Mathematisches Institut SS 2002 der Universität München Set 6 Prof. Dr. B. Pareigis

Problem set for Quantum Groups and Noncommutative Geometry

(21) Let H be a Hopf algebra. Then S is an antihomomorphism of algebras and coalgebras i.e. S “inverts the order of the multiplication and the comultiplication”.

(22) Let H and K be Hopf algebras and let f : H ⟶ K be a homomorphism of bialgebras. Then fS_{H} = S_{K}f, i.e. f is compatible with the antipode.

(23) Let 𝕂 be a field. Show that an element x ∈ 𝕂G satisfies Δ(x) = x ⊗ x and ε(x) = 1 if and only if x = g ∈ G.

(24) (a) Show that the grouplike elements of a Hopf algebra form a group under

multiplication of the Hopf algebra.

(b) Show that the set of primitive elements P (H) = {x ∈ H|Δ(x) = x ⊗ 1 + 1 ⊗ x} of a Hopf algebra H is a Lie subalgebra of H^{L}.

Due date: Tuesday, 04.06.2002, 16:15 in Lecture Hall E41