Читать книгу Algebra and Applications 2 - Группа авторов - Страница 21

A filtered bialgebra on k is a k-vector space together with an increasing ℕ-indexed filtration:

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 bialgebra, and such that:

[1.11]

[1.12]

It is easy (and left to the reader) to show that the unit u and the counit ε are of degree zero, if we consider the filtration on the base field k given by k⁰ = {0} and kn = k for any n ≥ 1. Namely, u(kn) ⊂ ℋⁿ and ε(ℋⁿ) ⊂ kn for any n ≥ 0.

If we ask for the existence of an antipode S, we get the definition of a filtered Hopf algebra if the antipode is of degree zero, that is, if:

[1.13]

for any n ⊂ 0. Contrarily to the graded case, it is not likely 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.