scholarly journals The Guillemin–Sternberg conjecture for noncompact groups and spaces

2008 ◽  
Vol 1 (3) ◽  
pp. 473-533 ◽  
Author(s):  
P. Hochs ◽  
N.P. Landsman
Keyword(s):  
Normal Subgroup ◽  
Dirac Operator ◽  
Lie Groups ◽  
Group Action ◽  
Symplectic Manifolds ◽  
Lie Group Action ◽  
Compact Case ◽  
Assembly Map ◽  
High Degree ◽  
Special Case

AbstractThe Guillemin–Sternberg conjecture states that “quantisation commutes with reduction” in a specific technical setting. So far, this conjecture has almost exclusively been stated and proved for compact Lie groups G acting on compact symplectic manifolds, and, largely due to the use of Spinc Dirac operator techniques, has reached a high degree of perfection under these compactness assumptions. In this paper we formulate an appropriate Guillemin–Sternberg conjecture in the general case, under the main assumptions that the Lie group action is proper and cocompact. This formulation is motivated by our interpretation of the “quantisation commuates with reduction” phenomenon as a special case of the functoriality of quantisation, and uses equivariant K-homology and the K-theory of the group C*-algebra C*(G) in a crucial way. For example, the equivariant index – which in the compact case takes values in the representation ring R(G) – is replaced by the analytic assembly map – which takes values in K0(C*(G)) – familiar from the Baum–Connes conjecture in noncommutative geometry. Under the usual freeness assumption on the action, we prove our conjecture for all Lie groups G having a discrete normal subgroup Γ with compact quotient G/Γ, but we believe it is valid for all unimodular Lie groups.

FACTORISATION OF EQUIVARIANT SPECTRAL TRIPLES IN UNBOUNDED -THEORY

2018 ◽  
Vol 107 (02) ◽  
pp. 145-180
Author(s):  
IAIN FORSYTH ◽  
ADAM RENNIE
Keyword(s):  
Fixed Point ◽  
Lie Groups ◽  
Group Action ◽  
Principal Bundle ◽  
Sufficient Conditions ◽  
Total Space ◽  
Spectral Triple ◽  
Spectral Triples ◽  
Dirac Type ◽  
Special Case

We provide sufficient conditions to factorise an equivariant spectral triple as a Kasparov product of unbounded classes constructed from the group action on the algebra and from the fixed point spectral triple. We show that if factorisation occurs, then the equivariant index of the spectral triple vanishes. Our results are for the action of compact abelian Lie groups, and we demonstrate them with examples from manifolds and $\unicode[STIX]{x1D703}$ -deformations. In particular, we show that equivariant Dirac-type spectral triples on the total space of a torus principal bundle always factorise. Combining this with our index result yields a special case of the Atiyah–Hirzebruch theorem. We also present an example that shows what goes wrong in the absence of our sufficient conditions (and how we get around it for this example).

scholarly journals Homogeneous Ricci solitons

2015 ◽  
Vol 2015 (699) ◽  
Author(s):  
Michael Jablonski
Keyword(s):  
Lie Groups ◽  
Ricci Flow ◽  
Isometry Group ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Solvable Lie Groups ◽  
Compact Case ◽  
Group Of Isometries ◽  
Simply Connected ◽  
Special Case

AbstractIn this work, we study metrics which are both homogeneous and Ricci soliton. If there exists a transitive solvable group of isometries on a Ricci soliton, we show that it is isometric to a solvsoliton. Moreover, unless the manifold is flat, it is necessarily simply-connected and diffeomorphic to ℝIn the general case, we prove that homogeneous Ricci solitons must be semi-algebraic Ricci solitons in the sense that they evolve under the Ricci flow by dilation and pullback by automorphisms of the isometry group. In the special case that there exists a transitive semi-simple group of isometries on a Ricci soliton, we show that such a space is in fact Einstein. In the compact case, we produce new proof that Ricci solitons are necessarily Einstein.Lastly, we characterize solvable Lie groups which admit Ricci soliton metrics.

scholarly journals Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation

2020 ◽  
Author(s):  
Jun Jiang ◽  
◽  
Satyendra Kumar Mishra ◽  
Yunhe Sheng ◽  
◽  
...  
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Group Action ◽  
Lie Group ◽  
Smooth Manifold ◽  
Adjoint Representation ◽  
Lie Group Action

In this paper, we introduce the notion of a (regular) Hom-Lie group. We associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of this Hexp map. We also describe a Hom-Lie group action on a smooth manifold. Subsequently, we give the notion of an adjoint representation of a Hom-Lie group on its Hom-Lie algebra. At last, we integrate the Hom-Lie algebra (gl(V),[.,.],Ad), and the derivation Hom-Lie algebra of a Hom-Lie algebra.

