Poisson Brackets Of Hydrodynamic Type Research Articles

On the central extensions of Poisson brackets of hydrodynamic type

Poisson brackets of hydrodynamic type were introduced to construct a theory of conservation system of hydrodynamic type. They lead to the algebraic and differentially geometrical theory of local translationally invariant Lie algebras of first order, Frobenius-type algebras and Novikov algebras. One of the main ways to construct new examples of these algebras and their corresponding Poisson brackets is through the central extensions. In Balinskii A A and Novikov S P (1985 Sov. Math. Dokl. 32 228–31), there is a general theory of the central extensions. In this paper, we give a further detailed study of two (non-trivial) types of Novikov algebras which directly decide an important kind of central extension of the Poisson brackets. We find that the Novikov algebras in the case τ = 3 are in the case τ = 0 and these two cases coincide in dimension ≤ 3. Moreover, they also coincide for the transitive cases in dimension 4.

Derivations on Novikov Algebras

Novikov algebras are nonassociative algebras introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. As one of the most important topics in the study of Novikov algebras, the derivation is related to many fields such as the vector fields, the automorphisms, the cohomology theory, and so on. In this paper, we study the derivations and inner derivations of Novikov algebras. We also give their classification in low dimensions.

On fermionic Novikov algebras

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in formal variational calculus. They are a class of left-symmetric algebras with commutative right multiplication operators, which can be viewed as bosonic. Fermionic Novikov algebras are a class of left-symmetric algebras with anti-commutative right multiplication operators. They correspond to a certain Hamiltonian superoperator in a supervariable. In this paper, we commence a study on fermionic Novikov algebras from the algebraic point of view. We will show that any fermionic Novikov algebra in dimension ≤3 must be bosonic. Moreover, we give the classification of real fermionic Novikov algebras on four-dimensional nilpotent Lie algebras and some examples in higher dimensions. As a corollary, we obtain kinds of four-dimensional real fermionic Novikov algebras which are not bosonic. All of these examples will serve as a guide for further development including the application in physics.

A Lie algebraic approach to Novikov algebras

Novikov algebras were introduced in connection with Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. The commutator of a Novikov algebra is a Lie algebra. Thus it is useful to relate the study of Novikov algebras to the theory of Lie algebras. In this paper, we will try to realize Novikov algebras through a Lie algebraic approach. Such a realization could be important in physics and geometry. We find that all transitive Novikov algebras in dimension ≤3 can be realized as the Novikov algebras obtained through Lie algebras and their compatible linear (global) deformations.

On the Novikov algebra structures adapted to the automorphism structure of a Lie group

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. The commutator of a Novikov algebra is a Lie algebra in which there exists a special affine structure (connection with zero curvature and torsion) defined by the Novikov algebra. For ensuring the consequences for the group structure, we need consider the more intrinsic connections defined by Novikov algebra structures, that is, the connections which are adapted to the automorphism structure of a Lie group. The resultant Novikov algebra is called a derivation algebra which satisfies every left multiplication operator is a derivation of its sub-adjacent Lie algebra. In this paper, we commence a study of the Novikov derivation algebras and as a consequence, we can construct Novikov algebras on some 2-solvable Lie algebras.

Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them

We construct integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type and solve the problem of the canonical form for a pair of compatible nonlocal Poisson brackets of hydrodynamic type. A system of equations describing compatible nonlocal Poisson brackets of hydrodynamic type is derived. This system can be integrated by the inverse scattering problem method. Any solution of this integrable system generates integrable bi-Hamiltonian systems of hydrodynamic type according to explicit formulas. We construct a theory of Poisson brackets of the special Liouville type. This theory plays an important role in the construction of integrable hierarchies.

On the compatible weakly nonlocal Poisson brackets of hydrodynamic type

We consider the pairs of general weakly nonlocal Poisson brackets of hydrodynamic type (Ferapontov brackets) and the corresponding integrable hierarchies. We show that, under the requirement of the nondegeneracy of the corresponding “first” pseudo‐Riemannian metric and also some nondegeneracy requirement for the nonlocal part, it is possible to introduce a “canonical” set of “integrable hierarchies” based on the Casimirs, momentum functional and some “canonical Hamiltonian functions.” We prove also that all the “higher” “positive” Hamiltonian operators and the “negative” symplectic forms have the weakly nonlocal form in this case. The same result is also true for “negative” Hamiltonian operators and “positive” symplectic structures in the case when both pseudo‐Riemannian metrics and are nondegenerate.

Bilinear Forms on Novikov Algebras

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. On the other hand, there can be geometry and Lagrangian mechanics on homogenous spaces related to Novikov algebras. The nondegenerate symmetric bilinear forms on Novikov algebras can be regarded as the pseudometrics, and some additional identities for these forms correspond to some “conserved quantities.” In particular, there is an important kind of “conserved” nondegenerate symmetric bilinear forms that correspond to the pseudo-Riemannian connections such that parallel translation preserves the bilinear form on the tangent spaces. Moreover, the fact that the left multiplication operators form a Lie algebra for a Novikov algebra is compatible with such a form. However, we show in this note that there are no such forms on most Novikov algebras in low dimensions.

INTEGRABILITY OF THE EQUATIONS FOR NONSINGULAR PAIRS OF COMPATIBLE FLAT METRICS

We solve the problem of describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) in the general N-component case. This problem is equivalent to the problem of describing all compatible Dubrovin–Novikov brackets (compatible nondegenerate local Poisson brackets of hydrodynamic type) playing an important role in the theory of integrable systems of hydrodynamic type and also in modern differential geometry and field theory. We prove that all nonsingular pairs of compatible flat metrics are described by a system of nonlinear differential equations that is a special nonlinear differential reduction of the classical Lame equations, and we present a scheme for integrating this system by the method of the inverse scattering problem. The integration procedure is based on using the Zakharov method for integrating the Lame equations (a version of the inverse scattering method).

On a special class of compatible Poisson structures of hydrodynamic type

We consider the problem of classifying all compatible Poisson brackets of hydrodynamic type, one of which is linear in one of the coordinates and the other its derivative with respect to that coordinate.

Compatible Poisson brackets of hydrodynamic type

Some general properties of compatible Poisson brackets of hydrodynamic type are discussed, in particular: (1) an invariant differential-geometric criterion of the compatibility based on the Nijenhuis tensor; (2) the Lax pair with a spectral parameter governing compatible Poisson brackets in the diagonalizable case; (3) the connection of this problem with the class of surfaces in Euclidean space which possess nontrivial deformations preserving the Weingarten operator.

Hamiltonian systems of hydrodynamic type and constant curvature metrics

We consider the one-dimensional systems of hydrodynamic type. The general relations for the nonlocal Poisson brackets of hydrodynamic type are presented. For a nondegenerate nonlocal Poisson bracket of hydrodynamic type the relations mean exactly that the bracket is generated by a metric of constant Riemannian curvature.

ON SOME CONNECTIONS AND EXTENSIONS OF W-ALGEBRAS

In this note we briefly discuss the connections of W-algebras with the local characteristic integrals of the generalized Toda lattices, with the Poisson brackets of hydrodynamic type, as well as the supersymmetric extensions of these algebras and their generalizations associated with non-Abelian Toda systems.

Differential-geometric Poisson brackets on a lattice

The concept of differential-geometric Poisson brackets (DGPB) was introduced in [i] in connection with an investigation of the properties of Poisson brackets of hydrodynamic type [2] and their generalizations. Recall that homogeneous DGPBs of m-th order on a phase space of fields u i @),i = i ..... N,x ~ ~ (in this note we confine attention to the spatially onedimensional case), taking values in a manifold~ N, are defined by