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1.
对已知的Lie代数An-1作直接推广得到一类新的Lie代数gl(n,C).为应用方便,本文只考虑Lie代数gl(3,C)情形.构造了gl(3,C)的一个子代数,通过对阶数的规定,得到了一类新的loop代数.作为其应用,设计了一个新的等谱问题,得到了一个新的Lax对.利用屠格式获得了一族新的可积系统,具有双Hamilton结构,且是Liouville可积系.作为该方程族的约化情形,得到了新的耦合广义Schrdinger方程.
关键词:
Lie代数
可积系
Hamilton结构 相似文献
2.
A. Zuevsky 《Czechoslovak Journal of Physics》2004,54(5):501-507
Noncommutative counterparts of exactly solvable models are proposed on the basis of *-product continual Lie algebras. Examples of noncommutative Liouville and sine/sh-Gordon equations are discussed. 相似文献
3.
Using the technique of classical -matrices with spectral parameters we construct a general form of quantum Lax operators of interacting boson systems corresponding to an arbitrary simple (or reductive) Lie algebra. We prove quantum integrability of these models in the physically important case of and “diagonal” in the root basis classical -matrices. We consider in detail two classes of non-skew-symmetric classical -matrices with spectral parameters and obtain the corresponding quantum Lax operators and quantum integrable many-boson hamiltonians that generalize Bose–Hubbard dimer hamiltonians. 相似文献
4.
T. Skrypnyk 《Czechoslovak Journal of Physics》2003,53(11):1119-1124
Using special quasigraded Lie algebras we obtain new hierarchies of integrable equations in partial derivatives admitting zero-curvature representations. In particular, we obtain new type of so(3) anisotropic chiral-field equation along with its higher rank generalization. 相似文献
5.
We consider a vector space V over K=R or C, equipped with a skew symmetric bracket [⋅,⋅]:V×V→V and a 2-form ω:V×V→K. A simple change of the Jacobi identity to the form [A,[B,C]]+[C,[A,B]]+[B,[C,A]]=ω(B,C)A+ω(A,B)C+ω(C,A)B opens up new possibilities, which shed new light on the Bianchi classification of three-dimensional Lie algebras. 相似文献
6.
We consider special quasi-graded so(n)-valued Lie algebras on higher genus curves. Using them we find new Lie theoretical interpretation of the integrability of generalized Manakov tops and generalized Clebsch systems and explicitly construct their isomorphism on trajectories. 相似文献
7.
A new (in)finite-dimensional algebra which is a fundamental dynamical symmetry of a large class of (continuum or lattice) quantum integrable models is introduced and studied in details. Finite-dimensional representations are constructed and mutually commuting quantities—which ensure the integrability of the system—are written in terms of the fundamental generators of the new algebra. Relation with the deformed Dolan–Grady integrable structure recently discovered by one of the authors and Terwilliger's tridiagonal algebras is described. Remarkably, this (in)finite-dimensional algebra is a “q-deformed” analogue of the original Onsager's algebra arising in the planar Ising model. Consequently, it provides a new and alternative algebraic framework for studying massive, as well as conformal, quantum integrable models. 相似文献
8.
9.
Discrete integrable couplings associated with modified Korteweg——de Vries lattice and two hierarchies of discrete soliton equations
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A direct way to construct integrable couplings for discrete systems
is presented by use of two semi-direct sum Lie algebras. As their
applications, the discrete integrable couplings associated with
modified Korteweg--de Vries (m-KdV) lattice and two hierarchies of
discrete soliton equations are developed. It is also indicated that
the study of integrable couplings using semi-direct sums of Lie
algebras is an important step towards the complete classification of
integrable couplings. 相似文献
10.
George W. Bluman Omar Mrani-Zentar Deshin Finlay 《Journal of Nonlinear Mathematical Physics》2018,25(4):528-557
It is shown explicitly how one can obtain elements of Lie groups as compositions of products of other elements based on the commutator properties of associated Lie algebras. Problems of this kind can arise naturally in control theory. Suppose an apparatus has mechanisms for moving in a limited number of ways with other movements generated by compositions of allowed motions. Two concrete examples are: (1) the restricted parallel parking problem where the commutator of translations in y and rotations in the xy-plane yields translations in x. Here the control problem involves a vehicle that can only perform a series of translations in y and rotations with the aim of efficiently obtaining a pure translation in x; (2) involves an apparatus that can only perform rotations about two axes with the aim of performing rotations about a third axis. Both examples involve three-dimensional Lie algebras. In particular, the composition problem is solved for the nine three- and four-dimensional Lie algebras with non-trivial solutions. Three different solution methods are presented. Two of these methods depend on operator and matrix representations of a Lie algebra. The other method is a differential equation method that depends solely on the commutator properties of a Lie algebra. Remarkably, for these distinguished Lie algebras the solutions involve arbitrary functions and can be expressed in terms of elementary functions. 相似文献
11.
