Gelfand-Dickey Bracket Research Articles

Hamiltonian structures of generalized Manin–Radul super-KdV and constrained super KP hierarchies

A study of Hamiltonian structures associated with supersymmetric Lax operators is presented. Following a constructive approach, the Hamiltonian structures of Inami–Kanno super-KdV hierarchy and constrained modified super-KP hierarchy are investigated from the reduced supersymmetric Gelfand–Dickey brackets. By applying a gauge transformation on the Hamiltonian structures associated with these two nonstandard super-Lax hierarchies, we obtain the Hamiltonian structures of generalized Manin–Radul super-KdV and constrained super-KP hierarchies. We also work out a few examples and compare them with the known results.

Matrix formulation of Hamiltonian structures of constrained KP hierarchy

We give a matrix formulation of the Hamiltonian structures of constrained KP hierarchy. First, we derive from the matrix formulation the Hamiltonian structure of the one-constraint KP hierarchy, which was originally obtained by Oevel and Strampp. We then generalize the derivation to the multiconstraint case and show that the resulting bracket is actually the second Gelfand–Dickey bracket associated with the corresponding Lax operator. The matrix formulation of the Hamiltonian structure of the one-constraint KP hierarchy in the form introduced in the study of matrix model is also discussed.

ON THE MIURA AND BÄCKLUND TRANSFORMATIONS ASSOCIATED WITH THE SUPERSYMMETRIC GELFAND–DICKEY BRACKET

The supersymmetric version of the Miura and Bäcklund transformations associated with the supersymmetric Gelfand–Dickey bracket are investigated from the point of view of the Kupershmidt–Wilson theorem.

The generalized KP hierarchy, Hamiltonian structure and non-abelian two-boson realization

A matrix generalization of the KP hierarchy is formulated with the help of matrix pseudodifferential operator. While the first Gelfand-Dickey bracket is found to be a non-abelian generalization of the W1 + ∞ algebra, the corresponding second one is found to be nonlinear and nonlocal. Finally, the constrained system corresponding to this hierarchy is considered. In a particular case the resulting equations turn out to be a multicomponent generalization of the Nonlinear Schrödinger equation. It is also demonstrated that a matrix generalization of the usual two-boson realization exists in the present case.

Classical extended conformal algebras associated with constrained KP hierarchy

The conformal property of the second Hamiltonian structure of constrained KP hierarchy derived by Oevel and Strampp is examined. It is found that it naturally gives a family of nonlocal extended conformal algebras. Two examples of such algebras are given and it is found that they are similar to Bilal’s V algebra. By taking a gauge transformation one can map the constrained KP hierarchy to Kupershmidt’s nonstandard Lax hierarchy. The second Hamiltonian structure in this representation is considered. It is shown that after mapping the Lax operator to a pure differential operator the second structure becomes the sum of the second and the third Gelfand–Dickey brackets defined by this differential operator. The corresponding conformally covariant form of this differential operator is also given.

The constrained KP hierarchy and the generalised Miura transformation

Recently much attention has been paid to the restriction of KP to the submanifold of operators which can be represented as a ratio of two purely differetial operators L = AB f-1 . Whereas most of the aspects concerning this reduced hierarchy, like the Lax flows and the Hamiltonians, are by now well understood, there still lacks a clear and conclusive statement about the associated Poisson structure. We fill this gap by placing the problem in a more general framework and then showing how the required result follows from an interesting property of the second Gelfand-Dickey brackets under multiplication and inversion of Lax operators. As a byproduct we give an elegant and simple proof of the generalised Kuperschmidt-Wilson theorem.

NONLOCAL MATRIX GENERALIZATIONS OF N=2 SUPER VIRASORO ALGEBRA

We study the generalization of the second Gelfand-Dickey bracket to the superdifferential operators with matrix-valued coefficients. The associated matrix Miura transformation is derived. Using this bracket we work out a nonlocal and nonlinear N=2 superalgebra which contains the N=2 super Virasoro algebra as a subalgebra. The bosonic limit of this superalgebra is considered. We show that when the spin-1 fields in this bosonic algebra are set to zero the resulting Dirac bracket gives precisely the recently derived V2,2 algebra.

Affine Gelfand-Dickey brackets and holomorphic vector bundles

We define the (second) Adler-Gelfand-Dickey Poisson structure on differential operators over an elliptic curve and classify symplectic leaves of this structure. This problem leads to the problem of classification of coadjoint orbits for double loop algebras, conjugacy classes in loop groups, and holomorphic vector bundles over the elliptic curve. We show that symplectic leaves have a finite but (unlike the traditional case of operators on the circle) arbitrarily large codimension, and compute it explicitly.

On the Gelfand–Dickey algebra GD(SLn) and the W n-symmetry. I. The bosonic case

The algebra Ξ of nonlinear (local and nonlocal) differential operators, acting on the ring of analytic functions, is studied. It is shown in particular that this space splits into 3×2 special subalgebras ∑jr, j=0,±1, r=±1. Each subalgebra is completely specified by quantum numbers s and (p,q) describing the conformal spin, and the lowest and the highest degrees, respectively. The algebra ∑++ (and its dual ∑−−) of local (pure nonlocal) differential operators is used to calculate the general expression of the Gelfand–Dickey bracket and the Wn-symmetry Poisson, one in terms of a set of spin j canonical fields uj, 2≤j≤n and a nonlinear u-cubic dependent differential operator D(n,i,j;z,u). The explicit form of this operator is worked out. Other remarkable features are also discussed.

Superconformal covariantization of superdifferential operator on (1‖1) superspace and classical N=2 W superalgebras

A study of the superconformal covariantization of superdifferential operators defined on (1‖1) superspace is presented. It is shown that a superdifferential operator with a particular type of constraint can be covariantized only when it is of odd order. In such a case, the action of superconformal transformation on the superdifferential operator is nothing but a Hamiltonian flow defined by the corresponding supersymmetric second Gelfand–Dickey bracket. The covariant form of a superdifferential operator of odd order is given.

On diff(S1) covariantization of pseudodifferential operator

A study of diff(S1) covariant properties of a pseudodifferential operator of integer degree is presented. First, it is shown that the action of diff(S1) defines a Hamiltonian flow defined by the second Gelfand–Dickey bracket if and only if the pseudodifferential operator transforms covariantly. Second, the covariant form of a pseudodifferential operator of degree n ≠ 0,±1 is constructed by exploiting the inverse of the covariant derivative. This, in particular, implies the existence of a primary basis for WKP(n) (n ≠ 0,±1).

The second Gelfand-Dickey bracket as a bracket on a Poisson-Lie Grassmannian

We introduce a Poisson structure on a Grassmannian Gr k (V) on which the Poisson-Lie group GL(V) acts in a Poisson-Lie way. We discuss the analytic complications connected with the infinite-dimensional caseV=C ∞(ℝ) and show that an open subset of Gr k (V) with this Poisson structure is isomorphic to the Gelfand-Dickey manifold of differential operators of orderk with the second Gelfand-Dickey bracket. In fact we introduce as a consequence a Poisson-Lie action of an enormous group on the Gelfand-Dickey manifold generalizing (on the semiclassical level) the Sugavara inclusion.