Frobenius Algebra

Three Definitions of the Frobenius Structure

Frobenius Algebra

A Frobenius algebra is a finite-dimensional -algebra equipped with a linear functional whose nullspace contains no nontrivial left ideals. This linear functional is called the Frobenius form. Alternatively, it is a finite-dimensional -algebra equipped with an associative nondegenerate pairing , we call this pairing the Frobenius pairing. Moreover, it can be defined as a finite-dimensional -algebra equipped with left (or right) -isomorphism to its dual .

Remark

The Frobenius form/pairing is part of the structure. i.e., an algebra can carry various Frobenius structures. A concrete example will be illustrated later.

Symmetric Frobenius Algebra

A Frobenius algebra is symmetric if one and hence all of the following equivalent conditions hold:

1. The Frobenius form is central. i.e., ;
2. The Frobenius pairing is symmetric;
3. The left -isomorphism is also right -linear;
4. The right -isomorphism is also left -linear.

e.g.

• The trivial Frobenius algebra with Frobenius form being the identity map. Clearly, there is no nontrivial ideals in the kernel.
• The matrix algebra of all matrices over is a symmetric Frobenius algebra with the usual trace map. Moreover, if we precompose the trace with a noncentral invertible matrix, we can obtain another Frobenius structure on . For example, in the case of , precomposing the trace by gives another valid Frobenius form on .
• Consider the group algebra over some finite group . The Frobenius form can be defined as follows: