14 2. Basic notions of representation theory

Problem 2.3.15. Let V be a nonzero finite dimensional representa-

tion of an algebra A. Show that it has an irreducible subrepresen-

tation. Then show by example that this does not always hold for

infinite dimensional representations.

Problem 2.3.16. Let A be an algebra over a field k. The center

Z(A) of A is the set of all elements z ∈ A which commute with all

elements of A. For example, if A is commutative, then Z(A) = A.

(a) Show that if V is an irreducible finite dimensional representa-

tion of A, then any element z ∈ Z(A) acts in V by multiplication by

some scalar χV (z). Show that χV : Z(A) → k is a homomorphism.

It is called the central character of V .

(b) Show that if V is an indecomposable finite dimensional rep-

resentation of A, then for any z ∈ Z(A), the operator ρ(z) by which z

acts in V has only one eigenvalue χV (z), equal to the scalar by which z

acts on some irreducible subrepresentation of V . Thus χV : Z(A) → k

is a homomorphism, which is again called the central character of V .

(c) Does ρ(z) in (b) have to be a scalar operator?

Problem 2.3.17. Let A be an associative algebra, and let V be

a representation of A. By EndA(V ) one denotes the algebra of all

homomorphisms of representations V → V . Show that EndA(A) =

Aop, the algebra A with opposite multiplication.

Problem 2.3.18. Prove the following “infinite dimensional Schur

lemma” (due to Dixmier): Let A be an algebra over C and let V

be an irreducible representation of A with at most countable basis.

Then any homomorphism of representations φ : V → V is a scalar

operator.

Hint: By the usual Schur’s lemma, the algebra D := EndA(V ) is

an algebra with division. Show that D is at most countably dimen-

sional. Suppose φ is not a scalar, and consider the subfield C(φ) ⊂ D.

Show that C(φ) is a transcendental extension of C. Derive from this

that C(φ) is uncountably dimensional and obtain a contradiction.