Frobenius Algebra

Three Definitions of the Frobenius Structure

Frobenius Algebra

A Frobenius algebra is a finite-dimensional $\mathbb{K}$-algebra $A$ equipped with a linear functional $\epsilon :A\to \text{K}$ whose nullspace contains no nontrivial left ideals. This linear functional $\epsilon$ is called the Frobenius form.
Alternatively, it is a finite-dimensional $\mathbb{K}$-algebra equipped with an associative nondegenerate pairing $\beta :A\otimes A\to \text{K}$, we call this pairing the Frobenius pairing.
Moreover, it can be defined as a finite-dimensional $\mathbb{K}$-algebra $A$ equipped with left (or right) $A$-isomorphism to its dual $A^{\ast }$.

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

1. The Frobenius form $\epsilon :A\to \text{K}$ is central. i.e., $\epsilon (ab)=\epsilon (ba)$;
2. The Frobenius pairing $\beta$ is symmetric;
3. The left $A$-isomorphism is also right $A$-linear;
4. The right $A$-isomorphism is also left $A$-linear.
   

e.g.

• The trivial Frobenius algebra $\text{K}$ with Frobenius form being the identity map. Clearly, there is no nontrivial ideals in the kernel.
• The matrix algebra ${\mathrm{Mat}}_n(\text{K})$ of all $n\times n$ matrices over $\text{K}$ 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 ${\text{Mat}}_n(\text{K})$. For example, in the case of ${\text{Mat}}_n(\mathbb{R})$, precomposing the trace by $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ gives another valid Frobenius form on ${\text{Mat}}_n(\mathbb{R})$.
• Consider the group algebra $\text{K}G$ over some finite group $G=\{g_i{\}}_{0\le i\le n}$. The Frobenius form can be defined as follows: $\epsilon :\text{K}G\to \text{K},g_0\mapsto 1,g_j\mapsto 0\text{ for }j\ne 0.$
• Let $X$ be a compact oriented manifold. The cohomology ring $H^{\ast }(X;\text{K})$ with the cup product is a Frobenius algebra with the Frobenius form given by integration over the fundamental class of $X$: $\epsilon :H^{\ast }(X;\text{K})\to \text{K},\alpha \mapsto {\int }_{[X]}\alpha .$