The Frobenius–Virasoro algebra and Euler equations

We introduce an FF-valued generalization of the Virasoro algebra, called the Frobenius–Virasoro algebra virFvirF, where FF is a Frobenius algebra over RR. We also study Euler equations on the regular dual of virFvirF, including the FF-KdV equation and the FF-CH equation and the FF-HS equation, and discuss their Hamiltonian properties.