Frobenius algebra
A finite-dimensional algebra over a field
such that the left
-modules
and
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
-algebra
are equivalent:
1) is a Frobenius algebra;
2) there is a non-degenerate bilinear form such that
for all
;
3) if is a left and
is a right ideal of
, then (see Annihilator)
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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 to be Frobenius is that there is a linear form
on
such that if
for all
then
. If, moreover,
satisfies
for all
, then
is called a symmetric algebra.
Examples of symmetric algebras are semi-simple algebras and group algebras.
Frobenius algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_algebra&oldid=37655