An extension of Minkowski’s theorem to simply connected 2-step nilpotent groups

2010 ◽  
pp. 541-546
Author(s):  
Martin Moskowitz
Keyword(s):  
Nilpotent Groups ◽  
Simply Connected

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.

l′-Isolated Maps and Localizations

1983 ◽  
Vol 35 (2) ◽  
pp. 193-217
Author(s):  
Sara Hurvitz
Keyword(s):  
Finite Number ◽  
Free Algebra ◽  
Nilpotent Groups ◽  
Homotopy Groups ◽  
Homology Groups ◽  
Rational Cohomology ◽  
Locally Nilpotent Groups ◽  
Definition Of ◽  
Simply Connected ◽  
Path Connected

Let P be the set of primes, l ⊆ P a subset and l′ = P – l Recall that an H0-space is a space the rational cohomology of which is a free algebra.Cassidy and Hilton defined and investigated l′-isolated homomorphisms between locally nilpotent groups. Zabrodsky [8] showed that if X and Y are simply connected H0-spaces either with a finite number of homotopy groups or with a finite number of homology groups, then every rational equivalence f : X → Y can be decomposed into an l-equivalence and an l′-equivalence.In this paper we define and investigate l′-isolated maps between pointed spaces, which are of the homotopy type of path-connected nilpotent CW-complexes. Our definition of an l′-isolated map is analogous to the definition of an l′-isolated homomorphism. As every homomorphism can be decomposed into an l-isomorphism and an l′-isolated homomorphism, every map can be decomposed into an l-equivalence and an l′-isolated map.

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

Restricting Representations of Completely Solvable Lie Groups

1990 ◽  
Vol 42 (5) ◽  
pp. 790-824 ◽  
Author(s):  
R. L. Lipsman
Keyword(s):  
Lie Groups ◽  
Dual Problem ◽  
Nilpotent Groups ◽  
Solvable Lie Groups ◽  
Direct Integral ◽  
Induced Representations ◽  
Simply Connected ◽  
Direct Integral Decomposition ◽  
Integral Decomposition ◽  
Restriction Problem

We are concerned here with the problem of describing the direct integral decomposition of a unitary representation obtained by restriction from a larger group. This is the dual problem to the more commonly investigated problem of decomposing induced representations. In this paper we work in the context of completely solvable Lie groups—more general than nilpotent, but less general than exponential solvable. Moreover, the groups involved are simply connected. The restriction problem was considered originally in [2] and in [6] for nilpotent groups.

scholarly journals Functional calculus on non-homogeneous operators on nilpotent groups

2020 ◽  
Author(s):  
Mattia Calzi ◽  
Fulvio Ricci
Keyword(s):  
Functional Calculus ◽  
Nilpotent Groups ◽  
Plancherel Measure ◽  
Asymptotic Estimates ◽  
Local Contraction ◽  
Invariant Differential ◽  
Multiplier Theorems ◽  
Homogeneous Operators ◽  
Simply Connected ◽  
Left Invariant Differential Operator

AbstractWe study the functional calculus associated with a hypoelliptic left-invariant differential operator $$\mathcal {L}$$ L on a connected and simply connected nilpotent Lie group G with the aid of the corresponding Rockland operator $$\mathcal {L}_0$$ L 0 on the ‘local’ contraction $$G_0$$ G 0 of G, as well as of the corresponding Rockland operator $$\mathcal {L}_\infty $$ L ∞ on the ‘global’ contraction $$G_\infty $$ G ∞ of G. We provide asymptotic estimates of the Riesz potentials associated with $$\mathcal {L}$$ L at 0 and at $$\infty $$ ∞ , as well as of the kernels associated with functions of $$\mathcal {L}$$ L satisfying Mihlin conditions of every order. We also prove some Mihlin–Hörmander multiplier theorems for $$\mathcal {L}$$ L which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the ‘Plancherel measure’ associated with $$\mathcal {L}$$ L from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.

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)

scholarly journals Nilpotent Lie Algebras and Systolic Growth of Nilmanifolds

2017 ◽  
Vol 2019 (9) ◽  
pp. 2763-2799
Author(s):  
Yves Cornulier
Keyword(s):  
Lie Algebras ◽  
Lie Group ◽  
Polynomial Growth ◽  
Volume Growth ◽  
Growth Function ◽  
Nilpotent Groups ◽  
Upper Bounds ◽  
Nilpotent Lie Algebras ◽  
Open Questions ◽  
Simply Connected

Abstract Introduced by Gromov in the nineties, the systolic growth of a Lie group gives the smallest possible covolume of a lattice with a given systole. In a simply connected nilpotent Lie group, this function has polynomial growth, but can grow faster than the volume growth. We express this systolic growth function in terms of discrete cocompact subrings of the Lie algebra, making it more practical to estimate. After providing some general upper bounds, we develop methods to provide nontrivial lower bounds. We provide the first computations of the asymptotics of the systolic growth of nilpotent groups for which this is not equivalent to the volume growth. In particular, we provide an example for which the degree of growth is not an integer; it has dimension 7. Finally, we gather some open questions.

scholarly journals Orbital decompositions of representations of non-simply connected nilpotent groups

1990 ◽  
Vol 42 (2) ◽  
pp. 293-306
Author(s):  
Ronald L. Lipsman
Keyword(s):  
Symmetric Spaces ◽  
Integral Formula ◽  
Nilpotent Groups ◽  
Multiplicity Function ◽  
Qualitative Properties ◽  
Spectral Multiplicity ◽  
Induced Representation ◽  
Direct Integral ◽  
Orbital Parameters ◽  
Simply Connected

An orbital integral formula is proven for the direct integral decomposition of an induced representation of a connected nilpotent Lie group. Previous work required simple connectivity. An explicit description of the spectral measure and spectral multiplicity function is derived in terms of orbital parameters. It is also proven that connected (but not necessarily simply connected) exponential solvable symmetric spaces are multiplicity free. Finally, the qualitative properties of the spectral multiplicity function are examined via several illuminating examples.

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.

Equivariant prequantization and admissible coadjoint orbits

1993 ◽  
Vol 114 (1) ◽  
pp. 131-142
Author(s):  
P. L. Robinson
Keyword(s):  
Geometric Quantization ◽  
Classical Mechanics ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Coadjoint Orbits ◽  
Type I ◽  
Quantization Scheme ◽  
Orbit Method ◽  
Irreducible Unitary Representations ◽  
Simply Connected

The orbit method has as its primary goal the construction and parametrization of the irreducible unitary representations of a (simply-connected) Lie group in terms of its coadjoint orbits. This goal was achieved with complete success for nilpotent groups by Kirillov[8] and for type I solvable groups by Auslander and Kostant[l] but is known to encounter difficulties when faced with more general groups. Geometric quantization can be viewed as an outgrowth of the orbit method aimed at providing a geometric passage from classical mechanics to quantum mechanics. Whereas the original geometric quantization scheme due to Kostant[9] and Souriau[14] enabled such a passage in a variety of situations, it too encounters difficulties in broader contexts.

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