Mathematisches Institut SS 2002 der Universität München Set 9 Prof. Dr. B. Pareigis

Problem set for

Quantum Groups and Noncommutative Geometry

(33) (Linear Algebra) For U ⊆ V define U^{⊥} := {f ∈ V^{∗}|f(U) = 0}. For Z ⊆ V^{∗} define Z^{⊥} := {v ∈ V |Z(v) = 0}. Show that the following hold:

(a) U ⊆ V ⟹ U = U^{⊥⊥};

(b) Z ⊆ V^{∗} and dim Z < ∞ ⟹ Z = Z^{⊥⊥};

(c) {U ⊆ V | dim V/U < ∞} ≅ {Z ⊆ V^{∗}| dim Z < ∞} under the maps U ↦ U^{⊥} and Z ↦ Z^{⊥}.

(34) Let V = ⨁_{i=1}^{∞} 𝕂x_{i} be an infinite-dimensional vector space. Find an element g ∈ (V ⊗ V)^{∗} that is not in V^{∗} ⊗ V^{∗} (⊆ (V ⊗ V)^{∗}).

(35) For morphisms f : I ⟶ M and g : I ⟶ N in a monoidal category we define (f ⊗ 1 : N ⟶ M ⊗ N) := (f ⊗ 1_{I})ρ(I)^{−1} and (1 ⊗ g : M ⟶ M ⊗ N) := (1 ⊗ g)λ(I)^{−1}. Show that the diagram

[figure: N; M ⊗ N; f⊗1; I; M; f; g; 1⊗g]

commutes.

(36) Let G be a finite group and 𝕂^{G} := 𝕂[G]^{∗} the dual of the group algebra. Show that 𝕂^{G} is a Hopf algebra and that each module structure 𝕂[G] ⊗ M ⟶ M translates to the structure of a comodule M ⟶ 𝕂^{G} ⊗M and conversely. Show that this defines a monoidal equivalence of categories. Describe the group valued functor 𝕂-cAlg (𝕂^{G}, −) in terms of sets and their group structure.

Due date: Tuesday, 25.06.2002, 16:15 in Lecture Hall E41