On the algebraic structures connected with the linear Poisson brackets of hydrodynamics type

The generalized form of the Kac formula for Verma modules associated with linear brackets of hydrodynamics type is proposed. Second cohomology groups of the generalized Virasoro algebras are calculated. Connection of the central extensions with the problem of quntization of hydrodynamics brackets is demonstrated. 1. Poisson brackets of hydrodynamics type (PBHT) {u(x), u(y)} = g(u(x))δ(x− y) + ukxb ij k (u(x))δ(x− y) (1) (here and below we assumed a summation on repeat indexes) were introduced and studied in [1, 2] to construct a theory of conservative systems of hydrodynamics type and to develop a Bogolubov-Whitham method of averaging Hamiltonian fieldtheoretic systems. We refer to the recent expository article [3] and the extensive bibliography therein. In [4] S P Novikov and first author considered and gave a classification of these Poisson brackets depending linearly on the fields u relative to linear change u = Ajw . Some examples were discussed in [5, 6]. For the reader’s convenience we recall some construction from [4]. The simplest local Lie algebras arising from the brackets of hydrodynamics type are especially interesting, where, according to [4], in the case when all metrics are linear in u we have g = C k u k + g 0 b k = const, g ij 0 = const (2) C k = b ij k + b ji k . ∗Permanent address: Technion-Israel Institute of Technology, Department of Mathematics, 32000 Haifa, ISRAEL