scholarly journals THE STRATIFIED STRUCTURE OF SPACES OF SMOOTH ORBIFOLD MAPPINGS

2013 ◽  
Vol 15 (05) ◽  
pp. 1350018 ◽  
Author(s):  
JOSEPH E. BORZELLINO ◽  
VICTOR BRUNSDEN
Keyword(s):  
Lie Groups ◽  
Topological Groups ◽  
Diffeomorphism Groups ◽  
Stratified Space ◽  
Stratified Structure ◽  
Special Case

We consider four notions of maps between smooth C∞ orbifolds [Formula: see text], [Formula: see text] with [Formula: see text] compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of Cr maps between [Formula: see text] and [Formula: see text] with the Cr topology carries the structure of a smooth C∞ Banach (r finite)/Fréchet (r = ∞) manifold. For the notion of complete reduced orbifold map, the corresponding set of Cr maps between [Formula: see text] and [Formula: see text] with the Cr topology carries the structure of a smooth C∞ Banach (r finite)/Fréchet (r = ∞) orbifold. The remaining two notions carry a stratified structure: The Cr orbifold maps between [Formula: see text] and [Formula: see text] is locally a stratified space with strata modeled on smooth C∞ Banach (r finite)/Fréchet (r = ∞) manifolds while the set of Cr reduced orbifold maps between [Formula: see text] and [Formula: see text] locally has the structure of a stratified space with strata modeled on smooth C∞ Banach (r finite)/Fréchet (r = ∞) orbifolds. Furthermore, we give the explicit relationship between these notions of orbifold map. Applying our results to the special case of orbifold diffeomorphism groups, we show that they inherit the structure of C∞ Banach (r finite)/Fréchet (r = ∞) manifolds. In fact, for r finite they are topological groups, and for r = ∞ they are convenient Fréchet Lie groups.

scholarly journals Products of protopological groups

2001 ◽  
Vol 28 (7) ◽  
pp. 433-435
Author(s):  
Julie C. Jones
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Characterization Theorem ◽  
Invariant Subgroup ◽  
Topological Groups ◽  
Special Case

Montgomery and Zippin saied that a group is approximated by Lie groups if every neighborhood of the identity contains an invariant subgroupHsuch thatG/His topologically isomorphic to a Lie group. Bagley, Wu, and Yang gave a similar definition, which they called a pro-Lie group. Covington extended this concept to a protopological group. Covington showed that protopological groups possess many of the characteristics of topological groups. In particular, Covington showed that in a special case, the product of protopological groups is a protopological group. In this note, we give a characterization theorem for protopological groups and use it to generalize her result about products to the category of all protopological groups.

scholarly journals Stochastic processes on totally disconnected topological groups

2003 ◽  
Vol 2003 (48) ◽  
pp. 3067-3089 ◽  
Author(s):  
S. V. Ludkovsky
Keyword(s):  
Stochastic Processes ◽  
Banach Spaces ◽  
Lie Groups ◽  
Local Fields ◽  
Loop Groups ◽  
Topological Groups ◽  
Diffeomorphism Groups ◽  
Loop Spaces ◽  
Totally Disconnected ◽  
Continuous Representations

Stochastic processes on totally disconnected topological groups are investigated. In particular, they are considered for diffeomorphism groups and loop groups of manifolds on non-Archimedean Banach spaces. Theorems about a quasi-invariance and a pseudodifferentiability of transition measures are proved. Transition measures are used for the construction of strongly continuous representations including the irreducible ones of these groups. In addition, stochastic processes on general Banach-Lie groups, loop monoids, loop spaces, and path spaces of manifolds on Banach spaces over non-Archimedean local fields are also investigated.

