Uniform Finite Generation of Threedimensional Linear Lie Groups

1975 ◽  
Vol 27 (2) ◽  
pp. 396-417 ◽  
Author(s):  
R. M. Koch ◽  
Franklin Lowenthal
Keyword(s):  
Positive Integer ◽  
Lie Group ◽  
Finite Product ◽  
Connected Subgroup ◽  
Arcwise Connected ◽  
Finite Products ◽  
The One ◽  
Lie Subgroup ◽  
Infinitesimal Transformations ◽  
Connected Lie Group

A connected Lie group G is generated by one-parameter subgroups exp(tX1), … , exp(tXk) if every element of G can be written as a finite product of elements chosen from these subgroups. This happens just in case the Lie algebra of G is generated by the corresponding infinitesimal transformations X1, … , Xk ; indeed the set of all such finite products is an arcwise connected subgroup of G, and hence a Lie subgroup by Yamabe's theorem [9]. If there is a positive integer n such that every element of G possesses such a representation of length at most n, G is said to be uniformly finitely generated by the one-parameter subgroups.

Uniform Finite Generation of SU(2) and SL(2, R)

1972 ◽  
Vol 24 (4) ◽  
pp. 713-727 ◽  
Author(s):  
Franklin Lowenthal
Keyword(s):  
Positive Integer ◽  
Lie Algebra ◽  
Lie Group ◽  
Finitely Generated ◽  
Finite Product ◽  
Finite Generation ◽  
Arcwise Connected ◽  
Finite Products ◽  
Infinitesimal Transformations ◽  
Connected Lie Group

A connected Lie group H is generated by a pair of one-parameter subgroups if every element of H can be written as a finite product of elements chosen alternately from the two one-parameter subgroups, i.e., if and only if the subalgebra generated by the corresponding pair of infinitesimal transformations is equal to the whole Lie algebra h of H (observe that the subgroup of all finite products is arcwise connected and hence, by Yamabe's theorem [5], is a sub-Lie group). If, moreover, there exists a positive integer n such that every element of H possesses such a representation of length at most n, then H is said to be uniformly finitely generated by the pair of one-parameter subgroups. In this case, define the order of generation of H as the least such n ; otherwise define it as infinity.

Maximal Homotopy Lie Subgroups of Maximal Rank

1988 ◽  
Vol 40 (04) ◽  
pp. 769-787 ◽  
Author(s):  
John A. Frohliger
Keyword(s):  
Lie Group ◽  
Homogeneous Spaces ◽  
Maximal Rank ◽  
Homotopy Equivalent ◽  
Classifying Spaces ◽  
Loop Spaces ◽  
Connected Subgroup ◽  
Image Position ◽  
Lie Subgroup ◽  
Connected Lie Group

Let G be a compact connected Lie group with H a connected subgroup of maximal rank. Suppose there exists a compact connected Lie subgroup K with H ⊂ K ⊂ G. Then there exists a smooth fiber bundle G/H → G/K with K/H as the fiber. (See for example [13].) This can be incorporated into a diagram involving the classifying spaces as follows: (1) Here π, ϕ, ϕ 1 and ϕ 2 denote fibrations. We also know that the homogeneous spaces and the Lie groups, which are homotopy equivalent to the loop spaces of their respective classifying spaces, are homotopy equivalent to connected finite complexes. Now suppose H is a maximal subgroup. Can there still exist spaces, which we will call BK, K/H, and G/K, and fibrations so that diagram (1) is still valid? This paper will show that in many cases either G/K or K/H will be homo topically trivial.

Uniform Finite Generation of the Isometry Groups of Euclidean and Non-Euclidean Geometry

1971 ◽  
Vol 23 (2) ◽  
pp. 364-373 ◽  
Author(s):  
Franklin Lowenthal
Keyword(s):  
Positive Integer ◽  
Lie Group ◽  
Euclidean Geometry ◽  
Isometry Group ◽  
Spherical Geometry ◽  
Rotation Group ◽  
Finitely Generated ◽  
Finite Product ◽  
Non Euclidean Geometry ◽  
Connected Lie Group

A connected Lie group H is generated by a pair of oneparameter subgroups if every element of H can be written as a finite product of elements chosen alternately from the two one-parameter subgroups. If, moreover, there exists a positive integer n such that every element of H possesses such a representation of length at most n, then H is said to be uniformly finitely generated by the pair of one-parameter subgroups. In this case, define the order of generation of H as the least such n; otherwise define it as infinity.For the isometry group of the spherical geometry, or equivalently for the rotation group SO(3), the order of generation is always finite.

