scholarly journals POLYNOMIAL COHOMOLOGY AND POLYNOMIAL MAPS ON NILPOTENT GROUPS

2019 ◽  
Vol 62 (3) ◽  
pp. 706-736
Author(s):  
DAVID KYED ◽  
HENRIK DENSING PETERSEN
Keyword(s):  
Lie Group ◽  
Nilpotent Groups ◽  
Group Cohomology ◽  
Exponential Map ◽  
Nilpotent Lie Group ◽  
Polynomial Maps ◽  
Polynomial Coefficients ◽  
Classical Polynomials ◽  
Simply Connected ◽  
Refined Version

AbstractWe introduce a refined version of group cohomology and relate it to the space of polynomials on the group in question. We show that the polynomial cohomology with trivial coefficients admits a description in terms of ordinary cohomology with polynomial coefficients, and that the degree one polynomial cohomology with trivial coefficients admits a description directly in terms of polynomials. Lastly, we give a complete description of the polynomials on a connected, simply connected nilpotent Lie group by showing that these are exactly the maps that pull back to classical polynomials via the exponential map.

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 Proper Actions of Certain Nilpotent Affine Groups with Codimension One Orbits

2012 ◽  
Vol 4 (2) ◽  
pp. 315
Author(s):  
N. Salma
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Lie Group ◽  
Nilpotent Groups ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Codimension One ◽  
Affine Groups ◽  
Simply Connected

Criterion for proper actions has been established for a homogeneous space of reductive type by Kobayashi (Math. Ann. 1989, 1996). On the other hand, an analogous criterion to Kobayashi’s equivalent conditions was proposed by Lipsman (1995) for a nilpotent Lie group . Lipsman's Conjecture: Let  be a simply connected nilpotent Lie group. Then the following two conditions on connected subgroups  and  are equivalent: (i) the action of  on  is proper; (ii)  is compact for any  The condition (i) is important in the study of discontinuous groups for the homogeneous space , while the second condition (ii) can easily be checked. The implication (i)  (ii) is obvious, and the opposite implication (ii)  (i) was known only in some lower dimensional cases. In this paper we prove the equivalence (i) ? (ii) for certain affine nilpotent Lie groups . Keywords: Affine nilpotent groups; Homogeneous manifolds; Proper actions; Properly discontinuous actions; Simply connected nilpotent Lie groups; Compact isotropy property (CI);  Eigenvalues. © 2012 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v4i2.7889 J. Sci. Res. 4 (2), 315-326 (2012)

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 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 Equidistribution of joinings under off-diagonal polynomial flows of nilpotent Lie groups

2012 ◽  
Vol 33 (6) ◽  
pp. 1667-1708 ◽  
Author(s):  
TIM AUSTIN
Keyword(s):  
Lie Group ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Norm Convergence ◽  
Ergodic Averages ◽  
Multiple Recurrence ◽  
Polynomial Maps ◽  
Image Position ◽  
Polynomial Flows ◽  
Diagonal Subgroup

AbstractLet $G$ be a connected nilpotent Lie group. Given probability-preserving$G$-actions $(X_i,\Sigma _i,\mu _i,u_i)$, $i=0,1,\ldots ,k$, and also polynomial maps $\phi _i:\mathbb {R}\to G$, $i=1,\ldots ,k$, we consider the trajectory of a joining $\lambda $ of the systems $(X_i,\Sigma _i,\mu _i,u_i)$ under the ‘off-diagonal’ flow \[ (t,(x_0,x_1,x_2,\ldots ,x_k))\mapsto (x_0,u_1^{\phi _1(t)}x_1,u_2^{\phi _2(t)}x_2,\ldots ,u_k^{\phi _k(t)}x_k). \] It is proved that any joining $\lambda $ is equidistributed under this flow with respect to some limit joining $\lambda '$. This is deduced from the stronger fact of norm convergence for a system of multiple ergodic averages, related to those arising in Furstenberg’s approach to the study of multiple recurrence. It is also shown that the limit joining $\lambda '$ is invariant under the subgroup of $G^{k+1}$generated by the image of the off-diagonal flow, in addition to the diagonal subgroup.

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

Lattices in a split solvable Lie group

1997 ◽  
Vol 122 (2) ◽  
pp. 245-250 ◽  
Author(s):  
RICHARD MOSAK ◽  
MARTIN MOSKOWITZ
Keyword(s):  
Euclidean Space ◽  
Lie Group ◽  
Nilpotent Groups ◽  
Integer Lattice ◽  
Heisenberg Groups ◽  
Simply Connected ◽  
Positive Integers ◽  
Solvable Lie Group

Given a Lie group, it is often useful to have a parametrization of the set of its lattices. In Euclidean space ℝn, for example, each lattice corresponds to a basis, and any lattice is equivalent to the standard integer lattice under an automorphism in GL(n, ℝ). In the nilpotent case, the lattices of the Heisenberg groups are classified, up to automorphisms, by certain sequences of positive integers with divisibility conditions (see [1]). In this paper we will study the set of lattices in a class of simply connected, solvable, but not nilpotent groups G. The construction of G depends on a diagonal n×n matrix Δ with distinct non-zero eigenvalues, of trace 0; we will writeformula here

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.

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