Mathematisches Institut SS 2002 der Universität München Set 5 Prof. Dr. B. Pareigis

Problem set for Quantum Groups and Noncommutative Geometry

(17) Determine the coendomorphism bialgebra of A from problem 13.

(18) (*) Determine explicitly the dual coalgebra A^{∗} of A := 𝕂⟨x, y⟩/I where the ideal I is generated as a two-sided ideal by the polynomials

xy − q^{−1}yx, x^{2}, y^{2}.

(19) (*) Determine the coendomorphism bialgebra of A from problem 18.

(20) Let H be a bialgebra and S ∈ Hom(H, H). Then S is an antipode for H (and H is a Hopf algebra) iff S is a two sided inverse for id in the algebra (Hom(H, H), ∗, ηε) (see [AdvAlgebra] Proposition 2.21). In particular S is uniquely determined.

Due date: Tuesday, 28.05.2002, 16:15 in Lecture Hall E41