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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:
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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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f0417508.png" /> is a Frobenius algebra;
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1) $R$ 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" />;
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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 <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]])
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3) if $L$ is a left and $H$ is a right ideal of $R$, then (see [[Annihilator]])
 
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$$
<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>
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\mathfrak{Z}_{\mathrm{l}}(\mathfrak{Z}_{\mathrm{r}}(L)) = L,\ \ \ \mathfrak{Z}_{\mathrm{r}}(\mathfrak{Z}_{\mathrm{l}}(H)) = H \ ;
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041750/f04175016.png" /></td> </tr></table>
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$$
 +
\dim_K \mathfrak{Z}_{\mathrm{r}}(L) + \dim_K L = \dim_K R = \dim_K \mathfrak{Z}_{\mathrm{l}}(H) + \dim_K K \ \.
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$$
  
 
Frobenius algebras essentially first appeared in the papers of G. Frobenius [[#References|[3]]].
 
Frobenius algebras essentially first appeared in the papers of G. Frobenius [[#References|[3]]].
  
 
====References====
 
====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>
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<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>
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<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>
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</table>
  
  
  
 
====Comments====
 
====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.
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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.
 
Examples of symmetric algebras are semi-simple algebras and group algebras.
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 +
{{TEX|done}}

Latest revision as of 15:37, 5 February 2016

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.

How to Cite This Entry:
Frobenius algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_algebra&oldid=14845
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article