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1.
It is shown that the F
4 rational and trigonometric integrable systems are exactly solvable for arbitrary values of the coupling constants. Their spectra are found explicitly while eigenfunctions are obtained by pure algebraic means. For both systems new variables are introduced in which the Hamiltonian has an algebraic form being also (block)-triangular. These variables are a certain invariants of the F
4 Weyl group. Both Hamiltonians preserve the same (minimal) flag of spaces of polynomials, which is found explicitly. 相似文献
2.
An algebraic analysis of the Hamiltonian formulation of the model two-dimensional gravity is performed. The crucial fact is
an exact coincidence of the Poisson brackets algebra of the secondary constraints of this Hamiltonian formulation with the
SO(2,1)-algebra. The eigenvectors of the canonical Hamiltonian H
c
are obtained and explicitly written in closed form. 相似文献
3.
4.
G. C. Sherry 《International Journal of Theoretical Physics》1990,29(10):1115-1126
Hamiltonian dynamics can be formulated entirely in terms of a Poisson manifold, that is, one for which the algebra of smooth functions is a Poisson algebra. The latter is a commutative associative algebraA together with a skew-symmetric bracket which is a derivation onA. It is shown that a Poisson algebra can be generalized by replacingA by algebras which do not necessarily commute. These allow for algebraic generalizations of Hamiltonian dynamics in both classical and quantum forms. Particular examples are models of classical and quantum electrons. 相似文献
5.
6.
O.I. Bogoyavlenskij 《Communications in Mathematical Physics》2005,254(2):479-488
A hidden additional algebraic structure is discovered for the Lie algebra of symmetries of any dynamical system V. The structure is based on the properties of the Lie derivative operator LV and on a hidden canonical flag structure in the eigenspaces of any linear operator. 相似文献
7.
F. Finkel D. Gómez-Ullate A. González-López M.A. Rodríguez R. Zhdanov 《Communications in Mathematical Physics》2003,233(2):191-209
The B
N
hyperbolic Sutherland spin model is expressed in terms of a suitable set of commuting Dunkl operators. This fact is exploited
to derive a complete family of commuting integrals of motion of the model, thus establishing its integrability. The Dunkl
operators are shown to possess a common flag of invariant finite-dimensional linear spaces of smooth scalar functions. This
implies that the Hamiltonian of the model preserves a corresponding flag of smooth spin functions. The discrete spectrum of
the restriction of the Hamiltonian to this spin flag is explicitly computed by triangularization. The integrability of the
hyperbolic Sutherland spin chain of B
N
type associated with the dynamical model is proved using Polychronakos's ``freezing trick'.
Received: 14 February 2002 / Accepted: 19 June 2002 Published online: 10 December 2002
RID="*"
ID="*" Corresponding author. E-mail: artemio@fis.ucm.es
RID="**"
ID="**" On leave of absence from Institute of Mathematics, 3 Tereschenkivska St., 01601 Kyiv-4 Ukraine
Communicated by L. Takhtajan 相似文献
8.
Bruno Gruber 《Letters in Mathematical Physics》1982,6(5):329-334
In this article two theorems are given which permit, together with the concept of a representation vector diagram, to classify all (linear) finite-dimensional representations of the algebra and group E
2 which are induced by a master representation on the place of the universal enveloping algebra of the algebra E
2. Apart from a classification of the finite-dimensional representations, the two theorems make it possible to obtain the matrix elements of these representations for both, algebra and group, in explicit form. The material contained in this letter forms part of an analysis of indecomposable (finite- and infinite-dimensional) representations of the algebra and group E
2 which is contained in Reference [1]. No proofs will be given in this letter. We refer instead to [1]. 相似文献
9.
It is shown that every finite-dimensional irreducible module over the general linear Lie superalgebragl(n/1) can be deformed to an irreducible module ofU
q
[gl(n/1)], aq-analogue of the universal enveloping algebra ofgl(n/1). The results are extended also to all Kac modules, which in the atypical cases remain indecomposible. Within each module expressions for the transformations of the Gel'fand-Zetlin basis under the action of the algebra generators are written down. An analogoue of the Poincaré-Birkhoff-Witt theorem is formulated. 相似文献
10.
11.
We extend significantly previous works on the Hilbert space representations of the generalized uncertainty principle (GUP) in 3 + 1 dimensions of the form \([X_i,P_j] = i F_{ij}\) where \(F_{ij} = f({\mathbf {P}}^2) \delta _{ij} + g({\mathbf {P}}^2) P_i P_j\) for any functions f. However, we restrict our study to the case of commuting X’s. We focus in particular on the symmetries of the theory, and the minimal length that emerge in some cases. We first show that, at the algebraic level, there exists an unambiguous mapping between the GUP with a deformed quantum algebra and a quadratic Hamiltonian into a standard, Heisenberg algebra of operators and an aquadratic Hamiltonian, provided the boost sector of the symmetries is modified accordingly. The theory can also be mapped to a completely standard Quantum Mechanics with standard symmetries, but with momentum dependent position operators. Next, we investigate the Hilbert space representations of these algebraically equivalent models, and focus specifically on whether they exhibit a minimal length. We carry the functional analysis of the various operators involved, and show that the appearance of a minimal length critically depends on the relationship between the generators of translations and the physical momenta. In particular, because this relationship is preserved by the algebraic mapping presented in this paper, when a minimal length is present in the standard GUP, it is also present in the corresponding Aquadratic Hamiltonian formulation, despite the perfectly standard algebra of this model. In general, a minimal length requires bounded generators of translations, i.e. a specific kind of quantization of space, and this depends on the precise shape of the function f defined previously. This result provides an elegant and unambiguous classification of which universal quantum gravity corrections lead to the emergence of a minimal length. 相似文献
12.
