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.

40 pages; v2: minor corrections, updated references; v3: many corrections, added discussion, new examples; v4: minor corrections and clarifications, final version

References in corpus (11)

• Hopf Algebras in Combinatorics
• The On-Line Encyclopedia of Integer Sequences
• Hopf algebras, from basics to applications to renormalization
• Algebraic and combinatorial structures on pairs of twin binary trees
• Cambrian Hopf Algebras
• Involution words II: braid relations and atomic structures
• Brick polytopes, lattice quotients, and Hopf algebras
• Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions
• Hopf algebras of endomorphisms of Hopf algebras
• A lattice of combinatorial Hopf algebras, Application to binary trees with multiplicities
• Bialgebras for Stanley symmetric functions

Cited by in corpus (3)

• Enriched set-valued P-partitions and shifted stable Grothendieck polynomials
• Shifted combinatorial Hopf algebras from $K$-theory
• Bialgebras for Stanley symmetric functions