Mathematisches Institut SS 2002 der Universität München Set 4 Prof. Dr. B. Pareigis

Problem set for

Quantum Groups and Noncommutative Geometry

(13) Determine explicitly the dual coalgebra A^{∗} of A := 𝕂⟨x⟩/(x^{3}).

(14) Determine and describe the coendomorphism bialgebra of A from problem 12. (Hint: Determine first a set of algebra generators of M(A). Then describe the relations.)

(15) Let A be a finite dimensional 𝕂-algebra with universal bialgebra A ⟶ B ⊗ A.

Show

i) that A^{op} ⟶ B^{op} ⊗ A^{op} is universal (where A^{op} has the multiplication ∇τ : A ⊗ A ⟶ A ⊗ A ⟶ A);

ii) that A ≅ A^{op} implies B ≅ B^{op} (as bialgebras); iii) that for commutative algebras A the algebra B satisfies B ≅ B^{op} but

that B need not be commutative. iv) Find an isomorphism B ≅ B^{op} for the bialgebra B = 𝕂⟨a, b⟩/(a^{2}, ab+ba).

(compare problem 14).

(16) Consider the algebra K[ϵ]/(ϵ^{2}) the so called algebra of dual numbers over a field K. Consider the algebra B with (noncommuting) generators a, b, c, d and relations:

ac = acac ad = acad + adac bc = acbc = bcac
bd = acbd + adbc = bcad + bdac bcbc = bcbd + bdbc = 0
ac = 1 ad = 0

Show that B together with the cooperation

δ : A → B ⊗ A

with δ(ϵ) = bc ⊗ 1 + bd ⊗ ϵ is the coendomorphism bialgebra of A.

Due date: Tuesday, 14.05.2002, 16:15 in Lecture Hall E41