Mathematisches Institut WS 2001/02 der Universität München Set 5 Prof. Dr. B. Pareigis

Problem set for Advanced Algebra

(17) Show that V ⊗ V^{∗} is a coalgebra for every finite dimensional vector space V over a field 𝕂 if the comultiplication is defined by Δ(v ⊗ v^{∗}) := ∑_{i=1}^{n} v ⊗ v_{i}^{∗} ⊗ v_{i} ⊗ v^{∗} where {v_{i}} and {v_{i}^{∗}} are dual bases of V resp. V^{∗}.

(18) Show that a finite dimensional vector space V is a comodule over the coalgebra V ⊗ V^{∗} as defined in problem 17 with the coaction δ(v) := ∑ v ⊗ v_{i}^{∗} ⊗ v_{i} ∈ (V ⊗ V^{∗}) ⊗ V where ∑ v_{i}^{∗} ⊗ v_{i} is the dual basis of V in V^{∗} ⊗ V.

(19) Show that the free 𝕂-modules 𝕂X with the basis X and the comultiplication Δ(x) = x ⊗ x is a coalgebra. What is the counit? Is the counit unique?

(20) Show that the tensor product of two commutative 𝕂-algebras is a coproduct.

Due date: Tuesday, 20.11.2001, 16:15 in Lecture Hall 138