Frobenius algebra

In mathematics, a Frobenius algebra is an associative algebra A defined over a field K equipped with a special kind of bilinear form

$\sigma :A\times A\rightarrow K$,

then called a Frobenius form of the algebra. It is required to have the two following properties.

• Associativity: for every elements $a,b,c$ of $A$,

$\sigma (ab,c)=\sigma (a,bc)$

• Non-degeneracy:

$(\forall a\in A,\;\sigma (a,b)=0)\quad \Rightarrow \quad b=0$.

Example

Any matrix algebra defined over a field $K\,$ is a Frobenius algebra with Frobenius form given as

$\sigma (A,B)=\mathrm {tr} (AB)\,$,

where $\mathrm {tr}$ denotes the matrix trace.

Category-theoretical definition

In category theory, the notion of Frobenius object is an abstract definition of a Frobenius algebra in a category. A Frobenius object $(A,\mu ,\eta ,\delta ,\varepsilon )$ in a monoidal category $(C,\otimes ,I)$ consists of an object A of C together with four morphisms

$\mu :A\otimes A\to A,\qquad \eta :I\to A,\qquad \delta :A\to A\otimes A\qquad \mathrm {and} \qquad \varepsilon :A\to I$

such that

• $(A,\mu ,\eta )\,$ is a monoid object in $C\,$,

• $(A,\delta ,\varepsilon )$ is a comonoid object in $C\,$,

• the diagrams

and

commute (for simplicity the diagrams are given here in the case where the monoidal category $C\,$ is strict).

Applications

Frobenius algebras occur in the representation theory of algebras.

More recently, it has been seen that they play an important rôle in the algebraic treatment and axiomatic foundation of topological quantum field theory. A commutative Frobenius algebra namely determines uniquely (up to isomorphism) a 2-dimensional TQFT. More precisely, the category of commutative Frobenius K-algebras is equivalent to the category of symmetric strong monoidal functors from 2-Cob (the category of 2-dimensional cobordisms) to Vect_K (the category of vector spaces over K).

See also

• Frobenius norm
• Frobenius inner product