G}_0^{\mathrm{ab}}={G}^{\mathrm{ab}}. $$ G 0 ab = G ab . This gives useful information about S. As an application, we obtain a transparent proof of the following result of E. B. Vinberg and the author: either there is an epimorphism f : G → ℝ such that f (s) ≥ 0 for every s in S or the closure $$ \overline{S} $$ S ¯ of S is a subgroup of G and $$ G/\overline{S} $$ G / S ¯ is compact.

PROPER ACTIONS ON SOME EXPONENTIAL SOLVABLE HOMOGENEOUS SPACES

2005 ◽  
Vol 16 (09) ◽  
pp. 941-955 ◽  
Author(s):  
ALI BAKLOUTI ◽  
FATMA KHLIF
Keyword(s):  
Maximal Subgroup ◽  
Homogeneous Space ◽  
Lie Group ◽  
Homogeneous Spaces ◽  
Intersection Property ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Simply Connected

Let G be a connected, simply connected nilpotent Lie group, H and K be connected subgroups of G. We show in this paper that the action of K on X = G/H is proper if and only if the triple (G,H,K) has the compact intersection property in both cases where G is at most three-step and where G is special, extending then earlier cases. The result is also proved for exponential homogeneous space on which acts a maximal subgroup.

scholarly journals DISTAL ACTIONS OF AUTOMORPHISMS OF NILPOTENT GROUPS G ON SUB G AND APPLICATIONS TO LATTICES IN LIE GROUPS

2020 ◽  
pp. 1-20
Author(s):  
RAJDIP PALIT ◽  
RIDDHI SHAH
Keyword(s):  
Lie Group ◽  
Compact Subgroup ◽  
Nilpotent Groups ◽  
Discrete Groups ◽  
Nilpotent Lie Group ◽  
The Difference ◽  
Simply Connected ◽  
Chabauty Topology ◽  
Solvable Lie Group ◽  
Compactly Generated

Abstract For a locally compact group G, we study the distality of the action of automorphisms T of G on Sub G , the compact space of closed subgroups of G endowed with the Chabauty topology. For a certain class of discrete groups G, we show that T acts distally on Sub G if and only if T n is the identity map for some $n\in\mathbb N$ . As an application, we get that for a T-invariant lattice Γ in a simply connected nilpotent Lie group G, T acts distally on Sub G if and only if it acts distally on SubΓ. This also holds for any closed T-invariant co-compact subgroup Γ in G. For a lattice Γ in a simply connected solvable Lie group, we study conditions under which its automorphisms act distally on SubΓ. We construct an example highlighting the difference between the behaviour of automorphisms on a lattice in a solvable Lie group and that in a nilpotent Lie group. We also characterise automorphisms of a lattice Γ in a connected semisimple Lie group which act distally on SubΓ. For torsion-free compactly generated nilpotent (metrisable) groups G, we obtain the following characterisation: T acts distally on Sub G if and only if T is contained in a compact subgroup of Aut(G). Using these results, we characterise the class of such groups G which act distally on Sub G . We also show that any compactly generated distal group G is Lie projective.

scholarly journals Remarks on Hodge numbers and invariant complex structures of compact nilmanifolds

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Takumi Yamada
Keyword(s):  
Lie Group ◽  
Complex Structure ◽  
Complex Structures ◽  
Nilpotent Lie Group ◽  
Hodge Numbers ◽  
Compact Nilmanifolds ◽  
Simply Connected

AbstractIf N is a simply connected real nilpotent Lie group with a Γ-rational complex structure, where Γ is a lattice in N, then for each s, t.We study relations between invariant complex structures and Hodge numbers of compact nilmanifolds from a viewpoint of Lie algberas.

scholarly journals On the Discrete Subgroups and Homogeneous Spaces of Nilpotent Lie Groups

1951 ◽  
Vol 2 ◽  
pp. 95-110 ◽  
Author(s):  
Yozô Matsushima
Keyword(s):  
Homogeneous Space ◽  
Compact Space ◽  
Lie Group ◽  
Homogeneous Spaces ◽  
Discrete Subgroup ◽  
Nilpotent Lie Group ◽  
Discrete Subgroups ◽  
Connected Subgroup ◽  
Structure Constants ◽  
Simply Connected

