Algebra and Applications 2

and in that case, if x is a homogeneous element, x is of degree n if and only if | x | = n .

An important result is that any connected filtered bialgebra is indeed a filtered Hopf algebra, in the sense that the antipode comes for free. We give a proof of this fact as well as a recursive formula for the antipode using the convolution product : let ℋ be a (connected filtered) bialgebra, and let be any k -algebra (which will be called the target algebra ): the convolution product on is given by:

PROPOSITION 1.8.– The map , given by and e ( x ) = 0 for any x ∈ Ker ε, is a unit for the convolution product. Moreover, the set endowed with the convolution product is a group .

PROOF.– The first statement is straightforward. To prove the second, let us consider the formal series:

Using ( e — φ )( 1) = 0, we have ( e — φ ) *k( 1) = 0 immediately, and for any x ∈ Ker ε :

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When x ∈ ℋ n, this expression vanishes and then for k ≥ n + 1. The formal series then ends up with a finite number of terms for any x , which proves the result. □

COROLLARY 1.1.– Any connected filtered bialgebra ℋ is a filtered Hopf algebra. The antipode is defined by :

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It is given by S ( 1) = 1 and recursively by any of the two formulae for x ∈ Ker ε :

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PROOF.– The antipode, when it exists, is the inverse of the identity for the convolution product on . We then just need to apply Proposition 1.8 with . The two recursive formulas come directly from the two equalities:

fulfilled by any x ∈ Ker ε .

Let be the subspace of formed by the elements α , such that α ( 1) = 0. It is clearly a subalgebra of for the convolution product. We have:

From now on, we will suppose that the ground field k is of characteristic zero. For any x ∈ ℋ n, the exponential:

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is a finite sum (ending up at k = n ). It is a bijection from onto . Its inverse is given by:

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This sum again ends up at k = n for any x ∈ ℋ n. Let us introduce a decreasing filtration on :

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Clearly, and . We define the valuation val φ of an element φ of as the greatest integer k , such that φ is in . In the sequel we will consider the ultrametric distance on induced by the filtration:

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