Linear compactness and combinatorial bialgebras

TL;DR

This paper explores the structure of linearly compact vector spaces and their bialgebras, classifying relations and extending combinatorial Hopf algebra theory to broader contexts.

Contribution

It provides a classification of relations generating linearly compact bialgebras and extends combinatorial Hopf algebra theory to non-connected and infinite-dimensional cases.

Findings

Classified relations on words generating linearly compact bialgebras.

Extended combinatorial Hopf algebra theory to non-connected and infinite-dimensional bialgebras.

Discussed examples of quasi-symmetric functions beyond bounded degree.

Abstract

We present an expository overview of the monoidal structures in the category of linearly compact vector spaces. Bimonoids in this category are the natural duals of infinite-dimensional bialgebras. We classify the relations on words whose equivalence classes generate linearly compact bialgebras under shifted shuffling and deconcatenation. We also extend some of the theory of combinatorial Hopf algebras to bialgebras that are not connected or of finite graded dimension. Finally, we discuss several examples of quasi-symmetric functions, not necessarily of bounded degree, that may be constructed via terminal properties of combinatorial bialgebras.