Partial Algebraization Research Articles

Partial algebras, meaning categories and algebraization

Many approaches to natural language semantics are essentially model-theoretic, typically cast in type theoretic terms. Many linguists have adopted type theory or many-sorted algebras [see H. Hendriks, Compositionality and model-theoretic interpretation, J. Logic, Language Inform. 10 (2001) 29–48 and references therein]. However, recently Hodges [Formal features of compositionality, J. Logic, Language Inform. 10 (2001) 7–28] has offered an approach to compositionality using just partial algebras. An approach in terms of partial algebras seems at the outset more justified, since the typing is often just artificially superimposed on language (and makes many words massively polymorphic). On the other hand, many-sorted algebras are easier to handle than partial algebras, and are therefore generally preferred. This paper investigates the dialectics between partial algebras and many-sorted algebras and tries to set the background for an approach in the spirit of Hodges [Formal features of compositionality, J. Logic, Language Inform. 10 (2001) 7–28], which also incorporates insights from algebraic logic, in particular from Blok and Pigozzi [Algebraizable logics, Mem. Amer. Math. Soc. 77 (396) (1990)]. The analytic methods that we shall develop here shall also be applied to combinatory algebras and algebraizations of predicate logic.

Dolan–Grady relations and noncommutative quasi-exactly solvable systems

We investigate a U(1) gauge invariant quantum mechanical system on a 2D noncommutative space with coordinates generating a generalized deformed oscillator algebra. The Hamiltonian is taken as a quadratic form in gauge covariant derivatives obeying the nonlinear Dolan-Grady relations. This restricts the structure function of the deformed oscillator algebra to a quadratic polynomial. The cases when the coordinates form the su(2) and sl(2,R) algebras are investigated in detail. Reducing the Hamiltonian to 1D finite-difference quasi-exactly solvable operators, we demonstrate partial algebraization of the spectrum of the corresponding systems on the fuzzy sphere and noncommutative hyperbolic plane. A completely covariant method based on the notion of intrinsic algebra is proposed to deal with the spectral problem of such systems.

Nonlinear holomorphic supersymmetry on Riemann surfaces

We investigate the nonlinear holomorphic supersymmetry for quantum-mechanical systems on Riemann surfaces subjected to an external magnetic field. The realization is shown to be possible only for Riemann surfaces with constant curvature metrics. The cases of the sphere and Lobachevski plane are elaborated in detail. The partial algebraization of the spectrum of the corresponding Hamiltonians is proved by the reduction to one-dimensional quasi-exactly solvable sl(2, R) families. It is found that these families possess the “duality” transformations, which form a discrete group of symmetries of the corresponding 1D potentials and partially relate the spectra of different 2D systems. The algebraic structure of the systems on the sphere and hyperbolic plane is explored in the context of the Onsager algebra associated with the nonlinear holomorphic supersymmetry. Inspired by this analysis, a general algebraic method for obtaining the covariant form of integrals of motion of the quantum systems in external fields is proposed.

Quantum Deformation of the Non-Polynomial Interaction V(r) = r2 + λr2/(1 + g r2 )

Abstract A general scheme for studying q-deformed quasi-exactly solvable problems has been developed using the partial algebraization method. Exact solutions of the deformed non-polynomial interaction generated by the potential V (r) = r2 + λr2/( 1 + g r2) are obtained.

Exact bound states for the potentialV(r)=r 2+βr −4+λr −6 using partial algebraization technique

For the potential considered new sets of ground state and first few excited states are obtained. Also the defect of the wave function suggested by Kaushal and Parashar [Phys. Lett. A170 (1992) 335] and Gaurdiala and Ros [J. Phys. A25 (1992) 1351] has been shown.

Algebraization of difference eigenvalue equations related to Uq(sl2)

A class of second order difference (discrete) operators with a partial algebraization of the spectrum is introduced. The eigenfuncions of the algebraized part of the spectrum are polynoms (discrete polynoms). Such difference operators can be constructed by means of Uq(sl2) the quantum deformation of the sl2 algebra. The roots of the polynoms determine the spectrum and obey the Bethe ansatz equations. A particular case of difference equations for q-hypergeometric and Askey-Wilson polynoms is discussed. Applications to the problem of Bloch electrons in a magnetic field are outlined.

