Self-injective algebras: Comparison with Frobenius algebras. Lecture Notes, available at http://www.mathematik.uni-bielefeld.de/~sek/selected.html

left Λ-modules and the set of isoclasses of simple Λ modules will be denoted by modΛ and S(Λ), respectively. We will occasionally identify S(Λ) with a complete set of its representatives. Given a simple Λ-module S, we consider its projective cover P (S) and its injective envelope E(S). Recall that Top(P (S)) ∼ = S and Soc(E(S)) ∼ = S. In our lecture [2], we observed that Λ is self-injective if and only if S 7 → Soc(P (S)) defines a permutation ν: S(Λ) − → S(Λ), the so-called Nakayama permutation [2, Theorem]. In early articles, these algebras were referred to as quasi-Frobenius algebras (cf. [3]). The purpose of this lecture is to understand when a self-injective algebra is a Frobenius algebra. This class of algebras was introduced by Brauer and Nesbitt in [1], who provided the following chacterization: Lemma 1. Let π: Λ − → k be a linear map. Then the following statements are equivalent: (1) kerπ does not contain a non-zero left ideal. (2) The Λ-linear map Φpi: Λ − → Λ ∗ ; x 7 → x.π is an isomorphism. (3) kerπ does not contain a non-zero right ideal. Proof. (1) ⇒ (2). Consider the left ideal J: = ker Φpi. Since 0 = Φpi(x)(1) = π(x) ∀ x ∈ J, we obtain the injectivity and hence the bijectivity of Φpi. (2) ⇒ (3). Let J ⊂ kerπ be a right ideal. Given a linear form f ∈ Λ∗, condition (2) provides an element x ∈ Λ such that f = x.π. For y ∈ J we thus obtain f(y) = (x.π)(y) = π(yx) ∈ π(J) = (0). Consequently, J = (0). (3) ⇒ (1). Since (1) ⇒ (3) holds for every algebra, application to the opposite algebra Λop yields the desired conclusion. Definition. A k-algebra Λ is a Frobenius algebra if there exists a linear form π ∈ Λ ∗ such that ker π contains no non-zero left ideals. Given a linear form π: Λ − → k, we consider the bilinear form ( ,)pi: Λ × Λ − → k; (a, b): = π(ab) ∀ a, b ∈ Λ. This form is associative, that is, (ax, b)pi = (a, xb)pi ∀ a, b, x ∈ Λ