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.

scholarly journals Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics

2010 ◽  
Vol 2010 ◽  
pp. 1-35 ◽  
Author(s):  
Rudolf Schmid
Keyword(s):  
Lie Groups ◽  
Mathematical Physics ◽  
Quantum Field ◽  
Loop Groups ◽  
Diffeomorphism Groups ◽  
Gauge Groups ◽  
Infinite Dimensional ◽  
Lie Groups And Algebras ◽  
Volume Preserving ◽  
Infinite Dimensional Lie Groups

We give a review of infinite-dimensional Lie groups and algebras and show some applications and examples in mathematical physics. This includes diffeomorphism groups and their natural subgroups like volume-preserving and symplectic transformations, as well as gauge groups and loop groups. Applications include fluid dynamics, Maxwell's equations, and plasma physics. We discuss applications in quantum field theory and relativity (gravity) including BRST and supersymmetries.

scholarly journals Free products of topological groups: Corrigendum

1975 ◽  
Vol 12 (3) ◽  
pp. 480-480
Author(s):  
Sidney A. Morris
Keyword(s):  
Topological Groups ◽  
Free Products ◽  
Special Case

Professor Edward T. Ordman has pointed out to the author that the proof of Theorem 2.2 in [2] is incorrect. The theorem is, in fact, correct and was proved by Graev [1] The incorrect proof presented in my paper has been modified by Ordman [3] to provide a much simpler proof than Graev's in a special case.

scholarly journals Orthogonal Polynomials of Compact Simple Lie Groups

2011 ◽  
Vol 2011 ◽  
pp. 1-23 ◽  
Author(s):  
Maryna Nesterenko ◽  
Jiří Patera ◽  
Agnieszka Tereszkiewicz
Keyword(s):  
Simple Group ◽  
Lie Groups ◽  
Lie Group ◽  
Chebyshev Polynomials ◽  
Recurrence Relations ◽  
Uniform Method ◽  
Symmetric Polynomials ◽  
Semisimple Lie Group ◽  
Algebraic Construction ◽  
Special Case

Recursive algebraic construction of two infinite families of polynomials innvariables is proposed as a uniform method applicable to every semisimple Lie group of rankn. Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of typeA1. The obtained not Laurent-type polynomials are equivalent to the partial cases of the Macdonald symmetric polynomials. Recurrence relations are shown for the Lie groups of typesA1,A2,A3,C2,C3,G2, andB3together with lowest polynomials.

scholarly journals On the large-scale geometry of diffeomorphism groups of 1-manifolds

2018 ◽  
Vol 30 (1) ◽  
pp. 75-86
Author(s):  
Michael P. Cohen
Keyword(s):  
Banach Space ◽  
Large Scale ◽  
Local Property ◽  
Topological Groups ◽  
Diffeomorphism Groups ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Relative Property ◽  
Large Scale Geometry ◽  
Isometry Class

Abstract We apply the framework of Rosendal to study the large-scale geometry of the topological groups {\operatorname{Diff}_{+}^{k}(M^{1})} , consisting of orientation-preserving {C^{k}} -diffeomorphisms (for {1\leq k\leq\infty} ) of a compact 1-manifold {M^{1}} ( {=I} or {\mathbb{S}^{1}} ). We characterize the relative property (OB) in such groups: {A\subseteq\operatorname{Diff}_{+}^{k}(M^{1})} has property (OB) relative to {\operatorname{Diff}_{+}^{k}(M^{1})} if and only if {\sup_{f\in A}\sup_{x\in M^{1}}\lvert\log f^{\prime}(x)|<\infty} and {\sup_{f\in A}\sup_{x\in M^{1}}|f^{(j)}(x)|<\infty} for every integer j with {2\leq j\leq k} . We deduce that {\operatorname{Diff}_{+}^{k}(M^{1})} has the local property (OB), and consequently a well-defined non-trivial quasi-isometry class, if and only if {k<\infty} . We show that the groups {\operatorname{Diff}_{+}^{1}(I)} and {\operatorname{Diff}_{+}^{1}(\mathbb{S}^{1})} are quasi-isometric to the infinite-dimensional Banach space {C[0,1]} .

scholarly journals Lie groups in varieties of topological groups

1974 ◽  
Vol 30 (2) ◽  
pp. 229-235 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Lie Groups ◽  
Topological Groups
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