View source for Frobenius algebra

A finite-dimensional algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f0417501.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f0417502.png" /> such that the left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f0417503.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f0417504.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f0417505.png" /> 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|quasi-Frobenius ring]]. The converse is not true. The following properties of a finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f0417506.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f0417507.png" /> are equivalent:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f0417508.png" /> is a Frobenius algebra;

2) there is a non-degenerate bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f0417509.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175011.png" />;

3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175012.png" /> is a left and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175013.png" /> is a right ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175014.png" />, then (see [[Annihilator|Annihilator]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175015.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175016.png" /></td> </tr></table>

Frobenius algebras essentially first appeared in the papers of G. Frobenius [[#References|[3]]].

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



====Comments====
A criterion for an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175017.png" /> to be Frobenius is that there is a linear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175019.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175020.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175021.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175022.png" />. If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175023.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175024.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175025.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175026.png" /> is called a symmetric algebra.

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