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A graded Hopf algebra on k is a graded k-vector space:

endowed with a product m : ℋ ⊗ ℋ → ℋ, a coproduct Δ : ℋ ↑ ℋ ⊗ ℋ, a unit u : k → ℋ , a counit ε : ℋ → k and an antipode S : ℋ → ℋ, fulfilling the usual axioms of a Hopf algebra, and such that:

[1.1]

[1.2]

If we do not ask for the existence of an antipode ℋ, we get the definition of a graded bialgebra. In a graded bialgebra ℋ, we will consider the increasing filtration:

It is an easy exercise (left to the reader) to prove that the unit u and the counit ε are degree zero maps, that is, 1 ∈ ℋ₀ and ε(ℋ_n) = {0} for n ≥ 1. We can also show that the antipode S, when it exists, is also of degree zero, that is, S(ℋn) ⊂ ℋ_n. It can be proven as follows: let S′ : ℋ → ℋ be defined, so that S’(x) is the n^th homogeneous component of S(x) when x is homogeneous of degree n. We can write down the coproduct Δ(x) with Sweedler’s notation:

where x₁ and x₂ are the homogeneous of degree, say, k and n — k. We then have:

[1.3]

Similarly, m ∘ (Id ⊗ S′) ∘ Δ(x) = ε(x)1. By uniqueness of the antipode, we then get S’ = S.

Suppose that ℋ is connected, that is, ℋ₀ is one-dimensional. Then, we have:

PROPOSITION 1.6.– For any x ∈ ℋⁿ, n ≥ 1, we can write:

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

PROOF .– Thanks to connectedness we can clearly write:

with a,b ∈ k and ∈ Ker ε ⊗ Ker ε. The counit property then tells us that, with k ⊗ ℋ and ℋ ⊗ k canonically identified with ℋ:

[1.4]

hence, a = b = 1. We will use the following two variants of Sweedler’s notation:

[1.5]

[1.6]

the second being relevant only for x ⊗ Ker ε. If x is homogeneous of degree n, we can suppose that the components x₁, x₂, x′, x″ in the expressions above are homogeneous as well, and we then have |x₁| + |x₂| = n and |x′| + |x″| = n. We easily compute:

and

hence, the coassociativity of comes from the one of Δ. Finally, it is easily seen by induction on k that for any x ∈ ℋⁿ, we can write:

[1.7]

with |x^(j)| ≥ 1. The grading imposes:

so the maximum possible for any degree |x^(j)| is n — k. □