# Frobenius algebra

A finite-dimensional algebra $R$ over a field $K$ such that the left $R$-modules $R$ and $\mathrm{Hom}_K(R,K)$ are isomorphic. In the language of representations this means that the left and right regular representations are equivalent. Every group algebra of a finite group over a field is a Frobenius algebra. Every Frobenius algebra is a quasi-Frobenius ring. The converse is not true. The following properties of a finite-dimensional $K$-algebra $R$ are equivalent:

1) $R$ is a Frobenius algebra;

2) there is a non-degenerate bilinear form $F : R \times R \rightarrow K$ such that $f(ab,c) = f(a,bc)$ for all $a,b,c \in R$;

3) if $L$ is a left and $H$ is a right ideal of $R$, then (see Annihilator) $$\mathfrak{Z}_{\mathrm{l}}(\mathfrak{Z}_{\mathrm{r}}(L)) = L,\ \ \ \mathfrak{Z}_{\mathrm{r}}(\mathfrak{Z}_{\mathrm{l}}(H)) = H \ ;$$ $$\dim_K \mathfrak{Z}_{\mathrm{r}}(L) + \dim_K L = \dim_K R = \dim_K \mathfrak{Z}_{\mathrm{l}}(H) + \dim_K K \ \.$$

Frobenius algebras essentially first appeared in the papers of G. Frobenius .

How to Cite This Entry:
Frobenius algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_algebra&oldid=37655
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article