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

TL;DR

This paper explores algebraic structures related to linear Poisson brackets in hydrodynamics, proposing a generalized Kac formula, calculating cohomology groups, and linking central extensions to quantization issues.

Contribution

It introduces a generalized Kac formula for Verma modules and computes second cohomology groups of generalized Virasoro algebras, connecting algebraic extensions to hydrodynamics quantization.

Findings

Generalized Kac formula for hydrodynamics-type brackets

Calculation of second cohomology groups of generalized Virasoro algebras

Link between central extensions and quantization of hydrodynamics brackets

Abstract

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.