We construct an algebraic star product on the minimal nilpotent coadjoint orbit of a simple complex Lie group with a Lie algebra which is not of typeA
n. According to the deformation program, we study the representations of the Lie algebra associated to this orbit. 相似文献
13.
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, and the radical of a finite-dimensional Novikov algebra is transitive. In this paper, we give a classification of transitive Novikov algebras on four-dimensional nilpotent Lie algebras based on Kim (1986, Journal of Differential Geometry
24, 373–394). 相似文献
14.
A certain generalization of the algebra gl(N,
) of first-order differential operators acting on a space of inhomogeneous polynomials in
N−1 is constructed. The generators of this (non-) Lie algebra depend on permutation operators. It is shown that the Hamiltonian of the N-body Calogero model can be represented as a second-order polynomial in the generators of this algebra. The representation given implies that the Calogero Hamiltonian possesses infinitely-many finite-dimensional invariant subspaces with explicit bases, which are closely related to the finite-dimensional representations of the above algebra. This representation is an alternative to the standard representation of the Bargmann-Fock type in terms of creation and annihilation operators. 相似文献
15.
C. B. Collins 《General Relativity and Gravitation》1995,27(9):973-979
An equation of spinor algebra, which is specified by two positive integers,M andN, is solved by relating it to the problem of integrating a two-dimensional Hamiltonian homogeneous polynomial system of ordinary differential equations, whose degree isN}-1. The case in whichN=1 reduces to a well-known result of spinor algebra. The caseM=N=4 is of relevance in the study of symmetry operators of Maxwell's equations on a curved space-time. It is also shown, using spinor notation, that the first integral for a general two-dimensional Hamiltonian system of ordinary differential equations (whether polynomial or analytic) is determinable in a purely algebraic manner, i.e., by using no integration. 相似文献
16.
17.
A general construction of an sh Lie algebra (L
∞-algebra) from a homological resolution of a Lie algebra is given. It is applied to the space of local functionals equipped
with a Poisson bracket, induced by a bracket for local functions along the lines suggested by Gel'fand, Dickey and Dorfman.
In this way, higher order maps are constructed which combine to form an sh Lie algebra on the graded differential algebra
of horizontal forms. The same construction applies for graded brackets in field theory such as the Batalin-Fradkin-Vilkovisky
bracket of the Hamiltonian BRST theory or the Batalin-Vilkovisky antibracket.
Received: 5 March 1997 / Accepted: 21 May 1997 相似文献
18.
The classical Frobenius-Schur duality gives a correspondence between finite dimensional representations of the symmetric and
the linear groups. The goal of the present paper is to extend this construction to the quantum toroidal setup with only elementary
(algebraic) methods. This work can be seen as a continuation of [J, D1 and C2] (see also [C-P and G-R-V]) where the cases
of the quantum groups U
q
(sl(n)), Y(sl(n)) (the Yangian) and U
q
(sl(n)) are given. In the toroidal setting the two algebras involved are deformations of Cherednik's double affine Hecke algebra
introduced in [C1] and of the quantum toroidal group as given in [G-K-V]. Indeed, one should keep in mind the geometrical
construction in [G-R-V] and [G-K-V] in terms of equivariant K-theory of some flag manifolds. A similar K-theoretic construction
of Cherednik's algebra has motivated the present work. At last, we would like to lay emphasis on the fact that, contrary to
[J, D1 and C2], the representations involved in our duality are infinite dimensional. Of course, in the classical case, i.e.,q=1, a similar duality holds between the toroidal Lie algebra and the toroidal version of the symmetric group.
The authors would like to thank V. Ginzburg for a useful remark on a preceding version of this paper.
Communicated by M. Jimbo 相似文献
19.
The purely algebraic notion of CQG algebra (algebra of functions on a compact quantum group) is defined. In a straightforward algebraic manner, the Peter-Weyl theorem for CQG algebras and the existence of a unique positive definite Haar functional on any CQG algebra are established. It is shown that a CQG algebra can be naturally completed to aC
*-algebra. The relations between our approach and several other approaches to compact quantum groups are discussed. 相似文献
20.
N. Iorgov S. Pakuliak V. Shadura Y. Tykhyy G. von Gehlen 《Journal of statistical physics》2010,139(5):743-768
We derive spin operator matrix elements between general eigenstates of the superintegrable ℤ
N
-symmetric chiral Potts quantum chain of finite length. Our starting point is the extended Onsager algebra recently proposed
by Baxter. For each pair of spaces (Onsager sectors) of the irreducible representations of the Onsager algebra, we calculate
the spin matrix elements between the eigenstates of the Hamiltonian of the quantum chain in factorized form, up to an overall
scalar factor. This factor is known for the ground state Onsager sectors. For the matrix elements between the ground states
of these sectors we perform the thermodynamic limit and obtain the formula for the order parameters. For the Ising quantum
chain in a transverse field (N=2 case) the factorized form for the matrix elements coincides with the corresponding expressions obtained recently by the
Separation of Variables method. 相似文献