Читать книгу Algebra and Applications 2 - Группа авторов - Страница 15

There is a natural structure of cocommutative Hopf algebra on the tensor algebra T(V) of any vector space V. Namely, we define the coproduct Δ as the unique algebra morphism from T(V) into T(V) ⊗ T(V), such that:

We 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.