Questions: Semisimplicity and the Wedderburn-Artin Theorem

Question 1

The Artin-Wedderburn theorem says every semisimple ring is a product of matrix algebras over division rings. Over ℂ, what simplification occurs?

Model answer: Over ℂ (algebraically closed), the only finite-dimensional division algebra is ℂ itself (by Schur's lemma, the endomorphism ring of a simple module is a division algebra, and over an algebraically closed field this must be the field). So every semisimple ℂ-algebra decomposes as a product of matrix algebras M_{nᵢ}(ℂ).

Over ℝ, the division algebras are ℝ, ℂ, and ℍ (the quaternions), by Frobenius's theorem. This means real group algebras can have matrix algebras over ℂ or ℍ as summands, leading to representations with Frobenius-Schur indicators ±1 or 0. Over algebraically closed fields of appropriate characteristic, the theory simplifies maximally.

Question 2

If ℂ[G] ≅ M₁(ℂ) × M₁(ℂ) × M₂(ℂ), what is |G|?

• A. 4
• B. 5
• C. 6
• D. 8

Answer: (C)

dim(ℂ[G]) = |G| as a vector space. The dimension of the right side is dim(M₁(ℂ)) + dim(M₁(ℂ)) + dim(M₂(ℂ)) = 1 + 1 + 4 = 6. So |G| = 6. The irreducible representations have dimensions 1, 1, 2, and the sum-of-squares is 1 + 1 + 4 = 6. This matches the representation theory of S₃ (the only group of order 6 with this decomposition pattern).

Question 3

A semisimple ring has no nonzero nilpotent ideals.

• T. True
• F. False

Answer: True

In a semisimple ring R ≅ M_{n₁}(D₁) × ··· × M_{nₖ}(Dₖ), each matrix algebra M_{nᵢ}(Dᵢ) is simple (no proper two-sided ideals). The only ideals of the product are products of subsets of the factors. Since each factor is simple, the ideals are direct sums of some subset of factors, and none of these is nilpotent (each M_{nᵢ}(Dᵢ) contains the identity). This characterization is equivalent: a finite-dimensional algebra is semisimple if and only if its Jacobson radical is zero.

Question 4

Which of the following is NOT a consequence of the Artin-Wedderburn decomposition ℂ[G] ≅ ⊕ᵢ M_{dᵢ}(ℂ)?

• A. The number of irreducible representations equals the number of conjugacy classes
• B. |G| = Σ dᵢ²
• C. Every representation of G over ℂ is completely reducible
• D. Every normal subgroup of G is abelian

Answer: (D)

Options A, B, C all follow from Artin-Wedderburn. The number of simple components equals the number of simple modules (= irreps = conjugacy classes). Comparing dimensions gives |G| = Σ dᵢ². Complete reducibility is the definition of semisimplicity. Option D is false in general (S₄ has normal subgroup A₄, which is non-abelian) and has nothing to do with Artin-Wedderburn.