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 [3].

References

[1]	C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)
[2]	C. Faith, "Algebra: rings, modules and categories" , 1–2 , Springer (1973–1976)
[3]	G. Frobenius, "Theorie der hyperkomplexen Grössen" Sitzungsber. Königl. Preuss. Akad. Wiss. : 24 (1903) pp. 504–537; 634–645

Comments

A criterion for an algebra $A$ to be Frobenius is that there is a linear form $\phi$ on $A$ such that if $\phi(ab) = 0$ for all $a \in A$ then $b = 0$. If, moreover, $\phi$ satisfies $\phi(ab) = \phi(ba)$ for all $a,b \in A$, then $A$ is called a symmetric algebra.

Examples of symmetric algebras are semi-simple algebras and group algebras.