1,721,072 research outputs found
Non-local Hamiltonian structures and applications to the theory of integrable systems I
No full textWe develop a rigorous theory of non-local Hamiltonian structures, built on the notion of a non-local Poisson vertex algebra. As an application, we find conditions that guarantee applicability of the Lenard-Magri scheme of integrability to a pair of compatible non-local Hamiltonian structures
Finite W-algebras for gl_N
Full text linkWe study the quantum finite W-algebras W(gl_N,f), associated to the Lie
algebra gl_N, and its arbitrary nilpotent element f. We construct for such an
algebra an r_1 x r_1 matrix L(z) of Yangian type, where r_1 is the number of
maximal parts of the partition corresponding to f. The matrix L(z) is the
quantum finite analogue of the operator of Adler type which we introduced in
the classical affine setup. As in the latter case, the matrix L(z) is obtained
as a generalized quasideterminant. It should encode the whole structure of
W(gl_N,f), including explicit formulas for generators and the commutation
relations among them. We describe in all detail the examples of principal,
rectangular and minimal nilpotent elements
Multiplets of representations, twisted Dirac operators and Vogan's conjecture in affine setting
No full textWe extend classical results of Kostant et al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting an analogue of Vogan's conjecture on infinitesimal characters of Harish-Chandra modules in terms of Dirac cohomology, For Our calculations we use the machinery of Lie conformal and vertex algebras. (C) 2007 Elsevier Inc. All rights reserved
Irreducible modules over finite simple Lie pseudoalgebras II. Primitive pseudoalgebras of type K
No full textOne of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra. A Lie pseudoalgebra is a generalization of the notion of a Lie conformal algebra for which C[partial derivative] is replaced by the universal enveloping algebra H of a finite-dimensional Lie algebra. The finite (i.e., finitely generated over H) simple Lie pseudoalgebras were classified in our previous work (Bakalov et al., 2001) [2]. The present paper is the second in our series on representation theory of simple Lie pseudoalgebras. In the first paper we showed that any finite irreducible module over a simple Lie pseudoalgebra of type W or S is either an irreducible tensor module or the kernel of the differential in a member of the pseudo de Rham complex. In the present paper we establish a similar result for Lie pseudoalgebras of type K. with the pseudo de Rham complex replaced by a certain reduction called the contact pseudo de Rham complex. This reduction in the context of contact geometry was discovered by Rumin
Lie Conformal Algebra Cohomology and the Variational Complex
No full textWe find an interpretation of the complex of variational calculus in terms of the Lie conformal
algebra cohomology theory. This leads to a better understanding of both theories. In par-
ticular, we give an explicit construction of the Lie conformal algebra cohomology complex,
and endow it with a structure of a g-complex. On the other hand, we give an explicit con-
struction of the complex of variational calculus in terms of skew-symmetric poly-differential
operators
Calculus structure on the Lie conformal algebra complex and the variational complex
No full textWe construct a calculus structure on the Lie conformal algebra cochain complex. By restricting to degree one chains, we recover the structure of a \documentclass[12pt]{minimal}\begin{document}\end{document}g-complex introduced in [A. De Sole and V. G. Kac, Commun. Math. Phys. 292, 667 (2009)10.1007/s00220-009-0886-1]. A special case of this construction is the variational calculus, for which we provide explicit formulas.Alberto De Sole, Pedram Hekmati, and Victor G. Ka
Non-local Poisson structures and applications to the theory of integrable systems II
Full text linkWe develop further the Lenard-Magri scheme of integrability for a pair of compatible non-local Poisson structures, which we discussed in Part I. We apply this scheme to several such pairs, proving thereby integrability of various evolution equations, as well as hyperbolic equations. Some of these equations may be new
Poisson vertex algebras in the theory of Hamiltonian equations
No full textWe lay down the foundations of the theory of Poisson vertex algebras aimed at its
applications to integrability of Hamiltonian partial differential equations. Such an equation
is called integrable if it can be included in an infinite hierarchy of compatible Hamiltonian
equations, which admit an infinite sequence of linearly independent integrals of motion in
involution. The construction of a hierarchy and its integrals of motion is achieved by making
use of the so called Lenard scheme. We find simple conditions which guarantee that the
scheme produces an infinite sequence of closed 1-forms ωj , j ∈ Z+, of the variational complex Ω. If these forms are exact, i.e. ωj are variational derivatives of R some local functionals
hj, then the latter are integrals of motion in involution of the hierarchy formed by the
corresponding Hamiltonian vector fields. We show that the complex Ω
is exact, provided
that the algebra of functions V is “normal”; in particular, for arbitrary V, any closed form
in Ω
becomes exact if we add to V a finite number of antiderivatives. We demonstrate
on the examples of the KdV, HD and CNW hierarchies how the Lenard scheme works.
We also discover a new integrable hierarchy, which we call the CNW hierarchy of HD type.
Developing the ideas of Dorfman, we extend the Lenard scheme to arbitrary Dirac structures,
and demonstrate its applicability on the examples of the NLS, pKdV and KN hierarchies
On classification of poisson vertex algebras
No full textDedicated to Vladimir Morozov on the 100th anniversary of his birth.We describe a conjectural classification of Poisson vertex algebras of CFT type and of Poisson vertex algebras in one differential variable (= scalar Hamiltonian operators).Massachusetts Institute of Technology. Dept. of MathematicsNational Science Foundation (U.S.
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