der Universitat Munchen 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 ina canonical way again
a (co-)reprensentation. For two H-modules M and N describe the module
structure onM N if
(a) H =KG: g(mn)=::: for g 2G;
(b) H =U(g): g(mn)=::: for g 2g;
(c) H =U
q
(sl(2)):
(i) E(mn) =:::,
(ii) F(mn) =:::,
(iii) K(mn)=:::
for the elementsE;F;K 2U
q
(sl(2)).
(38) LetG be amonoid.
(a)Thecategoryof G-familiesofvectorspacesM G
= Q
g2G
Vec hasfamilies
of vector spaces (V
g
jg 2 G) as objects and families of linear maps (f
g : V
g
! W
g
jg 2 G) as morphisms. The composition is (f
g
jg 2 G)Æ(h
g
jg 2 G)=
(f
g Æh
g
jg2G). ShowthatM G
isamonoidalcategorywiththetensorproduct
(V
g
jg 2G)(W
g
jg2G):=(
h;k2G;hk=g V
h W
k
jg2G):
Where dounit and associativity laws of Genter the proof?
(b) A vector space V together with a family of subspaces (V
g
Vjg2 G)
iscalledG-graded, if V =
g2G V
g
holds. Let(V;(V
g
jg2G))and (W;(W
g jg2
G))be G-graded vector spaces. A linearmap f :V !W is calledG-graded,
iff(V
g )W
g
for allg 2G. The G-graded vector spacesand 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 dened by
(V W)
g :=
X
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
monoidallyequivalenttothemonoidalcategoryofKG-comodulesM KG
. (Hint:
Use the following constructions. For a G-family (V
g
jg 2 G) construct a G-
graded vector space
^
V :=
g2G V
g
(exterior direct sum) with the subspaces
^
V
g
:= Im(V
g
) in the direct sum. Conversely if (V;(V
g
jg 2 G)) is a G-graded
vector space then (V
g
jg 2 G) is a G-family of vector spaces. Fora G-graded
vector space(V;(V
g
jg 2G))construct theKG-comodule V withthe structure
map Æ : V ! V KG, Æ(v) :=vg for all (homogeneous elements) v 2 V
g
and for all g 2 G. Conversely let (V;Æ : V ! V KG) be a KG-comodule.
ThenconstructthevectorspaceV withdengraded(homogenous)components
V
g
:=fv 2VjÆ(v)=vgg).
(39) Let (D;!) be a diagram in Vec. Let D be a monoidal category and ! be a
monoidalfunctor. Then(D;!) iscalled a monoidal diagram.
Let (D;!) be a monoidal diagram Vec. Let A 2 Vec be an algebra. A
natural transformation ' : ! ! ! B is called monoidal monoidal if the
diagrams
!(X)!(Y) !(X)!(Y)BB
- '(X)'(Y)
?
? m
!(X Y) !(X Y)B
- '(XY)
and
K K K
-
=
? ?
!(I) !(I)B
- '(I)
commute.
Wedenote the set ofmonoidalnaturaltransformationsby Nat
(!;!B).
Show that Nat
(!;!B) is afunctor in B.
(40) Show that theadjoint action HH 3ha7!
P
h
(1) aS(h
(2)
)2H makesH
anH-module algebra.
