Mathematisches Institut SS 2002 der Universität München Set 3 Prof. Dr. B. Pareigis

Problem set for

Quantum Groups and Noncommutative Geometry

(9) Consider the following subset ℍ of the set of complex 2 × 2-matrices:

ℍ := \{( \begin{matrix}x & −y \\ ȳ & x̄ \end{matrix} ) ∈ M_{ℂ}(2 × 2)|x, y ∈ ℂ\}

We call ℍ the set of Hamiltonian quaternions. For

h = ( \begin{matrix}x & −y \\ ȳ & x̄ \end{matrix} )

we define:

h̄ := ( \begin{matrix}x̄ & y \\ −ȳ & x \end{matrix} )

Show:

(a) hh̄ = (|x|^{2} + |y|^{2})E (E the unit matrix),

(b) ℍ is a real subalgebra of the complex algebra of 2 × 2-matrices.

(c) ℍ is a division algebra, i. e. each element different from zero has an inverse under the multiplication.

(d) Let

I := ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) J := ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) K := ( \begin{matrix}0 & −i \\ −i & 0 \end{matrix} )

Then E, I, J, K is an ℝ-basis of ℍ an we have the following multiplication table:

I^{2} = J^{2} = K^{2} = −1
IJ = −JI = K JK = −KJ = I KI = −IK = J.

(10) Compute the ℍ-points A^{2|0}(ℍ) of the quantum plane.

(11) Definition: Let X be an geometric space with affine algebra A. Let D be an algebra. A natural transformation ρ : D × X ⟶ 𝔸 is called an algebra action if ρ(B)(-, p) : D ⟶ 𝔸(B) = B is an algebra homomorphism for all B and all p ∈ X (B). Give proofs for: Lemma: The natural transformation ψ : A × X ⟶ 𝔸 is an algebra action.

Theorem: Let D be an algebra and ρ : D×X (-) ⟶ 𝔸(-) be an algebra action. Then there exists a unique algebra homomorphism f : D ⟶ A such that the diagram

[figure: A × X (B); B; ψ(B); ρ(B); D × X (B); f×1]

commutes.

(12) Determine explicitly the dual coalgebra A^{∗} of the algebra A := 𝕂⟨x⟩/(x^{2}). (Hint: Find a basis for A.)

Due date: Tuesday, 07.05.2002, 16:15 in Lecture Hall E41