DIRECT APPROACH TO PARTIAL ALGEBRAIZATION

This paper presents an alternative approach to the method of partial algebraization. This approach does not make any use of Lie symmetries. All Quasi-Exactly-Solvable (QES) problems, constructed by the method presented in Refs. 1–3, can also be constructed in this approach.

PARTIAL ALGEBRAIZATION OF THE NON-POLYNOMIAL POTENTIAL

The solutions of the potential x2 + λ[x2/(1 + gx2)] are investigated using partial algebraization of the spectral problem and supersymmetric quantum mechanics.

—Generalization of Partial Algebraization Method for spectral Problems and Heisenberg Model11The project supported by National Natural Science Foundation of China.

In terms of the -multiple direct product of Lie algebra the method of partial algebraization for quantal spectrum problems is generalized to deal with the cases of many-mode coupling. Using this generalized method to the antiferromagnetic Heisenberg model, we explicitly obtain some analytic and numeral results for the eigenstates and eigenvalues.

Hill determinants, supersymmetry and partial algebraization

We investigate the relationship between the method of Hill determinants, supersymmetry and partial algebraization.

Quantal problems with partial algebraization of the spectrum

We discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly-solvable problems (e.g., harmonic oscillator) and all others. The problems belonging to this class are distinguished by the fact that a part of the eigenvalues, and eigenfunctions can be found algebraically, but not the whole spectrum. The reason explaining the existence of the quasi-exactly-solvable problems is a hidden dynamical symmetry present in the hamiltonian. For one-dimensional motion this hidden symmetry isSL(2,R). It is shown that other groups lead to a partial algebraization in multidimensional quantal problems. In particular,SL(2,R)×SL(2,R),SO(3) andSL(3,R) are relevant to two-dimensional motion inducing a class of quasi-exactly-solvable two-dimensional hamiltonians. Typically they correspond to systems in a curved space, but sometimes the curvature turns out to be zero. Graded algebras open the possibility of constructing quasi-exactlysolvable hamiltonians acting on multicomponent wave functions. For example, with a (non-minimal) superextension ofSL(2,R) we get a hamiltonian describing the motion of a spinor particle.

EXPANDING THE CLASS OF SPECTRAL PROBLEMS WITH PARTIAL ALGEBRAIZATION

The phenomenon of the partial algebraization in the Schrödinger-type equations with a hidden dynamical symmetry is discussed. A generalization of the Turbiner procedure is proposed which can expand the class of the quasi-exactly-solvable problems. A specific example relevant to a one-dimensional Hamiltonian with the SU(2) hidden symmetry is constructed.

Partial algebraization of spectral problems and supersymmetry

We study a class of problems for which an arbitrary part of the spectrum (but not the whole spectrum) can be found analytically. It is shown here that these quasi-exactly solvable problems, in addition to admitting a dynamical group symmetry, is also supersymmetric and exploiting the supersymmetric structure of the Hamiltonian the spectrum can be determined in a very simple way.

NEW FINDINGS IN QUANTUM MECHANICS (PARTIAL ALGEBRAIZATION OF THE SPECTRAL PROBLEM)

We discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly-solvable problems (like the famous harmonic oscillator) and all others. The problems belonging to this class are distinguished by the fact that an (arbitrary) part of the eigenvalues and eigenfunctions can be found algebraically, but not the whole spectrum. The reason explaining the existence of the quasi-exactly-solvable problems is a hidden dynamical symmetry present in the Hamiltonian. For one-dimensional motion, this hidden symmetry is SU(2). The simplest one-dimensional system admitting algebraization for a part of the spectrum is the anharmonic oscillator with the x6 anharmonicity and a relation between the coefficients in front of x2 and x6. We review also more complicated cases with the emphasis on pedagogical aspects. The groups SU (2)× SU (2), SO(3) and SU(3) generate two-dimensional problems with the partial algebraization of the spectrum. Typically we get Schrödinger-type equations in curved space. An intriguing relation between the algebraic structure of the Hamiltonian and the geometry of the space emerges. Another interesting development is the use of the graded algebras which allow one to construct multi-component quasi-exactly-solvable Hamiltonians.