[Paper Review] Infinitesimal and $B\_\infty$-algebras, finite spaces, and quasi-symmetric functions

Loïc Foissy, Claudia Malvenuto+1 more|arXiv (Cornell University)|Mar 28, 2014

Advanced Combinatorial Mathematics20 references4 citations

TL;DR

This paper establishes that the linear span of finite topological spaces forms a $B_{ u}$-algebra and admits a surjective Hopf algebra morphism to the algebra of quasi-symmetric functions. Using combinatorial Hopf algebra techniques, it links finite spaces to infinitesimal bialgebras and introduces Schur-Weyl categories to describe rigidity in associative algebras, revealing global algebraic structures underlying local topological invariants.

ABSTRACT

Finite topological spaces are in bijective correspondence with preorders on finite sets. We undertake their study using combinatorial tools that have been developed to investigate general discrete structures. A particular emphasis will be put on recent topological and combinatorial Hopf algebra techniques. We will show that the linear span of finite spaces carries generalized Hopf algebraic structures that are closely connected with familiar constructions and structures in topology (such as the one of cogroups in the category of associative algebras that has appeared e.g. in the study of loop spaces of suspensions). The most striking results that we obtain are certainly that the linear span of finite spaces carries the structure of the enveloping algebra of a $B\_\infty$--algebra, and that there are natural (Hopf algebraic) morphisms between finite spaces and quasi-symmetric functions. In the process, we introduce the notion of Schur-Weyl categories in order to describe rigidity theorems for cogroups in the category of associative algebras and related structures, as well as to account for the existence of natural operations (graded permutations) on them.

Motivation & Objective

• To investigate finite topological spaces using modern combinatorial and Hopf algebraic tools.
• To uncover global algebraic structures—specifically $B_{\infty}$-algebra and infinitesimal bialgebra structures—on the linear span of finite spaces.
• To establish a natural Hopf algebra morphism from finite spaces to the algebra of quasi-symmetric functions.
• To introduce Schur-Weyl categories to formalize rigidity theorems for cocommutative cogroups in associative algebras and related bialgebras.
• To clarify the interplay between local constructions (e.g., cup products) and global algebraic operations on the linear span of finite spaces.

Proposed method

• Utilizes the bijective correspondence between finite topological spaces and preorders on finite sets to model spaces algebraically.
• Defines a coproduct on the linear span of finite spaces via extraction of open subsets, forming an infinitesimal bialgebra structure.
• Introduces the $\succ_q$-product on quasi-symmetric functions and constructs a morphism $\phi_q$ from the linear span of finite spaces to $\mathbf{QSym}$.
• Employs standard linear orderings $Lin_{\text{Std}}(\mathcal{T})$ to encode topological data and define the map $\phi_q$ via weighted sums over labelings.
• Applies the notion of Schur-Weyl categories to describe rigidity in cocommutative cogroups in the category of associative algebras.
• Uses the $F_<$ and $F_=$ maps to define operations on labelings that preserve and reflect the $\succ_q$-product structure.

Experimental results

Research questions

• RQ1 How can the linear span of finite topological spaces be endowed with a $B_{\infty}$-algebra structure?
• RQ2 What is the nature of the Hopf algebra morphism from the linear span of finite spaces to the algebra of quasi-symmetric functions?
• RQ3 How do global algebraic operations on finite spaces relate to local topological invariants such as cup products?
• RQ4 What is the role of Schur-Weyl categories in characterizing rigidity of cogroup structures in associative algebras?
• RQ5 Under what conditions do two finite spaces induce the same image under the morphism $\phi_q$?

Key findings

• The linear span $\mathcal{F}$ of finite spaces carries the structure of the enveloping algebra of a $B_{\infty}$-algebra, as proven in Theorem 19.
• There exists a surjective, structure-preserving Hopf algebra morphism $\phi_q: \mathcal{F} \to \mathbf{QSym}$, as established in Theorem 21.
• The morphism $\phi_q$ respects the $\succ_q$-product, satisfying $\phi_q(\overline{\mathcal{T}}_1 \succ \overline{\mathcal{T}}_2) = \phi_q(\overline{\mathcal{T}}_1) \succ_q \phi_q(\overline{\mathcal{T}}_2)$, confirming compatibility with algebraic operations.
• The character $\zeta_q = \zeta_{\mathbf{QSym}} \circ \phi_q$ on $\mathcal{F}$ evaluates to $q^{\alpha((n))} = q^{\text{number of pairs } (i,j) \text{ with } i <_{\mathcal{T}} j}$, linking topological data to $q$-deformations.
• The map $\phi_q$ is not injective; distinct finite spaces $\overline{\mathcal{T}}$ and $\overline{\mathcal{T}}'$ may map to the same element in $\mathbf{QSym}$, as shown by a counterexample with 204 terms.
• Two topologies $\mathcal{T}$ and $\mathcal{T}'$ on the same set are equal if and only if $Lin(\mathcal{T}) = Lin(\mathcal{T}')$, providing a combinatorial criterion for topological isomorphism.