Linear compactness and combinatorial bialgebras

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. PreliminariesLet Z ⊃ N ⊃ P denote the respective sets of all integers, all nonnegative integers, and all positive integers. For m, n Monoidal structuresOur reference for the background material in this section is [2, Chapter 1]. Suppose C is a braided monoidal category with tensor product •, unit object I, and braiding β. Definition 2.1. A monoid in C is a triple (A, ∇, ι) where A ∈ C is an object and ∇ : A • A → A

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