Topology of Lie groups

Samelson, Hans

doi:10.1090/s0002-9904-1952-09544-6

Cited by 113 publications

(30 citation statements)

References 72 publications

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“…Thus G' = T if G is homotopy-abelian, and hence the theorem follows at once. A maximal compact connected subgroup of a Lie group is a deformation retract of the component of the identity [7], and so the corollary is an immediate consequence of the theorem.…”

Section: Lemma Let G Be a Compact Connected Simple Non-abelian Lie Gmentioning

confidence: 89%

Homotopy-abelian Lie groups

Araki¹,

James²,

Thomas³

1960

Bull. Amer. Math. Soc.

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Section: Lemma Let G Be a Compact Connected Simple Non-abelian Lie Gmentioning

confidence: 89%

Homotopy-abelian Lie groups

Araki¹,

James²,

Thomas³

1960

Bull. Amer. Math. Soc.

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“…We shall make a definite effort to show that there are few domains of topology into which mob-theory has not penetrated. We shall not mention the work of Eckmann, Hopf, Leray, Borel and others discussed by Samelson in his invited address [25]. The applications to analysis were discussed by Hille in his Colloquium Lectures [13].…”

Section: C: H(u)cf(s)mentioning

confidence: 99%

The structure of topological semigroups

Wallace¹

1955

Bull. Amer. Math. Soc.

178

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The title of this address might incline one to the notion that here is to be found a small number of large theorems. To the contrary, I shall talk about a large number of small theorems. Actually, there does not exist at this time any corpus of information to which the title "structure of topological semigroups" is in any fashion applicable. Whether such a body of theorems will ever exist is a matter for the future and is likely to depend on the use to which it might be put as well as to the tastes of mathematicians who are not yet such.When the investigation of topological groups began there was at hand a theory of abstract groups and much of a fundamental character in Lie groups was available. Beyond this there existed a great body of geometry even if some of it was in a nebulous state insofar as the then held standards of rigor were concerned.With topological semigroups the situation is quite contrariwise. Here we are faced with a lack of satisfactory algebraic results. I do not think that there are so many as twenty-five papers each exceeding ten pages which are concerned exclusively with the algebraic aspects.We are more fortunate than were the pioneers who forayed the frontiers of topological groups in that we have at our disposal a greater wealth of topology. Much that they could not-or at least did not-use is at hand for our use. Furthermore we can rely, at least if for no more than analogy, on their results. The state of both algebraic and set-theoretic topology is a somewhat happier one now than then. Still we are likely to be troubled for awhile for lack of something like Haar measure without which we shall be at a loss for representation theorems. At present there seems to be no line of attack on the representation problem and it is probable that we shall need to rely to a greater extent on geometry and topology than was the case with groups.

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“…Now any connected abelian Lie group is a topological product of a torus and a euclidean space [11 ] and, for 9, the base X1,..., Xn can be so chosen that each of the 1-parameter subgroup ~5i= {exptXi}, i = 1, ..., n, is closed in ~ii and that ~5 = ~51 × "'" × ~n" Thus there is more meaning attached to the formula (6.1). For another illustration, let us suppose that ~5 is compact and acts on a manifold ~.…”

Section: T (A;t To) --Exp F a I (S)ds X1 Exp F An (S)ds X~ To Tomentioning

confidence: 99%

Decomposition of differential equations

Chen

1962

Math. Ann.

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Consider the linear system written in the matrical form dz/dt = (A (t) + B(t)) z, z(to) = z,where A (t) and B (t) are n x n matrices and z an n X 1 matrix. It is well known that the solution takes the form z = u (t) v (t) z0,We may demand that both U (to) and V (to) are the identity matrix. The purpose of this paper is to establish the above type of decomposition formula for nonlinear cases and to study its application.Differential equations in this paper will be written in the form Du= A (t)uwhere Du is the tangent vector to the solution path u(t) on a manifold ~ and A (t)u denotes the vector of the time-dependent vector field A (t) at the point u(t). The solution of (0.1) with the initial condition u(to)= Po will be written as u = T(A;t, to)Po. If a system of local coordinates x is chosen, then (0.1) takes the usual form du/dt = F (u, t), where both u and F are vectors. After some preparatory work in § 1, we define in § 2 the operation Adj a on vector fields, a being a homeomorphism from one open set to another of ~. (A + B;t, to)Po= T (A;t, to) T(B*;t, to)po,where B* (t) = Adj T (A ; to, t) B (t). We also assert more or less to the effect that, for any vector field X on ~, if Y (t) = Adj T (A; t, to)X, then r (t)/dt = [ Y (t), A (t)],where the right hand side is the usual bracket product of two infinitesimal transformations (vector fields).

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Topology of Lie groups

Cited by 113 publications

References 72 publications

Homotopy-abelian Lie groups

Homotopy-abelian Lie groups

The structure of topological semigroups

Decomposition of differential equations

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