der Universitat Munchen Set 3
Prof. Dr. B. Pareigis
Problem set for
Quantum Groups and Noncommutative Geometry
(9) Consider the followingsubset H of the set of complex22-matrices:
H :=
x y
y x
2M
C
(22)jx;y2C
We callH the set of Hamiltonian quaternions. For
h =
x y
y x
wedene:
h:=
x y
y x
Show:
(a) h
h =(jxj 2
+jyj 2
)E (E the unit matrix),
(b) H is a real subalgebra of the complex algebraof 22-matrices.
(c) H isadivisionalgebra,i.e.eachelementdierentfromzerohasaninverseunder
the multiplication.
(d) Let
I :=
i 0
0 i
J :=
0 1
1 0
K :=
0 i
i 0
Then E;I;J;K isanR-basis ofH anwehavethe followingmultiplicationtable:
I 2
=J 2
=K 2
= 1
IJ = JI =K JK = KJ =I KI = IK =J:
(10) Compute the H-points A 2j0
(H) of the quantum plane.
(11) Denition: LetX beangeometricspacewithaÆnealgebraA. LetDbeanalgebra.
Anaturaltransformation :DX !A iscalledanalgebraaction if (B)(-;p):D
!A(B) =B is analgebra homomorphismfor allB and all p2X(B).
Give proofsfor:
Theorem: LetDbeanalgebraand:DX(-) !A(-)beanalgebraaction. Then
there exists a unique algebrahomomorphismf :D !A such that the diagram
AX(B) B -
(B) (B)
@
@
@
@
@ R DX(B)
? f1
commutes.
(12) Determine explicitly the dual coalgebra A
of the algebra A := Khxi=(x 2
). (Hint:
Finda basis for A.)