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 The stratified spaces of a symplectic lie group action

2006 ◽  
Vol 58 (1) ◽  
pp. 51-75 ◽  
Author(s):  
Juan-Pablo Ortega ◽  
Tudor S. Ratiu
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Stratified Spaces ◽  
Lie Group Action

SINGULAR SWITCHED LINEAR SYSTEMS: SOME GEOMETRIC ASPECTS

2012 ◽  
Vol 26 (25) ◽  
pp. 1246006
Author(s):  
H. DIEZ-MACHÍO ◽  
J. CLOTET ◽  
M. I. GARCÍA-PLANAS ◽  
M. D. MAGRET ◽  
M. E. MONTORO
Keyword(s):  
Linear Systems ◽  
Group Action ◽  
Lie Group ◽  
Equivalence Relation ◽  
Geometric Approach ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Switched Linear Systems ◽  
Lie Group Action

We present a geometric approach to the study of singular switched linear systems, defining a Lie group action on the differentiable manifold consisting of the matrices defining their subsystems with orbits coinciding with equivalence classes under an equivalence relation which preserves reachability and derive miniversal (orthogonal) deformations of the system. We relate this with some new results on reachability of such systems.

scholarly journals Certain Types of Complex Lie Group Action

2018 ◽  
Vol 10 (3) ◽  
Author(s):  
Ahmed Khalaf Radhi ◽  
Taghreed Hur Majeed
Keyword(s):  
Tensor Product ◽  
Group Action ◽  
Lie Group ◽  
The Other ◽  
Smooth Representation ◽  
Equivalent Relation ◽  
Complex Group ◽  
Lie Group Action ◽  
Group 10 ◽  
Definition Of

     The main aim in this paper is to look for a novel action with new properties on       from the  , the Literature are concerned with studying the action of  of two representations , one is usual and the other is the dual, while our  interest in this work  is focused on some actions on complex Lie group[10] . Let G be a matrix complex  group , and  is representation of   In this study we will present and analytic  the  concepts of action of complex  group on    We recall the definition of  tensor  product of two representations of  group and construct  the definition of action of   group on , then by using the equivalent  relation   between  and  , we get a new action : The two actions are forming smooth  representation of    This  we have new action which called     denoted by    which acting on      This  is smooth representation of   The theoretical Justifications are developed and prove supported by some concluding  remarks and illustrations.

scholarly journals The use of parallel computing for the high-resolution determination of earthquake source parameters

2019 ◽  
pp. 68-75
Author(s):  
A. S. Fomochkina ◽  
V. G. Bukchin
Keyword(s):  
Parallel Computing ◽  
Focal Mechanism ◽  
Source Parameters ◽  
Nodal Plane ◽  
Earthquake Parameters ◽  
Earthquake Source Parameters ◽  
Detailed Search ◽  
High Degree ◽  
Special Case

Alongside the determination of the focal mechanism and source depth of an earthquake by direct examination of their probable values on a grid in the parameter space, also the resolution of these determinations can be estimated. However, this approach requires considerable time in the case of a detailed search. A special case of a shallow earthquake whose one nodal plane is subhorizontal is an example of the sources that require the use of a detailed grid. For studying these events based on the records of the long-period surface waves, the grids with high degree of detail in the angles of the focal mechanism are required. We discuss the application of the methods of parallel computing for speeding up the calculations of earthquake parameters and present the results of studying the strongest aftershock of the Tohoku, Japan, earthquake by this approach.

scholarly journals Central Extensions of Lie Groups Preserving a Differential Form

2020 ◽  
Author(s):  
Tobias Diez ◽  
Bas Janssens ◽  
Karl-Hermann Neeb ◽  
Cornelia Vizman
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Differential Form ◽  
Integral Form ◽  
Symplectic Manifolds ◽  
Central Extensions ◽  
Mapping Space ◽  
Canonical Class ◽  
Differential Characters

Abstract Let $M$ be a manifold with a closed, integral $(k+1)$-form $\omega $, and let $G$ be a Fréchet–Lie group acting on $(M,\omega )$. As a generalization of the Kostant–Souriau extension for symplectic manifolds, we consider a canonical class of central extensions of ${\mathfrak{g}}$ by ${\mathbb{R}}$, indexed by $H^{k-1}(M,{\mathbb{R}})^*$. We show that the image of $H_{k-1}(M,{\mathbb{Z}})$ in $H^{k-1}(M,{\mathbb{R}})^*$ corresponds to a lattice of Lie algebra extensions that integrate to smooth central extensions of $G$ by the circle group ${\mathbb{T}}$. The idea is to represent a class in $H_{k-1}(M,{\mathbb{Z}})$ by a weighted submanifold $(S,\beta )$, where $\beta $ is a closed, integral form on $S$. We use transgression of differential characters from $ S$ and $ M $ to the mapping space $ C^\infty (S, M) $ and apply the Kostant–Souriau construction on $ C^\infty (S, M) $.

Sign in / Sign up

Export Citation Format

Share Document