[Paper Review] Free and cofree Hopf algebras

Loïc Foissy|arXiv (Cornell University)|Oct 26, 2010

Advanced Topics in Algebra16 references4 citations

TL;DR

This paper establishes that graded, connected, free and cofree Hopf algebras over a field of characteristic zero are self-dual and classified up to isomorphism by their Poincaré-Hilbert series. It proves the Lie algebra of primitive elements is free, and two such Hopf algebras are isomorphic as non-graded Hopf algebras if and only if their abelianizations of primitive elements have the same dimension.

ABSTRACT

We first prove that a graded, connected, free and cofree Hopf algebra is always self-dual; then that two graded, connected, free and cofree Hopf algebras are isomorphic if, and only if, they have the same Poincaré-Hilbert formal series. If the characteristic of the base field is zero, we prove that the Lie algebra of the primitive elements of such an object is free, and we deduce a characterization of the formal series of free and cofree Hopf algebras by a condition of growth of the coefficients. We finally show that two graded, connected, free and cofree Hopf algebras are isomorphic as (non graded) Hopf algebras if, and only if, the Lie algebra of their primitive elements have the same number of generators.

Motivation & Objective

• To determine whether free and cofree Hopf algebras are always self-dual.
• To investigate whether two such Hopf algebras with the same Poincaré-Hilbert series are isomorphic.
• To characterize the structure of the Lie algebra of primitive elements in free and cofree Hopf algebras when the base field has characteristic zero.
• To establish isomorphism criteria for free and cofree Hopf algebras as non-graded Hopf algebras.

Proposed method

• Construct a non-degenerate, symmetric Hopf pairing on a free and cofree Hopf algebra using a pairing on the space of indecomposable primitive elements.
• Prove self-duality by extending a symmetric pairing on $\frac{\mathfrak{g}}{\mathfrak{g} \cap H^{+2}}$ to the entire algebra $H$.
• Use the existence of a non-degenerate symmetric Hopf pairing to show that two such Hopf algebras are isomorphic if and only if they have the same Poincaré-Hilbert series.
• Introduce a bigraduation on the Hopf algebra to refine the grading and analyze the structure of the Lie algebra of primitive elements.
• Leverage the rigidity of non-commutative Connes-Kreimer Hopf algebras to show that the formal series $S(h)$ of indecomposable primitive elements can be arbitrarily chosen.
• Apply the fact that the Lie algebra of primitive elements is free in characteristic zero to derive growth conditions on the coefficients of the Poincaré-Hilbert series.

Experimental results

Research questions

• RQ1 Is every graded, connected, free and cofree Hopf algebra self-dual?
• RQ2 Are two graded, connected, free and cofree Hopf algebras isomorphic if they have the same Poincaré-Hilbert series?
• RQ3 What is the structure of the Lie algebra of primitive elements in a free and cofree Hopf algebra over a field of characteristic zero?
• RQ4 When are two free and cofree Hopf algebras isomorphic as non-graded Hopf algebras?
• RQ5 Can the formal series of free and cofree Hopf algebras be characterized by growth conditions on their coefficients?

Key findings

• A graded, connected, free and cofree Hopf algebra is self-dual via a non-degenerate, symmetric Hopf pairing constructed from a pairing on $\frac{\mathfrak{g}}{\mathfrak{g} \cap H^{+2}}$.
• Two such Hopf algebras are isomorphic as graded Hopf algebras if and only if they have the same Poincaré-Hilbert series.
• In characteristic zero, the Lie algebra of primitive elements of a free and cofree Hopf algebra is free.
• The formal series of free and cofree Hopf algebras are characterized by growth conditions on the coefficients, as expressed in Corollary 19.
• Two free and cofree Hopf algebras are isomorphic as non-graded Hopf algebras if and only if the dimensions of the abelianizations $\frac{\mathfrak{g}}{[\mathfrak{g}, \mathfrak{g}]}$ and $\frac{\mathfrak{g}'}{[\mathfrak{g}', \mathfrak{g}']}$ are equal.
• The Hopf algebras $\mathbf{FQSym}$, $\mathbf{PQSym}$, $\mathcal{H}_{LR}$, $\mathbf{YSym}$, $\mathcal{H}_{NCK}$, and their decorated versions are all isomorphic as non-graded Hopf algebras because they have infinite-dimensional abelianizations of primitive elements.