der Universitat Munchen Set 8
Prof. Dr. B. Pareigis
Problem set for
Quantum Groups and Noncommutative Geometry
(29) Determinethe structure ofacovectorspaceonavectorspace V fromthefact
that Hom(V;W) isa vector space for allvector spacesW.
(30) The real unit circle S 1
(R) carriesthe structure of agroup by the addition of
angles. Is it possible to make S 1
with the aÆne algebra K[c;s]=(s 2
+c 2
1)
into an aÆne algebraic group? (Hint: How can you add two points (x
1
;y
1 )
and(x
2
;y
2
)onthe unitcircle,suchthatyougetthe additionoftheassociated
angles?)
Finda group structure onthe torus T.
(31) Let V be a vector space. Show that there is a universal vector space E and
homomorphism: EV !V (such that for each vector space Z and each
homomorphismf :Z V !V there is aunique homomorphismg : Z !E
such that
EV V
-
f
@
@
@
@
@ R ZV
? g1
commutes). We call E and :E V !V a vector space acting universally
onV.
(32) Let E and : EV ! V be a vector space acting universally onV. Show
thatE hasauniquelydeterminedstructureofanalgebrasuchthatV becomes
aleft E-module.
