Exponentiality of Certain Real Solvable Lie Groups

1998 ◽  
Vol 41 (3) ◽  
pp. 368-373 ◽  
Author(s):  
Martin Moskowitz ◽  
Michael Wüstner
Keyword(s):  
Lie Groups ◽  
Semidirect Product ◽  
Lie Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Solvable Lie Groups ◽  
Necessary And Sufficient ◽  
Solvable Lie Group

AbstractIn this article, making use of the second author’s criterion for exponentiality of a connected solvable Lie group, we give a rather simple necessary and sufficient condition for the semidirect product of a torus acting on certain connected solvable Lie groups to be exponential.

scholarly journals The Ricci pinching functional on solvmanifolds

2019 ◽  
Author(s):  
Jorge Lauret ◽  
Cynthia E Will
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Invariant Metrics ◽  
Solvable Lie Groups ◽  
Left Invariant Metrics ◽  
Solvable Lie Group

Abstract We study the natural functional $F=\frac {\operatorname {scal}^2}{|\operatorname {Ric}|^2}$ on the space of all non-flat left-invariant metrics on all solvable Lie groups of a given dimension $n$. As an application of properties of the beta operator, we obtain that solvsolitons are the only global maxima of $F$ restricted to the set of all left-invariant metrics on a given unimodular solvable Lie group, and beyond the unimodular case, we obtain the same result for almost-abelian Lie groups. Many other aspects of the behavior of $F$ are clarified.

scholarly journals Cylindrical representations of some infinite dimensional nuclear Lie groups

1992 ◽  
Vol 46 (2) ◽  
pp. 295-310 ◽  
Author(s):  
Jean Marion
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Additive Group ◽  
Test Functions ◽  
Necessary And Sufficient Condition ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations ◽  
Direct Product Group ◽  
Necessary And Sufficient

Let Γ.𝒜 be the semi-direct product group of a nuclear Lie group Γ with the additive group 𝒜 of a real nuclear vector space. We give an explicit description of all the continuous representations of Γ.𝒜 the restriction of which to 𝒜 is a cyclic unitary representation, and a necessary and sufficient condition for the unitarity of such cylindrical representations is stated. This general result is successfully used to obtain irreducible unitary representations of the nuclear Lie groups of Riemannian motions, and, in the setting of the theory of multiplicative distributions initiated by I.M. Gelfand, it is proved that for any connected real finite dimensional Lie groupGand for any strictly positive integerkthere exist non located and non trivially decomposable representations of orderkof the nuclear Lie group(M;G) of all theG-valued test-functions on a given paracompact manifoldM.

∗-Ricci tensor on almost Kenmotsu 3-manifolds

2020 ◽  
Vol 17 (13) ◽  
pp. 2050196
Author(s):  
Dibakar Dey ◽  
Pradip Majhi
Keyword(s):  
Lie Group ◽  
Ricci Tensor ◽  
Three Dimensional ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Addition Formula ◽  
Kenmotsu Manifold ◽  
Ricci Operator ◽  
Necessary And Sufficient ◽  
Parallel Ricci Tensor

In this paper, we obtain the expressions of the ∗-Ricci operator of a three-dimensional almost Kenmotsu manifold [Formula: see text] and find that the ∗-Ricci tensor is not symmetric for [Formula: see text]. We obtain a necessary and sufficient condition for the ∗-Ricci tensor to be symmetric and proved that if the ∗-Ricci tensor of a non-Kenmotsu almost Kenmotsu 3-[Formula: see text]-manifold [Formula: see text] is symmetric, then [Formula: see text] is locally isometric to a three-dimensional non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure. Further, it is shown that the ∗-Ricci tensor of a non-Kenmotsu almost Kenmotsu 3-manifold [Formula: see text] is parallel if and only if [Formula: see text] is ∗-Ricci flat. In addition, [Formula: see text] satisfying [Formula: see text] is locally isometric to [Formula: see text]. Finally, we discuss about [Formula: see text]-parallel ∗-Ricci tensor on almost Kenmotsu 3-manifolds.

