We analyze the algebraic structures based on a classifying space of a compact Lie group. We construct the connected graded free Lie algebra structure by considering the rationally nontrivial indecomposable and decomposable generators of homotopy groups and the cohomology cup products, and we show that the homomorphic image of homology generators can be expressed in terms of the Lie brackets in rational homology. By using the Milnor-Moore theorem, we also investigate the concrete primitive elements in the Pontrjagin algebra.

A Lie group is a differentiable manifold

As usual we let

Let

In this paper all spaces are based and have the based homotopy type of based, connected CW-complexes. All maps and homotopies preserve the base point. Unless otherwise stated, we do not distinguish notationally between a map and its homotopy class.

The main purpose of this paper is to investigate the algebraic explanation based on a classifying space of the compact Lie group

Let

A

If

The Eckmann-Hilton dual of a co-H-group is an H-group (see [

The principle examples of a co-H-group and an H-group are the suspension

We note that

From now on, we denote

The cofibration sequence

In the above definition, we note that

We now define the following.

We define a rationally nontrivial homotopy element

Now we take the self-map

We recall that

We now consider a wedge of spheres

We end this section with the following Hilton’s formula [

Let the ordered basic products of

We note that the direct sum is finite for each

By using the addition of a co-H-group in Section

We define a

Let

Similarly, we define the following.

The map

Let

Note that the weak category of

We recall that the Samelson product gives

Let

Let

If

the Samelson product makes

the Hurewicz homomorphism for

the Hurewicz homomorphism extends to an isomorphism of graded Hopf algebras

It is natural to ask what are the rationally nontrivial indecomposable and decomposable generators of the graded Lie algebra for

The connected graded Lie algebra for

It suffices to show that the iterated Samelson products

We first note that the Eckmann-Hilton dual of the Hopf-Thom theorem (see [

Let

For induction, we now suppose that the

Similarly, a cofibration shows that

Finally by taking the adjointness, we complete the proof.

Let

The graded rational homotopy group

We note that the iterated Samelson products

It is well known that the Hurewicz homomorphism

Let

(1) The adjointness shows that, for

(2) For the second part, if

By applying the rational homology to the above diagram, we have

Here,

Let

We first show that the following diagram is strictly commutative:

Indeed, the composition

On the other hand,

By adjointness, we have

We now consider the cell structure on

Since the restrictions

The Milnor-Moore theorem asserts that the image of the Hurewicz homomorphism is primitive. The following is another expression of the primitive elements in the Pontrjagin algebra.

The image of a homomorphism

Since

Since the restriction to the skeleton

We note that the self-map

The author is grateful to an anonymous referee for a careful reading and many helpful suggestions that improved the quality of the paper. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0007611).