On Central Extension Of Two Kinds Of Infinite-dimensional Lie Algebras And Their Related Integrable Systems

We compute the second continuous cohomology group of Frobenius-Virasoro alge-bra and Heisenberg-Virasoro algebra.We show that the dimensions of these two groups are N and(N+1)(N+2)/2.Following this we obtain the central extension of these two Lie algebras.We also study related integrable systems with local bi-hamiltonian structures.