Frobenius algebra

In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Brauer and Nesbitt and were named after Frobenius. Nakayama discovered the beginnings of a rich duality theory in his (Nakayama 1939) and especially in his (Nakayama 1941). Dieudonné used this to characterize Frobenius algebras in his (Dieudonné 1958) where he called this property of Frobenius algebras a perfect duality. Frobenius algebras were generalized to quasi-Frobenius rings, those Noetherian rings whose right regular representation is injective. In recent times, interest has been renewed in Frobenius algebras due to connections to topological quantum field theory.

A finite-dimensional, unital, associative algebra A defined over a field k is said to be a Frobenius algebra if A is equipped with a nondegenerate bilinear form σ:A × A → k that satisfies the following equation: σ(a·b,c)=σ(a,b·c). This bilinear form is called the Frobenius form of the algebra.

...
Wikipedia