Algebra and Applications 2

1 1) S ∘ u = u and ε o S = ε.

2 2) S is an algebra antimorphism and a coalgebra antimorphism, that is, if denotes the flip, we have:

3 3) If ℋ is commutative or cocommutative, then S2 = I.

For a detailed proof, see Kassel (1995).

PROPOSITION 1.5.–

1 1) If x is a primitive element, then S(x) = –x.

2 2) The linear subspace Prim ℋ of primitive elements in ℋ is a Lie algebra.

PROOF.– If x is primitive, then ( ε ⊗ ε ) ∘ Δ( x ) = 2 ε ( x ). On the contrary, ( ε ⊗ ε ) ∘ Δ( x ) = ε ( x ), so ε ( x ) = 0. Then:

Now let x and y be the primitive elements of ℋ. Then, we can easily compute:

We introduce the crucial property of connectedness for bialgebras. The main interest resides in the possibility of implementing recursive procedures in connected bialgebras, the induction taking place with respect to a filtration or a grading. An important example of these techniques is the recursive construction of the antipode, which then “comes for free”, showing that any connected bialgebra is in fact a connected Hopf algebra.

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∈ ℋ 0and 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 (ℋ n ) ⊂ ℋ n. It can be proven as follows: let S ′ : ℋ → ℋ be defined, so that S ’( x ) is the n thhomogeneous component of S ( x ) when x is homogeneous of degree n . We can write down the coproduct Δ( x ) with Sweedler’s notation:

where x 1and x 2are 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, ℋ 0is one-dimensional. Then, we have:

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

The map is coassociative on Ker ε and sends ℋ n 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:

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the second being relevant only for x ⊗ Ker ε . If x is homogeneous of degree n , we can suppose that the components x 1, x 2, x ′, x ″ in the expressions above are homogeneous as well, and we then have | x 1| + | x 2| = 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 ∈ ℋ n, we can write:

[1.7]