When are Hopf algebras determined by integer sequences?

TL;DR

This paper characterizes when graded Hopf algebras are uniquely determined by their sequences of graded dimensions, providing criteria based on a specific sequence transformation and exploring homomorphisms and subalgebras.

Contribution

It introduces conditions on dimension sequences that determine the existence and structure of graded Hopf algebras and their relationships.

Findings

Existence of Hopf algebras corresponds to nonnegativity of the INVERTi transformation.

Criteria for surjective homomorphisms between Hopf algebras based on dimension sequences.

Conditions for embedding one Hopf algebra as a subalgebra of another.

Abstract

We study the category of graded Hopf algebras that are free noncommutative, cocommutative, graded and connected from the perspective of the sequences of dimensions of the graded pieces. We show that a Hopf algebra exists with a given sequence of graded dimensions if and only if the ``INVERTi'' transformation of the sequence is nonnegative. We give conditions on the sequences of graded dimensions for two Hopf algebras $H$ and $K$ in this category under which there exists a surjective homomorphism from $H$ to $K$. We also give conditions such that an isomorphic copy of $H$ occurs as a Hopf subalgebra of $K$.