Mathematisches Institut SS 2002 der Universität München Set 8 Prof. Dr. B. Pareigis

Problem set for

Quantum Groups and Noncommutative Geometry

(29) Determine the structure of a covector space on a vector space V from the fact that Hom(V, W) is a vector space for all vector spaces W.

(30) The real unit circle S^{1}(ℝ) carries the structure of a group by the addition of angles. Is it possible to make S^{1} with the affine algebra 𝕂[c, s]/(s^{2} + c^{2} − 1) into an affine algebraic group? (Hint: How can you add two points (x_{1}, y_{1}) and (x_{2}, y_{2}) on the unit circle, such that you get the addition of the associated angles?) Find a group structure on the torus T .

(31) Let V be a vector space. Show that there is a universal vector space E and homomorphism ρ : E ⊗ V ⟶ V (such that for each vector space Z and each homomorphism f : Z ⊗ V ⟶ V there is a unique homomorphism g : Z ⟶ E such that

[figure: E ⊗ V; V; ρ; f; Z ⊗ V; g⊗1]

commutes). We call E and ρ : E ⊗ V ⟶ V a vector space acting universally on V.

(32) Let E and ρ : E ⊗ V ⟶ V be a vector space acting universally on V. Show that E has a uniquely determined structure of an algebra such that V becomes a left E-module.

Due date: Tuesday, 18.06.2002, 16:15 in Lecture Hall E41