We apply the algebraic Bethe ansatz developed in our previous paper [C.S. Melo, M.J. Martins, Nucl. Phys. B 806 (2009) 567] to three different families of U(1) integrable vertex models with arbitrary N bond states. These statistical mechanics systems are based on the higher spin representations of the quantum group Uq[SU(2)] for both generic and non-generic values of q as well as on the non-compact discrete representation of the algebra. We present for all these models the explicit expressions for both the on-shell and the off-shell properties associated to the respective transfer matrices eigenvalue problems. The amplitudes governing the vectors not parallel to the Bethe states are shown to factorize in terms of elementary building blocks functions. The results for the non-compact model are argued to be derived from those obtained for the compact systems by taking suitable N→∞ limits. This permits us to study the properties of the non-compact model starting from systems with finite degrees of freedom. 相似文献
12.
A systematic construction for an action describing a class of supersymmetric integrable models as well as for pure fermionic theories is discussed in terms of the gauged WZNW model associated to half integer graded affine Kac–Moody algebras. Explicit examples of the N=1,2 super-sinh(sine)-Gordon models are discussed in detail. Pure fermionic theories arises for cosets sl(p,1)/sl(p)u(1) when a maximal kernel condition is fulfilled. The integrability condition for such models is discussed and it is shown that the simplest example when p=2 leads to the constrained Bukhvostov–Lipatov, Thirring, scalar massive and pseudo-scalar massless Gross–Neveu models. 相似文献
13.
In this paper we construct two free field realizations of the elliptic affine Lie algebra where . The first realization provides an analogue of Wakimoto’s construction for Affine Kac–Moody algebras, but in the setting of the elliptic affine Lie algebra. The second realization gives new types of representations analogous to Imaginary Verma modules in the Affine setting. 相似文献
14.
Matrix Lie Algebras and Integrable Couplings 总被引:2,自引:0,他引:2
ZHANG Yu-Feng GUO Fu-Kui 《理论物理通讯》2006,46(5):812-818
Three kinds of higher-dimensional Lie algebras are given which can be used to directly construct integrable couplings of the soliton integrable systems. The relations between the Lie algebras are discussed. Finally, the integrable couplings and the Hamiltonian structure of Giachetti-Johnson hierarchy and a new integrable system are obtained, respectively. 相似文献
15.
The Lie algebras of order have important applications for the fractional supersymmetry, and on the other hand the filiform Lie (super)algebras have very important properties into the Lie Theory. Thus, the aim of this work is to study filiform Lie algebras of order which were introduced in Navarro (2014). In this work we obtain new families of filiform Lie algebras of order 3, in which the complexity of the problem rises considerably respecting to the cases considered in Navarro (2014). 相似文献
16.
ZHANG Yu-Feng GUO Fu-Kui 《理论物理通讯》2006,46(11)
Three kinds of higher-dimensional Lie algebras are given which can be used to directly construct integrable couplings of the soliton integrable systems. The relations between the Lie algebras are discussed. Finally, the integrable couplings and the Hamiltonian structure of Giachetti-Johnson hierarchy and a new integrable system are obtained, respectively. 相似文献
17.
An integrable Hamiltonian hierarchy, a high-dimensional loop algebra and associated integrable coupling system
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A subalgebra of loop algebra ?_2 is established. Therefore, a new isospectral problem is designed. By making use of Tu's scheme, a new integrable system is obtained, which possesses bi-Hamiltonian structure. As its reductions, a formalism similar to the well-known Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and a generalized standard form of the Schr?dinger equation are presented. In addition, in order for a kind of expanding integrable system to be obtained, a proper algebraic transformation is supplied to change loop algebra ?_2 into loop algebra ?_1. Furthermore, a high-dimensional loop algebra is constructed, which is different from any previous one. An integrable coupling of the system obtained is given. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented. 相似文献
18.
In the paper,we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly.By the approach the various loop algebras of the Lie algebra A_1are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained,respectively.A reduction of the later hierarchy is just right the famous Ablowitz-Ladik hierarchy.Finally,via two different enlarging Lie algebras of the Lie algebra A_1,we derive two resulting differential-difference integrable couplings of the Toda hierarchy,of course,they are all various discrete expanding integrable models of the Toda hierarchy.When the introduced spectral matrices are higher degrees,the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple. 相似文献
19.
We revisit the notion of possible relativity or kinematic symmetries mutually connected through Lie algebra contractions under a new perspective on what constitutes a relativity symmetry. Contractions of an SO(m,n) symmetry as an isometry on an m+n dimensional geometric arena which generalizes the notion of spacetime are discussed systematically. One of the key results is five different contractions of a Galilean-type symmetry G(m,n) preserving a symmetry of the same type at dimension m+n−1, e.g. a G(m,n−1), together with the coset space representations that correspond to the usual physical picture. Most of the results are explicitly illustrated through the example of symmetries obtained from the contraction of SO(2,4), which is the particular case for our interest on the physics side as the proposed relativity symmetry for “quantum spacetime”. The contractions from G(1,3) may be relevant to real physics. 相似文献
20.
A 3? 3 matrix Lie algebra is first introduced, its subalgebras and the generated Lie algebras are obtained, respectively. Applications of a few Lie subalgebras give rise to two integrable nonlinear hierarchies of evolution equations from their reductions we obtain the nonlinear Schrödinger equations, the mKdV equations, the Broer-Kaup (BK) equation and its generalized equation, etc. The linear and nonlinear integrable couplings of one integrable hierarchy presented in the paper are worked out by casting a 3? 3 Lie subalgebra into a 2? 2 matrix Lie algebra. Finally, we discuss the elliptic variable solutions of a generalized BK equation. 相似文献