Mathematisches Institut SS 2002 der Universität München Set 1 Prof. Dr. B. Pareigis

Problem set for

Quantum Groups and Noncommutative Geometry

(1) Determine the affine algebra of the functor “unit sphere” S^{n−1} in 𝔸^{n}.

(2) Determine the affine algebra of the functor “torus” T and find an “embedding” of T into 𝔸^{3}.

(3) Let U denote the plane curve xy = 1. Then U is not isomorphic to the affine line. (Hint: An isomorphism 𝕂[x, x^{−1}] ⟶ 𝕂[y] sends x to a polynomial p(y) which must be invertible. Consider the highest coefficient of p(y) and show that p(y) ∈ 𝕂. But that means that the map cannot be bijective.) U is also called the unit functor. Can you explain, why?

(4) Let 𝕂 = ℂ be the field of complex numbers. Show that the unit functor U : 𝕂-cAlg ⟶ Set in Problem (3) is naturally isomorphic to the unit circle S^{1}. (Hint: There is an algebra isomorphism between the representing algebras 𝕂[e, e^{−1}] and 𝕂[c, s]/(c^{2} + s^{2} − 1).)

Due date: Tuesday, 23.04.2002, 16:15 in Lecture Hall E41