Feynman approximation to integrals with respect to Brownian sheet on Lie groups

2015 ◽  
Vol 18 (01) ◽  
pp. 1550008 ◽  
Author(s):  
A. A. Kalinichenko
Keyword(s):  
Brownian Motion ◽  
Lie Groups ◽  
Lie Group ◽  
Functional Space ◽  
Brownian Sheet ◽  
Functional Integrals ◽  
Finite Dimensional ◽  
Approximation Formulas ◽  
The One ◽  
Connected Lie Group

We consider the Feynman-type approximations to functional integrals over the distribution of the Brownian sheet on a compact connected Lie group M, which give a representation of the integrals over the functional space C([0, 1] × [0, 1], M) as the limit of integrals over the finite-dimensional manifolds M × ⋯ × M. The known approximation formulas for the one-parameter Brownian motion are generalized to the case of the Brownian sheet.

On Lie groups and their homotopy groups

1959 ◽  
Vol 55 (3) ◽  
pp. 244-247 ◽  
Author(s):  
I. M. James
Keyword(s):  
Topological Group ◽  
Positive Integer ◽  
Lie Group ◽  
Infinite Order ◽  
Bilinear Pairing ◽  
Homotopy Groups ◽  
Homotopy Classification ◽  
Classification Of Maps ◽  
Connected Lie Group

We prove a theorem which facilitates homotopy classification of maps into a topological group G. Some information about homotopy groups of G is obtained, including the following two results. Consider the Samelson product, as defined in (7), which constitutes a bilinear pairing of πp(G) with πq(G) to πp+q(G). The product of a α ∈ πp(G) with β ∈ πq(G) is written in the form 〈α, β〉. There exist groups having Samelson products of infinite order. Homotopy-commutative groups have zero Samelson products. We shall proveTheorem (1·1). If G is a connected Lie group then there exists a positive integer n such that n〈α, β〉 = 0 for every pair α, β of elements in the homotopy groups of G.

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.

Equidimensional immersions of locally compact groups

1989 ◽  
Vol 105 (2) ◽  
pp. 253-261 ◽  
Author(s):  
K. H. Hofmann ◽  
T. S. Wu ◽  
J. S. Yang
Keyword(s):  
Exponential Function ◽  
Lie Group ◽  
Compact Groups ◽  
Locally Compact ◽  
Lie Group Theory ◽  
Group A ◽  
Image Position ◽  
Analytic Subgroup ◽  
Algebra Level ◽  
Connected Lie Group

Dense immersions occur frequently in Lie group theory. Suppose that exp: g → G denotes the exponential function of a Lie group and a is a Lie subalgebra of g. Then there is a unique Lie group ALie with exponential function exp:a → ALie and an immersion f:ALie→G whose induced morphism L(j) on the Lie algebra level is the inclusion a → g and which has as image an analytic subgroup A of G. The group Ā is a connected Lie group in which A is normal and dense and the corestrictionis a dense immersion. Unless A is closed, in which case f' is an isomorphism of Lie groups, dim a = dim ALie is strictly smaller than dim h = dim H.

HIDA DISTRIBUTIONS ON COMPACT LIE GROUPS

2000 ◽  
Vol 03 (03) ◽  
pp. 337-362 ◽  
Author(s):  
THOMAS DECK
Keyword(s):  
Lie Group ◽  
Noise Analysis ◽  
Growth Conditions ◽  
Nuclear Space ◽  
Compact Lie Groups ◽  
Space Of Analytic Functions ◽  
S Transform ◽  
Tensor Norm ◽  
Taylor Map ◽  
Connected Lie Group

We show that a nuclear space of analytic functions on K is associated with each compact, connected Lie group K. Its dual space consists of distributions (generalized functions on K) which correspond to the Hida distributions in white noise analysis. We extend Hall's transform to the space of Hida distributions on K. This extension — the S-transform on K — is then used to characterize Hida distributions by holomorphic functions satisfying exponential growth conditions (U-functions). We also give a tensor description of Hida distributions which is induced by the Taylor map on U-functions. Finally we consider the Wiener path group over a complex, connected Lie group. We show that the Taylor map for square integrable holomorphic Wiener functions is not isometric w.r.t. the natural tensor norm. This indicates (besides other arguments) that there might be no generalization of Hida distribution theory for (noncommutative) path groups equipped with Wiener measure.

scholarly journals A note on primes in certain residue classes

2018 ◽  
Vol 14 (08) ◽  
pp. 2219-2223
Author(s):  
Paolo Leonetti ◽  
Carlo Sanna
Keyword(s):  
Positive Integer ◽  
Asymptotic Density ◽  
Residue Classes ◽  
Euler Totient Function ◽  
The One ◽  
Positive Integers

Given positive integers [Formula: see text], we prove that the set of primes [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density relative to the set of all primes which is at least [Formula: see text], where [Formula: see text] is the Euler totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density which is at least [Formula: see text].

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