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- W3098749090 abstract "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 $mathcal{RP}=(mathbb Zvarnothingoplusmathcal{P},divideontimes)$ 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 $RtoQsotimesR$, where $R$ is $mathcal{P}$ or $mathcal{RP}$. Composing this homomorphism with the characters $P^ntoalpha^n$ of $mathcal{P}$, $P^ntoalpha^{n+1}$ of $mathcal{RP}$, and with the counit we obtain the ring homomorphisms $fcolonmathcal{P}toQs[alpha]$, $f_{mathcal{RP}}colonmathcal{RP}toQs[alpha]$, and $F^*:mathcal{RP}toQs$, 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." @default.
- W3098749090 created "2020-11-23" @default.
- W3098749090 creator A5012742710 @default.
- W3098749090 creator A5088971896 @default.
- W3098749090 date "2011-04-30" @default.
- W3098749090 modified "2023-09-23" @default.
- W3098749090 title "Polytopes, Fibonacci numbers, Hopf algebras, and quasi-symmetric functions" @default.
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- W3098749090 doi "https://doi.org/10.1070/rm2011v066n02abeh004741" @default.
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