der Universitat Munchen Set 9
Prof. Dr. B. Pareigis
Problem set for
Quantum Groups and Noncommutative Geometry
(33) (Linear Algebra) For U V dene U
?
:=ff 2 V
jf(U) =0g. For Z V
deneZ
?
:=fv 2VjZ(v)=0g. Showthat the following hold:
(a) U V =) U =U
??
;
(b) Z V
and dimZ <1 =) Z =Z
??
;
(c) fU VjdimV=U < 1g
=
fZ V
jdimZ < 1g under the maps
U 7!U
?
and Z 7!Z
?
.
(34) Let V = L
1
i=1 Kx
i
be an innite-dimensionalvector space. Find an element
g 2(V V)
that isnot in V
V
((V V)
).
(35) Formorphisms f :I ! M and g :I ! N in amonoidal category we dene
(f 1 : N ! M N) := (f 1
I )(I)
1
and (1g : M ! M N) :=
(1g)(I) 1
. Showthat the diagram
N M N
-
f1
I M
- f
? g
? 1g
commutes.
(36) LetG beanitegroupand K G
:=K[G]
thedualof thegroupalgebra. Show
that K G
is a Hopf algebra and that each module structure K[G]M ! M
translatestothestructure ofacomoduleM !K G
M and conversely. Show
that this denes a monoidalequivalence of categories.
Describe the group valued functor K-cAlg(K G
; ) in terms of sets and their
groupstructure.
