The Frobenius-Virasoro algebra and Euler equations

We introduce an $\mathfrak{F}$-valued generalization of the Virasoro algebra, called the Frobenius-Virasoro algebra $\mathfrak{vir_F}$, where $\mathfrak{F}$ is a Frobenius algebra over $\mathbb{R}$. We also study Euler equations on the regular dual of $\mathfrak{vir_F}$, including the $\mathfrak{F}$-$\mathrm{KdV}$ equation and the $\mathfrak{F}$-$\mathrm{CH}$ equation and the $\mathfrak{F}$-$\mathrm{HS}$ equation, and discuss their Hamiltonian properties.