scholarly journals AFFINE MAXIMAL TORUS FIBRATIONS OF A COMPACT LIE GROUP

2002 ◽  
Vol 13 (03) ◽  
pp. 217-225 ◽  
Author(s):  
MARCOS SALVAI
Keyword(s):  
Lie Group ◽  
Maximal Torus ◽  
General Procedure ◽  
Great Circle ◽  
Compact Lie Group ◽  
Semisimple Lie Group ◽  
Infinite Dimensional ◽  
Three Sphere ◽  
Maximal Tori

By a generalization of the method developed by Gluck and Warner to characterize the oriented great circle fibrations of the three-sphere, we give, for any compact connected semisimple Lie group G, a general procedure to obtain the continuous fibrations of G by Weyl-oriented affine maximal tori, find conditions for smoothness and provide infinite dimensional spaces of concrete examples.

scholarly journals Projectively flat Finsler manifolds with infinite dimensional holonomy

2015 ◽  
Vol 27 (2) ◽  
Author(s):  
Zoltán Muzsnay ◽  
Péter T. Nagy
Keyword(s):  
Lie Group ◽  
Holonomy Group ◽  
Finsler Manifold ◽  
Flag Curvature ◽  
Compact Lie Group ◽  
Finsler Manifolds ◽  
Infinite Dimensional ◽  
Constant Flag Curvature ◽  
Projectively Flat

AbstractRecently, we developed a method for the study of holonomy properties of non-Riemannian Finsler manifolds and obtained that the holonomy group cannot be a compact Lie group if the Finsler manifold of dimension >2 has non-zero constant flag curvature. The purpose of this paper is to move further, exploring the holonomy properties of projectively flat Finsler manifolds of non-zero constant flag curvature. We prove in particular that projectively flat Randers and Bryant–Shen manifolds of non-zero constant flag curvature have infinite dimensional holonomy group.

scholarly journals THE FUNDAMENTAL GROUP OF G-MANIFOLDS

2013 ◽  
Vol 15 (03) ◽  
pp. 1250056 ◽  
Author(s):  
HUI LI
Keyword(s):  
Fixed Point ◽  
Fundamental Group ◽  
Lie Group ◽  
Maximal Torus ◽  
Moment Map ◽  
Compact Lie Group ◽  
Identity Component ◽  
Connected Subgroup ◽  
Stabilizer Group ◽  
Simply Connected

Let G be a connected compact Lie group, and let M be a connected Hamiltonian G-manifold with equivariant moment map ϕ. We prove that if there is a simply connected orbit G ⋅ x, then π1(M) ≅ π1(M/G); if additionally ϕ is proper, then π1(M) ≅ π1 (ϕ-1(G⋅a)), where a = ϕ(x). We also prove that if a maximal torus of G has a fixed point x, then π1(M) ≅ π1(M/K), where K is any connected subgroup of G; if additionally ϕ is proper, then π1(M) ≅ π1(ϕ-1(G⋅a)) ≅ π1(ϕ-1(a)), where a = ϕ(x). Furthermore, we prove that if ϕ is proper, then [Formula: see text] for all a ∈ ϕ(M), where [Formula: see text] is any connected subgroup of G which contains the identity component of each stabilizer group; in particular, π1(M/G) ≅ π1(ϕ-1(G⋅a)/G) for all a ∈ ϕ(M).

STOCHASTIC EQUATIONS AND QUASI-INVARIANCE ON INFINITE PRODUCT GROUPS

1999 ◽  
Vol 02 (02) ◽  
pp. 283-288 ◽  
Author(s):  
SERGIO ALBEVERIO ◽  
ALEXEI DALETSKII
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Lie Group ◽  
Existence And Uniqueness ◽  
Infinite Product ◽  
Stochastic Equations ◽  
Compact Lie Group ◽  
Infinite Dimensional ◽  
Countable Power ◽  
Infinite Dimensional Lie Group

A stochastic differential equation on an infinite-dimensional Lie group G constructed as the countable power of a compact Lie group G is considered. The existence and uniqueness of the solutions and quasi-invariance of their distribution are proved.

scholarly journals q-Independence of the Jimbo–Drinfeld Quantization

2020 ◽  
Vol 376 (3) ◽  
pp. 1737-1765
Author(s):  
Olof Giselsson
Keyword(s):  
Lie Group ◽  
Maximal Torus ◽  
Compact Lie Group ◽  
The Right

AbstractLet $${\mathrm {G}}$$G be a connected semi-simple compact Lie group and for $$0<q<1$$0<q<1, let $$({\mathbb {C}}[\mathrm {G]_q},\varDelta _q)$$(C[G]q,Δq) be the Jimbo–Drinfeld q-deformation of $${\mathrm {G}}$$G. We show that the $$C^*$$C∗-completions of $$\mathrm {C}[\mathrm {G]_q}$$C[G]q are isomorphic for all values of q. Moreover, these isomorphisms are equivariant with respect to the right-actions of the maximal torus.

scholarly journals Central lacunary sets for Lie groups

1988 ◽  
Vol 45 (1) ◽  
pp. 30-45 ◽  
Author(s):  
A. H. Dooley
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Maximal Torus ◽  
Infinite Subset ◽  
Semisimple Lie Group ◽  
Infinite Sets ◽  
Connected Lie Group

AbstractIf G is a compact connected Lie group every infinite subset of Ĝ contains an infinite central Λ(p) set, for p < 2 + 2 rank G/(dim G - rank G). A subset R of Ĝ is of type central Λ(2) if and only if the associated set of characters on the maximal torus is of type Λ(2). The dual of a compact connected semisimple Lie group contains infinite sets which are central p-Sidon for all p > 1. Every infinite subset of the dual of Su(2) contains such a set.

scholarly journals Homogeneous bundles and the trace of the heat kernel

1986 ◽  
Vol 40 (1) ◽  
pp. 71-82
Author(s):  
H. D. Fegan
Keyword(s):  
Heat Equation ◽  
Heat Kernel ◽  
Lie Group ◽  
Maximal Torus ◽  
Direct Calculation ◽  
Compact Lie Group ◽  
Main Method ◽  
Homogeneous Bundle

AbstractWe study the heat equation on a homogeneous bundle over a compact Lie group. The trace of the heat kernel is explicitly calculated. By comparing this with the formula constructed form the eigenvalues (with multiplicities) of the Laplacian we obtain and unusual formula involving the Clebsch-Gordan numbers. The main method is to use invariance under conjugation to pass from the group to its maximal torus, where a direct calculation can be carried out.

scholarly journals RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

2021 ◽  
pp. 1-29
Author(s):  
DREW HEARD
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Algebraic Model ◽  
Compact Lie Group ◽  
Loop Spaces ◽  
Local Systems ◽  
Special Case ◽  
Finite Loop ◽  
Finite Loop Space

Abstract Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group W G K is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

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