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 (Nakayama 1939), (Nakayama 1941). Dieudonné used this to characterize Frobenius algebras (Dieudonné 1958). 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. Equivalently, one may equip A with a linear functional λ : A → k such that the kernel of λ contains no nonzero left ideal of A. A Frobenius algebra is called symmetric if σ is symmetric, or equivalently λ satisfies λ(a·b) = λ(b·a). There is also a different, mostly unrelated notion of the symmetric algebra of a vector space. In category theory, the notion of Frobenius object is an abstract definition of a Frobenius algebra in a category. A Frobenius object ( A , μ , η , δ , ε ) {displaystyle (A,mu ,eta ,delta ,varepsilon )} in a monoidal category ( C , ⊗ , I ) {displaystyle (C,otimes ,I)} consists of an object A of C together with four morphisms