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Journal Articles Journal of Pure and Applied Algebra Year : 2012

Free and cofree Hopf algebras

Loïc Foissy

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.
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Dates and versions

hal-00529811 , version 1 (26-10-2010)
hal-00529811 , version 2 (26-01-2011)
hal-00529811 , version 3 (22-06-2011)

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Loïc Foissy. Free and cofree Hopf algebras. Journal of Pure and Applied Algebra, 2012. ⟨hal-00529811v3⟩

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