Algebra and Applications 2. Группа авторов

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that a filtered bialgebra with antipode is automatically a filtered Hopf algebra (see, for example, Montgomery (1993, Lemma 5.2.8), Andruskiewitsch and Schneider (2002) and Andruskiewitsch and Cuadra (2013)). The antipode is, however, of degree zero in the connected case: for any x ∈ ℋ, we set

      [1.14]

      We say that the filtered bialgebra ℋ is connected if ℋ⁰ is one-dimensional. There is an analogue of Proposition 1.6 in the connected filtered case, the proof of which is very similar:

      PROPOSITION 1.7.– For any x ∈ ℋⁿ, n ≥ 1, we can write:

      [1.15]

      The map is coassociative on Ker ε and sends ℋⁿ into (ℋ^n-k)^{⊗k + 1}.

As an easy corollary, the degree of the antipode is also zero in the connected case, that is, S(ℋⁿ) ⊆ ℋⁿ for any n. This is an immediate consequence of the recursive formulae [1.19] and [1.20] below.

      Any graded bialgebra, or the Hopf algebra, is obviously filtered by the canonical filtration associated with the grading:

      [1.16]

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

1.3.4. The convolution product

      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:

      [1.17]

      When x ∈ ℋⁿ, 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:

      [1.18]

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

[1.19]

[1.20]

      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 ∈ Kerε.

      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 ∈ ℋⁿ, the exponential:

      [1.21]

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

      [1.22]

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

      [1.23]

Clearly, and . We define

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