scholarly journals Conformal Killing forms on 2-step nilpotent Riemannian Lie groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Viviana del Barco ◽  
Andrei Moroianu
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Conformal Killing ◽  
Nilpotent Lie Groups ◽  
Killing Form ◽  
Invariant Metric ◽  
Nilpotent Lie Algebra ◽  
Simply Connected ◽  
Connected Lie Group

Abstract We study left-invariant conformal Killing 2- or 3-forms on simply connected 2-step nilpotent Riemannian Lie groups. We show that if the center of the group is of dimension greater than or equal to 4, then every such form is automatically coclosed (i.e. it is a Killing form). In addition, we prove that the only Riemannian 2-step nilpotent Lie groups with center of dimension at most 3 and admitting left-invariant non-coclosed conformal Killing 2- and 3-forms are the following: The Heisenberg Lie groups and their trivial 1-dimensional extensions, endowed with any left-invariant metric, and the simply connected Lie group corresponding to the free 2-step nilpotent Lie algebra on 3 generators, with a particular 1-parameter family of metrics. The explicit description of the space of conformal Killing 2- and 3-forms is provided in each case.

scholarly journals Strictification of étale stacky Lie groups

2011 ◽  
Vol 148 (3) ◽  
pp. 807-834 ◽  
Author(s):  
Giorgio Trentinaglia ◽  
Chenchang Zhu
Keyword(s):  
Fundamental Group ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Crossed Module ◽  
Simply Connected ◽  
The Given ◽  
Connected Lie Group

AbstractWe define stacky Lie groups to be group objects in the 2-category of differentiable stacks. We show that every connected and étale stacky Lie group is equivalent to a crossed module of the form (Γ,G) where Γ is the fundamental group of the given stacky Lie group and G is the connected and simply connected Lie group integrating the Lie algebra of the stacky group. Our result is closely related to a strictification result of Baez and Lauda.

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 Homogeneous geodesics in pseudo-Riemannian nilmanifolds

2016 ◽  
Vol 16 (2) ◽  
Author(s):  
Viviana del Barco
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Orbit Space ◽  
Nilpotent Lie Groups ◽  
Invariant Metric ◽  
Natural Parameter ◽  
Geodesic Orbit Space ◽  
Affine Parameter ◽  
Homogeneous Geodesics

AbstractWe study the geodesic orbit property for nilpotent Lie groups N endowed with a pseudo-Riemannian left-invariant metric. We consider this property with respect to different groups acting by isometries. When N acts on itself by left-translations we show that it is a geodesic orbit space if and only if the metric is bi-invariant. Assuming N is 2-step nilpotent and with non-degenerate center we give algebraic conditions on the Lie algebra n of N which imply that every geodesic is the orbit of a one-parameter subgroup of N.Auto(N). In addition we present an example of an almost g.o. space such that for null homogeneous geodesics, the natural parameter of the orbit is not always the affine parameter of the geodesic.

LEFT INVARIANT COMPLEX STRUCTURES ON NILPOTENT SIMPLY CONNECTED INDECOMPOSABLE 6-DIMENSIONAL REAL LIE GROUPS

2007 ◽  
Vol 17 (01) ◽  
pp. 115-139 ◽  
Author(s):  
L. MAGNIN
Keyword(s):  
Automorphism Group ◽  
Normal Form ◽  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Normal Forms ◽  
Equivalence Classes ◽  
Complex Structures ◽  
Simply Connected ◽  
Connected Lie Group

Integrable complex structures on indecomposable 6-dimensional nilpotent real Lie algebras have been computed in a previous paper, along with normal forms for representatives of the various equivalence classes under the action of the automorphism group. Here we go to the connected simply connected Lie group G0 associated to such a Lie algebra 𝔤. For each normal form J of integrable complex structures on 𝔤, we consider the left invariant complex manifold G = (G0, J) associated to G0 and J. We explicitly compute a global holomorphic chart for G and we write down the multiplication in that chart.

Self-Maps of Low Rank Lie Groups at Odd Primes

2010 ◽  
Vol 62 (2) ◽  
pp. 284-304 ◽  
Author(s):  
Jelena Grbić ◽  
Stephen Theriault
Keyword(s):  
Lie Groups ◽  
Structural Properties ◽  
Lie Group ◽  
Low Rank ◽  
Rank Condition ◽  
Homotopy Classes ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a simple, compact, simply-connected Lie group localized at an odd prime p. We study the group of homotopy classes of self-maps [G, G] when the rank of G is low and in certain cases describe the set of homotopy classes ofmultiplicative self-maps H[G, G]. The low rank condition gives G certain structural properties which make calculations accessible. Several examples and applications are given.

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 Diophantine properties of nilpotent Lie groups

2015 ◽  
Vol 151 (6) ◽  
pp. 1157-1188 ◽  
Author(s):  
Menny Aka ◽  
Emmanuel Breuillard ◽  
Lior Rosenzweig ◽  
Nicolas de Saxcé
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Nilpotent Lie Groups ◽  
Finitely Generated ◽  
Derived Length ◽  
Simple Lie Groups ◽  
The Ideal

A finitely generated subgroup ${\rm\Gamma}$ of a real Lie group $G$ is said to be Diophantine if there is ${\it\beta}>0$ such that non-trivial elements in the word ball $B_{{\rm\Gamma}}(n)$ centered at $1\in {\rm\Gamma}$ never approach the identity of $G$ closer than $|B_{{\rm\Gamma}}(n)|^{-{\it\beta}}$. A Lie group $G$ is said to be Diophantine if for every $k\geqslant 1$ a random $k$-tuple in $G$ generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most $5$, or derived length at most $2$, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non-Diophantine nilpotent and solvable (non-nilpotent) Lie groups.

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.

On the ‘strange formula’ of Freudenthal and de Vries

1989 ◽  
Vol 105 (2) ◽  
pp. 249-252 ◽  
Author(s):  
H. D. Fegan ◽  
B. Steer
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Weyl Chamber ◽  
Inner Product ◽  
Killing Form ◽  
De Vries ◽  
Positive Roots ◽  
Connected Lie Group

Suppose that G is a semi-simple, compact, connected Lie group. Endow g, its Lie algebra, with the inner product which is the negative of the Killing form. Choose a fundamental Weyl Chamber and let R+ denote the positive roots so determined.

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