Exercise set 3

Throughout k denotes a field and A a k-algebra.

(1)

(Difficulty level: 2) Prove or disprove: a k-vector space V is a simple module for End_k(V ).

(2)

(Difficulty level: 2) Every simple module for a finite dimensional k-algebra is finite dimensional. (In particular, every irreducible representation of a finite group is finite dimensional.)

(3)

(Difficulty level: 2) Show that the group ring kG is isomorphic to its opposite (for any field k and any group G).

(4)

(Difficulty level: 3) Show that every simple linear representation of a finite abelian group over an algebraically closed field is one dimensional. (Hint: Use Schur’s lemma.)

(5)

(Difficulty level: 2) Show by means of an example that the hypothesis in item (4) above of the field being algebraically closed cannot be omitted.

(6)

(Difficulty level: 2) Let k be algebraically closed and let V and W be finite dimensional semisimple A-modules. Show that if dimEnd_AV = dimHom_A(V,W) = dimEnd_A(W), then V is isomorphic to W.

(7)

(Difficulty level: 2) Let k be algebraically closed and M be a semisimple A-module. Show the following:

• M is simple if and only if End_A(M) ≃ k
• M is multiplicity free (that is, no simple component of M occurs with multiplicity more than 1) if and only if End_A(M) is commutative.

(8)

(Difficulty level: 3) Let M be a multiplicity free semisimple module (over some ring). Determine the submodules of M.

(9)

(Difficulty level: 2) (Converse of Maschke) Let k be a field, G a group, and kG the group ring. We can turn k into an kG-module by letting each g in G act as identity and extending linearly: (∑ λ_gg) ⋅ μ = (∑ λ_g)μ. This module is called the trivial kG-module. Let H := {∑ _g∈Gλ_gg ∈ kG|∑ _g∈Gλ_g = 0}. Then H is a codimension 1 subspace of kG and is a kG-submdoule. Moreover, kG∕H is trivial.

Now assume that G is finite. Show that the span of ∑ _g∈Gg is the only 1-dimensional trivial kG-submodule of kG. Conclude that H does not have a complemtary submodule if the characteristic of k is positive and divides |G|.

(10)

(Difficulty level: 3) Let F be the finite field ℤ∕pℤ and M₃(F) the ring of 3 × 3 matrices with coefficients in F. What are the possible dimensions of left ideals in M₃(F)? How many left ideals are there of each such dimension?