The Frobenius-Virasoro algebra and Euler equations-II: Multi-component cases

Let WA be the vector field Lie algebra with values in the Frobneius algebra A of rank N. We show that the second continuous cohomology group is H2(WA,K)=⨁k=1NKωk,K=ℂorA,where ω1,…,ωN are ℂ-valued basic Gelfand-Fuchs cocycles. As an application, we present a generalized Virasoro algebra virA[α1,…,αk]. We also consider another kind of generalization and propose a multi-component Frobenius-Virasoro algebra virA[n] and study its related Euler-type equations.