Recently A, Malcev has shown that the homogeneous space of a connected nilpotent Lie group G is the direct product of a compact space and an Euclidean-space and that the compact space of this direct decomposition is also a homogeneous space of a connected subgroup of G. Any compact homogeneous space M of a connected nilpotent Lie group is of the form where is a connected simply connected nilpotent group whose structure constants are rational numbers in a suitable coordinate system and D is a discrete subgroup of G.

scholarly journals Some examples of quasiisometries of nilpotent Lie groups

2016 ◽  
Vol 2016 (718) ◽  
Author(s):  
Xiangdong Xie
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Simply Connected

AbstractWe construct quasiisometries of nilpotent Lie groups. In particular, for any simply connected nilpotent Lie group

scholarly journals VARIANTS OF MIYACHI’S THEOREM FOR NILPOTENT LIE GROUPS

2010 ◽  
Vol 88 (1) ◽  
pp. 1-17 ◽  
Author(s):  
ALI BAKLOUTI ◽  
SUNDARAM THANGAVELU
Keyword(s):  
Fourier Transform ◽  
Lie Groups ◽  
Lie Group ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Group Fourier Transform ◽  
Simply Connected ◽  
Miyachi’S Theorem

AbstractWe formulate and prove two versions of Miyachi’s theorem for connected, simply connected nilpotent Lie groups. This allows us to prove the sharpness of the constant 1/4 in the theorems of Hardy and of Cowling and Price for any nilpotent Lie group. These theorems are proved using a variant of Miyachi’s theorem for the group Fourier transform.

scholarly journals Equidistribution of dense subgroups on nilpotent Lie groups

2009 ◽  
Vol 30 (1) ◽  
pp. 131-150 ◽  
Author(s):  
EMMANUEL BREUILLARD
Keyword(s):  
Random Walk ◽  
Lie Group ◽  
Analogous Result ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Uniform Measure ◽  
Dense Subgroup ◽  
Symmetric Set ◽  
Dense Subgroups ◽  
Simply Connected

AbstractLet Γ be a dense subgroup of a simply connected nilpotent Lie group G generated by a finite symmetric set S. We consider the n-ball Sn for the word metric induced by S on Γ. We show that Sn (with uniform measure) becomes equidistributed on G with respect to the Haar measure as n tends to infinity. We also prove the analogous result for random walk averages.

scholarly journals Generalized Ricci flow on nilpotent Lie groups

2021 ◽  
Vol 33 (4) ◽  
pp. 997-1014
Author(s):  
Fabio Paradiso
Keyword(s):  
Heisenberg Group ◽  
Lie Group ◽  
Ricci Flow ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Geometric Flows ◽  
Courant Algebroid ◽  
New Family ◽  
Lie Brackets ◽  
Simply Connected

Abstract We define solitons for the generalized Ricci flow on an exact Courant algebroid. We then define a family of flows for left-invariant Dorfman brackets on an exact Courant algebroid over a simply connected nilpotent Lie group, generalizing the bracket flows for nilpotent Lie brackets in a way that might make this new family of flows useful for the study of generalized geometric flows such as the generalized Ricci flow. We provide explicit examples of both constructions on the Heisenberg group. We also discuss solutions to the generalized Ricci flow on the Heisenberg group.

scholarly journals One more method to construct the irreducible unitary representations of nipotent Lie groups

1986 ◽  
Vol 40 (1) ◽  
pp. 89-94
Author(s):  
A. A. Astaneh
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Nilpotent Lie Group ◽  
Irreducible Unitary Representations ◽  
Simply Connected

AbstractIn this paper one more canonical method to construct the irreducible unitary representations of a connected, simply connected nilpotent Lie group is introduced. Although we used Kirillov' analysis to deduce this procedure, the method obtained differs from that of Kirillov's, in that one does not need to consider the codjoint representation of the group in the dual of its Lie algebra (in fact, neither does one need to consider the Lie algebra of the group, provided one knows certain connected subgroups and their characters). The method also differs from that of Mackey's as one only needs to induce characters to obtain all irreducible representations of the group.

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