scholarly journals ON GENERATORS AND PRESENTATIONS OF SEMIDIRECT PRODUCTS IN INVERSE SEMIGROUPS

2009 ◽  
Vol 79 (3) ◽  
pp. 353-365 ◽  
Author(s):  
E. R. DOMBI ◽  
N. RUŠKUC
Keyword(s):  
Semidirect Product ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Semidirect Products ◽  
Necessary And Sufficient ◽  
Finitely Presented

AbstractIn this paper we prove two main results. The first is a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated. The second result is a necessary and sufficient condition for such a semidirect product to be finitely presented.

scholarly journals On the component group of the automorphism group of a Lie group

1994 ◽  
Vol 57 (3) ◽  
pp. 365-385 ◽  
Author(s):  
P. B. Chen ◽  
T. S. Wu
Keyword(s):  
Automorphism Group ◽  
Lie Group ◽  
Faithful Representation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Connected Component ◽  
Component Group ◽  
Necessary And Sufficient

AbstractLet G be a Lie group, Go the connected component of G that contains the identity, and Aut G the group of all topological automorphisms of G. In the case when G/Go is finite and G has a faithful representation, we obtain a necessary and sufficient condition for G so that Aut G has finitely many components in terms of the maximal central torus in Go.

Topology on the unitary dual of completely solvable Lie groups

2015 ◽  
Vol 6 (4) ◽  
Author(s):  
Detlev Poguntke
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Orbit Space ◽  
Orbit Method ◽  
Solvable Lie Groups ◽  
Bijective Correspondence ◽  
Unitary Dual ◽  
Exponential Lie Group ◽  
Solvable Lie Group

AbstractIt was one of great successes of Kirillov's orbit method to see that the unitary dual of an exponential Lie group is in bijective correspondence with the orbit space associated with the linear dual of the Lie algebra of the group in question. To show that this correspondence is an homeomorphism turned out to be unexpectedly difficult. Only in 1994 H. Leptin and J. Ludwig gave a proof using the notion of variable groups. In this article their proof in the case of completely solvable Lie group is reorganized, some “philosophy” and some new arguments are added. The purpose is to contribute to a better understanding of this proof.

scholarly journals Existence of Cocompact Lattices in Lie Groups With a Bi-invariant Metric of Index 2

2020 ◽  
Author(s):  
Ines Kath
Keyword(s):  
Lie Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Invariant Metric ◽  
One Dimensional ◽  
Salem Numbers ◽  
Index 2 ◽  
Necessary And Sufficient ◽  
Simply Connected ◽  
Cocompact Lattices

Abstract We study the existence of cocompact lattices in Lie groups with bi-invariant metric of signature $(2,n-2)$. We assume in addition that the Lie groups under consideration are simply-connected, indecomposable, and solvable. Then their centre is one- or two-dimensional. In both cases, a parametrisation of the set of such Lie groups is known. We give a necessary and sufficient condition for the existence of a lattice in terms of these parameters. For groups with one-dimensional centre this problem is related to Salem numbers.

scholarly journals Topological Loops with Decomposable Solvable Multiplication Groups

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Ameer Al-Abayechi ◽  
Ágota Figula
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Solvable Lie Groups ◽  
3 Dimensional ◽  
Multiplication Group ◽  
Simply Connected ◽  
Solvable Lie Group

AbstractIn this paper we deal with the class $$\mathcal {C}$$ C of decomposable solvable Lie groups having dimension six. We determine those Lie groups in $$\mathcal {C}$$ C and their subgroups which are the multiplication groups Mult(L) and the inner mapping groups Inn(L) for three-dimensional connected simply connected topological loops L. This result completes the classification of the at most 6-dimensional solvable multiplication Lie groups of the loops L. Moreover, we obtain that every at most 3-dimensional connected topological proper loop having a solvable Lie group of dimension at most six as its multiplication group is centrally nilpotent of class two.

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