Mathematisches Institut SS 2002 der Universität München Set 10 Prof. Dr. B. Pareigis

Problem set for

Quantum Groups and Noncommutative Geometry

(37) We have seen that in representation theory and in corepresentation theory of quantum groups H such as 𝕂G, U(𝔤), SL_{q}(2), U_{q}(sl(2)) the ordinary tensor product (in 𝕂-Mod ) of two (co-)reprensentations is in a canonical way again a (co-)reprensentation. For two H-modules M and N describe the module structure on M ⊗ N if

(a) H = 𝕂G: g(m ⊗ n) = … for g ∈ G;

(b) H = U(𝔤): g(m ⊗ n) = … for g ∈ 𝔤;

(c) H = U_{q}(sl(2)): (i) E(m ⊗ n) = …, (ii) F (m ⊗ n) = …, (iii) K(m ⊗ n) = … for the elements E, F, K ∈ U_{q}(sl(2)).

(38) Let G be a monoid.

(a) The category of G-families of vector spaces M^{G} = ∏_{g∈G} Vec has families of vector spaces (V_{g}|g ∈ G) as objects and families of linear maps (f_{g} : V_{g} ⟶ W_{g}|g ∈ G) as morphisms. The composition is (f_{g}|g ∈ G) ∘ (h_{g}|g ∈ G) = (f_{g} ∘h_{g}|g ∈ G). Show that M^{G} is a monoidal category with the tensor product

(V_{g}|g ∈ G) ⊗ (W_{g}|g ∈ G) := (⊕_{h,k∈G,hk=g}V_{h} ⊗ W_{k}|g ∈ G).

Where do unit and associativity laws of G enter the proof?

(b) A vector space V together with a family of subspaces (V_{g} ⊆ V |g ∈ G) is called G-graded, if V = ⊕_{g∈G}V_{g} holds. Let (V, (V_{g}|g ∈ G)) and (W, (W_{g}|g ∈ G)) be G-graded vector spaces. A linear map f : V ⟶ W is called G-graded, if f(V_{g}) ⊆ W_{g} for all g ∈ G. The G-graded vector spaces and G-graded linear maps form the category M^{[G]} of G-graded vector spaces. Show that M^{[G]} is a monoidal category with the tensor product V ⊗ W, where the subspaces (V ⊗ W)_{g} are defined by

(V ⊗ W)_{g} := ∑_{h,k∈G,hk=g} V_{h} ⊗ W_{k}.

(c) Show that the monoidal category M^{G} of G-families of vector spaces is monoidally equivalent to the monoidal category M^{[G]} of G-graded vector spaces.

(d) Show that the monoidal category M^{[G]} of G-graded vector spaces is monoidally equivalent to the monoidal category of 𝕂G-comodules M^{𝕂G}. (Hint: Use the following constructions. For a G-family (V_{g}|g ∈ G) construct a Ggraded vector space V̂ := ⊕_{g∈G}V_{g} (exterior direct sum) with the subspaces V̂_{g} := Im(V_{g}) in the direct sum. Conversely if (V, (V_{g}|g ∈ G)) is a G-graded vector space then (V_{g}|g ∈ G) is a G-family of vector spaces. For a G-graded vector space (V, (V_{g}|g ∈ G)) construct the 𝕂G-comodule V with the structure map δ : V ⟶ V ⊗ 𝕂G, δ(v) := v ⊗ g for all (homogeneous elements) v ∈ V_{g} and for all g ∈ G. Conversely let (V, δ : V ⟶ V ⊗ 𝕂G) be a 𝕂G-comodule. Then construct the vector space V with den graded (homogenous) components V_{g} := {v ∈ V |δ(v) = v ⊗ g}).

(39) Let (D, ω) be a diagram in Vec . Let D be a monoidal category and ω be a

monoidal functor. Then (D, ω) is called a monoidal diagram.

Let (D, ω) be a monoidal diagram Vec . Let A ∈ Vec be an algebra. A natural transformation φ : ω ⟶ ω ⊗ B is called monoidal monoidal if the diagrams

[figure: ω(X) ⊗ ω(Y); ω(X) ⊗ ω(Y) ⊗ B ⊗ B; φ(X)⊗φ(Y); ρ; ρ⊗m; ω(X ⊗ Y); ω(X ⊗ Y) ⊗ B; φ(X⊗Y)]

and

[figure: 𝕂; 𝕂 ⊗ 𝕂; ≅; ω(I); ω(I) ⊗ B; φ(I)]

commute.

We denote the set of monoidal natural transformations by Nat^{⊗}(ω, ω ⊗ B). Show that Nat^{⊗}(ω, ω ⊗ B) is a functor in B.

(40) Show that the adjoint action H ⊗ H ∋ h ⊗ a ↦ ∑ h_{(1)}aS(h_{(2)}) ∈ H makes H an H-module algebra.

Due date: Tuesday, 02.07.2002, 16:15 in Lecture Hall E41