Mathematisches Institut SS 2002 der Universität München Set 7 Prof. Dr. B. Pareigis

Problem set for

Quantum Groups and Noncommutative Geometry

(25) Let the set Z together with the multiplication m : Z × Z ⟶ Z be a monoid. Show that the unit element e ∈ Z is uniquely determined. Let (Z, m) be a group. Show that also the inverse i : Z ⟶ Z is uniquely determined. Show that unit element and inverses of groups are preserved by maps that are compatible with the multiplication.

(26) Find an example of monoids Y and Z and a map f : Y ⟶ Z with f(y_{1}y_{2}) = f(y_{1})f(y_{2}) for all y_{1}, y_{2} ∈ Y, but f(e_{Y}) ≠ e_{Z}.

(27) Let (G, m) be a group in C and i_{X} : G(X) ⟶ G(X) be the inverse for all X ∈ C. Show that i is a natural transformation. Show that the Yoneda Lemma provides a morphism S : G ⟶ G such that i_{X} = Mor_{C}(X, S) = S(X) for all X ∈ C. Formulate and prove properties of S of the type S ∗ id = . . ..

(28) Let C be a category with finite products. Show that a morphism f : G ⟶ G′ in C is a homomorphism of groups if and only if

[figure: G × G; G; m; G′ × G′; G′; m′; f×f; f]

commutes.

Due date: Tuesday, 11.06.2002, 16:15 in Lecture Hall E41