Frobenius algebra

Remarks

Example: the matrix algebra $M_n(k)$ with $\sigma(a, b) = \tr(ab)$. Clause 2 holds because both sides equal $\tr(abc)$. Clause 1 holds because $\tr(a E_{ji}) = a_{ij}$ for the matrix unit $E_{ji}$, so $\sigma(a, \cdot) = 0$ forces $a = 0$. Another example is the group algebra $k[G]$ of a finite group $G$, with $\sigma(a, b)$ the coefficient of the identity element of $G$ in $ab$. Non-example: $A = k[x, y]/(x^2, xy, y^2)$, with basis $1, x, y$, is not Frobenius. If $\sigma$ satisfies clause 2, then $\sigma(a, b) = \lambda(ab)$ for the linear form $\lambda = \sigma(1, \cdot)$. The plane spanned by $x$ and $y$ meets $\ker \lambda$ in some nonzero $u = \alpha x + \beta y$. Since $ub \in k u$ for every $b \in A$, $\sigma(u, b) = 0$ for all $b$, and clause 1 fails. Equivalent forms: $A$ is Frobenius if and only if some linear form $\lambda \colon A \to k$ has a kernel that contains no nonzero left ideal. This holds if and only if $A \cong \Hom_k(A, k)$ as left $A$-modules. If moreover $\sigma$ is symmetric, $A$ is called a symmetric algebra.