Algebra and Applications 2. Группа авторов

define the counit as the algebra morphism, such that ε(1) = 1 and . This endows T(V) with a cocommutative bialgebra structure. We claim that the principal anti-automorphism:

      verifies the axioms of an antipode, so that T(V) is indeed a Hopf algebra. For x ∈ V, we have S(x) = –x; hence, S * I(x) = I * S(x) = 0. As V generates T(V) as an algebra, it is easy to conclude.

      1.2.5.3. Enveloping algebras

      Let be a Lie algebra. The universal enveloping algebra is the quotient of the tensor algebra by the ideal J generated by x ⊗ y — y ⊗ x — [x, y], .

      LEMMA 1.1.– J is a Hopf ideal, that is, Δ(J) ⊂ ℋ ⊗ J + J ⊗ ℋ and S(J) ⊂ J.

      PROOF.– The ideal J is generated by primitive elements (according to Proposition 1.5 below), and any ideal generated by primitive elements is a Hopf ideal (very easy and left to the reader). □

      The quotient of a Hopf algebra by a Hopf ideal is a Hopf algebra. Hence, the universal enveloping algebra is a cocommutative Hopf algebra.

      1.2.6. Some basic properties of Hopf algebras

      In the proposition below we summarize the main properties of the antipode in a Hopf algebra:

      PROPOSITION 1.4.– (see Sweedler (1969, Proposition 4.0.1)). Let ℋ be a Hopf algebra with multiplication m, comultiplication Δ, unit u : 1 ↦ 1, counit ε and antipode S. Then:

      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) = 2ε(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:

1.3. Connected Hopf algebras

      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.

      1.3.1. Connected graded bialgebras

      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]