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.

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.

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.

Ergodic elements for actions of Lie groups

1996 ◽  
Vol 16 (4) ◽  
pp. 703-717
Author(s):  
K. Robert Gutschera
Keyword(s):  
Invariant Measure ◽  
Simple Group ◽  
Lie Groups ◽  
Lie Group ◽  
Isometry Group ◽  
Group Structure ◽  
Structure Theory ◽  
Ergodic Action ◽  
Finite Invariant Measure ◽  
Connected Lie Group

AbstractGiven a connected Lie group G acting ergodically on a space S with finite invariant measure, one can ask when G will contain single elements (or one-parameter subgroups) that still act ergodically. For a compact simple group or the isometry group of the plane, or any group projecting onto such groups, an ergodic action may have no ergodic elements, but for any other connected Lie group ergodic elements will exist. The proof uses the unitary representation theory of Lie groups and Lie group structure theory.

scholarly journals The invariant polynomial algebras for the groups IU(n) and ISO(n)

1984 ◽  
Vol 94 ◽  
pp. 43-59 ◽  
Author(s):  
Hitoshi Kaneta
Keyword(s):  
Lie Algebras ◽  
Lie Group ◽  
Dual Space ◽  
Polynomial Algebra ◽  
Invariant Polynomial ◽  
Coadjoint Representation ◽  
Finitely Generated ◽  
Polynomial Algebras ◽  
Enveloping Algebras ◽  
Connected Lie Group

By the coadjoint representation of a connected Lie group G with the Lie algebra g we mean the representation CoAd(g) = tAd(g-1) in the dual space g*. Imitating Chevalley’s argument for complex semi-simple Lie algebras, we shall show that the CoAd (G)-invariant polynomial algebra on g* is finitely generated by algebraically independent polynomials when G is the inhomogeneous linear group IU(n) or ISO(n). In view of a well-known theorem [8, p. 183] our results imply that the centers of the enveloping algebras for the (or the complexified) Lie algebras of these groups are also finitely generated. Recently much more inhomogeneous groups have been studied in a similar context [2]. Our results, however, are further reaching as far as the groups IU(n) and ISO(n) are concerned [cf. 3, 4, 6, 7, 9].

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.

scholarly journals “I have Learned Non-Euclidean Geometry Just from this Book” - Some facets of the correspondence between Friedrich Engel and David Hilbert

2021 ◽  
Vol 76 (3) ◽  
Author(s):  
Peter Ullrich
Keyword(s):  
Present Article ◽  
19Th Century ◽  
Euclidean Geometry ◽  
Academic Life ◽  
David Hilbert ◽  
Non Euclidean Geometry ◽  
The 19Th Century

AbstractFriedrich Engel and David Hilbert learned to know each other at Leipzig in 1885 and exchanged letters in particular during the next 15 years which contain interesting information on the academic life of mathematicians at the end of the 19th century. In the present article we will mainly discuss a statement by Hilbert himself on Moritz Pasch’s influence on his views of geometry, and on personnel politics concerning Hermann Minkowski and Eduard Study but also Engel himself.

scholarly journals Langevin simulations of protoplasmic streaming in non-Euclidean geometry

2021 ◽  
Vol 1730 (1) ◽  
pp. 012037
Author(s):  
Shuta Noro ◽  
Masahiko Okumura ◽  
Satoshi Hongo ◽  
Shinichiro Nagahiro ◽  
Toshiyuki Ikai ◽  
...  
Keyword(s):  
Euclidean Geometry ◽  
Protoplasmic Streaming ◽  
Non Euclidean Geometry

Practical Non-Euclidean Geometry

1925 ◽  
Vol 12 (177) ◽  
pp. 422 ◽  
Author(s):  
T. C. J. Elliott
Keyword(s):  
Euclidean Geometry ◽  
Non Euclidean Geometry
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