When are Hopf algebras determined by integer sequences?
Nicolas Andrews, Lucas Gagnon, Félix Gélinas, Eric Schlums, Mike Zabrocki
TL;DR
The paper develops a complete combinatorial framework for free graded connected cocommutative Hopf algebras (FGCCHAs) by encoding their structure in integer dimension sequences. A central result is that a sequence $\vec{h}$ is realizable as the graded dimensions of some FGCCHA if and only if the INVERTi transform of $\vec{h}$ is nonnegative, linking algebraic existence to a concrete transform condition. It provides a canonical representative for each isomorphism class, showing $H \cong \mathcal{U}(\mathfrak{L}(\vec{a}))$ with $\vec{a} = \phi_{h,a}(\vec{h})$, and gives explicit isomorphisms via ordered primitive generating sets and associated matrices. The authors also classify morphisms: surjections between FGCCHAs occur exactly when graded generator arrays satisfy $\vec{a} \ge \vec{b}$, and subalgebras are classified by primitive-dimension data, yielding a precise criterion $\phi_{a,p}(\vec{b}) \le \vec{p}$ for realizable subalgebras. Collectively, these results connect dimension data, primitive structure, and Lie-theoretic constructions to provide a complete, computable picture of the category of FGCCHAs and their subobjects.
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 $K$.