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.
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