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der Universitat Munchen Set 6

Prof. Dr. B. Pareigis

Problem set for

Quantum Groups and Noncommutative Geometry

(21) Let H be a Hopf algebra. Then S is an antihomomorphism of algebras and

coalgebras i.e. S \inverts the order of the multiplication and the comultipli-

cation".

(22) Let H and K be Hopf algebras and let f : H ! K be a homomorphism of

bialgebras. ThenfS

H

=S

K

f, i.e. f is compatible with the antipode.

(23) Let K be a eld. Show that an element x 2 KG satises (x) =xx and

"(x)=1 if and onlyif x=g 2G.

(24) (a) Show that the grouplike elements of a Hopf algebra form a group under

multiplicationof the Hopf algebra.

(b) Show thatthe set of primitiveelementsP(H)=fx2Hj(x)=x1+

1xg of a Hopfalgebra H isa Liesubalgebra of H L

.

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