In this paper we use the technique of Hopf algebras and quasi-symmetric
functions to study the combinatorial polytopes. Consider the free abelian group
$\mathcal{P}$ generated by all combinatorial polytopes. There are two natural
bilinear operations on this group defined by a direct product $\times$ and a
join $\divideontimes$ of polytopes. $(\mathcal{P},\times)$ is a commutative
associative bigraded ring of polynomials, and $\mathcal{RP}=(\mathbb
Z\varnothing\oplus\mathcal{P},\divideontimes)$ is a commutative associative
threegraded ring of polynomials. The ring $\mathcal{RP}$ has the structure of a
graded Hopf algebra. It turns out that $\mathcal{P}$ has a natural Hopf
comodule structure over $\mathcal{RP}$. Faces operators $d_k$ that send a
polytope to the sum of all its $(n-k)$-dimensional faces define on both rings
the Hopf module structures over the universal Leibnitz-Hopf algebra
$\mathcal{Z}$. This structure gives a ring homomorphism \R\to\Qs\otimes\R,
where $\R$ is $\mathcal{P}$ or $\mathcal{RP}$. Composing this homomorphism with
the characters $P^n\to\alpha^n$ of $\mathcal{P}$, $P^n\to\alpha^{n+1}$ of
$\mathcal{RP}$, and with the counit we obtain the ring homomorphisms
f\colon\mathcal{P}\to\Qs[\alpha],
f_{\mathcal{RP}}\colon\mathcal{RP}\to\Qs[\alpha], and
\F^*:\mathcal{RP}\to\Qs, where $F$ is the Ehrenborg transformation. We
describe the images of these homomorphisms in terms of functional equations,
prove that these images are rings of polynomials over $\mathbb Q$, and find the
relations between the images, the homomorphisms and the Hopf comodule
structures. For each homomorphism $f,\;f_{\mathcal{RP}}$, and \F the images
of two polytopes coincide if and only if they have equal flag $f$-vectors.
Therefore algebraic structures on the images give the information about flag
$f$-vectors of polytopes.Comment: 61 page