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 Richard Brauer and Cecil Nesbitt and were named after Georg Frobenius. Tadashi Nakayama discovered the beginnings of a rich duality theory , . Jean Dieudonné used this to characterize Frobenius algebras . 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. Definition 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 that satisfies the following equation: . This bilinear form is called the Frobenius form of the algebra. Equivalently, one may equip A with a linear functional 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 . There is also a different, mostly unrelated notion of the symmetric algebra of a vector space. Nakayama automorphism For a Frobenius algebra A with σ as above, the automorphism ν of A such that is Nakayama automorphism associated to A and σ. Examples Any matrix algebra defined over a field k is a Frobenius algebra with Frobenius form σ(a,b)=tr(a·b) where tr denotes the trace. Any finite-dimensional unital associative algebra A has a natural homomorphism to its own endomorphism ring End(A). A bilinear form can be defined on A in the sense of the previous example. If this bilinear form is nondegenerate, then it equips A with the structure of a Frobenius algebra. Every group ring k[G] of a finite group G over a field k is a symmetric Frobenius algebra, with Frobenius form σ(a,b) given by the coefficient of the identity element in a·b. For a field k, the four-dimensional k-algebra k[x,y]/ (x2, y2) is a Frobenius algebra. This follows from the characterization of commutative local Frobenius rings below, since this ring is a local ring with its maximal ideal generated by x and y, and unique minimal ideal generated by xy. For a field k, the three-dimensional k-algebra A=k[x,y]/ (x, y)2 is not a Frobenius