When are Hopf algebras determined by integer sequences?

2025/05/11 by Andrews, Nicolas, Gagnon, Lucas, Gélinas, Félix +2
#16T30 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.06941

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.

Citations

• Hopf structures in the representation theory of direct products
• Pattern groups and a poset based Hopf monoid
• Free and cofree Hopf algebras
• The primitives and antipode in the Hopf algebra of symmetric functions in noncommuting variables
• Combinatorial Hopf algebras
• Hopf Algebras of Heap Ordered Trees and Permutations
• The primitives of the Hopf algebra of noncommutative symmetric functions
• Symmetric Functions in Noncommuting Variables
• Structure of the Malvenuto-Reutenauer Hopf algebra of permutations
• A Chromatic Symmetric Function in Noncommuting Variables
• Noncommutative symmetric functions

Related

• On the structure of abelian Hopf algebras
• On Hopf algebras of dimension pq
• Free and cofree Hopf algebras
• Hopf algebras of dimension 14
• Injective Hopf bimodules, cohomologies of infinite dimensional Hopf algebras and graded-commutativity of the Yoneda product