On Frobenius Algebras

Abstract

An algebra A, with identity, finite dimensional over a field K, in which the right regular representation is equivalent to the left regular representation, is called a Frobenius algebra. Through the years these and related algebras have been thoroughly studied by a number of authors [1, 4, 5, 12, 13, 14, 15, 16, 17]; outstanding among such works are the classic papers of Nakayama [10, 11]. In [1] Eilenberg and Nakayama extended the definition of Frobenius algebra to finitely generated algebras A over a commutative ring K where A is K-projective. Many of the nice properties carry over to this larger class of algebras. In this paper we generalize the notion of Frobenius algebra in a different direction. We require that the coefficient ring K be a field, but we discard finiteness of dimension. We replace the condition about right and left regular representations by supplying the algebra with an inner product (x, y) satisfying the relation (xy, z) = (x, yz) for all x, y, z in the algebra. We also assume that the algebra possesses Nakayama's automorphism . A Forbenius algebra is symmetric if Nakayama's automorphism is the identity automorphism. We are led to this approach by the definition used in [1] which states that a Frobenius algebra A, as a left module over itself, be isomorphic to the module HomK(A, K). That is, A must in some sense be isomorphic to its dual. But we do not need the whole dual; indeed, if the dimension of A over K is infinite we cannot expect it. We need only enough of the dual to distinguish elements; that is, we require that the inner product be non-degenerate. The non-degenerate inner product induces two topologies on A, see [6; IV]. That these two topologies are the same is equivalent to the existence of Nakayama's automorphism; this is the content of Theorem 2.1. In ? 2 we outline some of the properties of this topology as developed in [6; II, IV]. It is at this point that we need the fact that K, the ring of coefficients, is a field. If the properties of the topology which we need could be deduced for a more general class of commutative rings, then the definition of Frobenius algebra could be extended a little farther. We note that the topology induced by the inner product is discrete if and only if the algebra is finite dimensional. Thus, as might be expected, 392