Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota–Baxter Operators

We will introduce an associative (or quantum) version of Poisson structure tensors. This object is defined as an operator satisfying a “generalized” Rota–Baxter identity of weight zero. Such operators are called generalized Rota–Baxter operators. We will show that generalized Rota–Baxter operators are characterized by a cocycle condition so that Poisson structures are so. By analogy with twisted Poisson structures, we propose a new operator “twisted Rota–Baxter operators,” which is a natural generalization of generalized Rota–Baxter operators. It is known that classical Rota–Baxter operators are closely related with dendriform algebras. We will show that twisted Rota–Baxter operators induce NS-algebra, which is a twisted version of dendriform algebra. The twisted Poisson condition is considered as a Maurer–Cartan equation up to homotopy. We will show the twisted Rota–Baxter condition also is so. And we will study a Poisson-geometric reason, how the twisted Rota–Baxter condition arises.     

A Dirac–Dunkl Equation on S2 and the Bannai–Ito Algebra

The Dirac–Dunkl operator on the two-sphere associated to the \({{\mathbb{Z}_{2}^{3}}}\) reflection group is considered. Its symmetries are found and are shown to generate the Bannai–Ito algebra. Representations of the Bannai–Ito algebra are constructed using ladder operators. Eigenfunctions of the spherical Dirac–Dunkl operator are obtained using a Cauchy–Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the Bannai–Ito algebra.     

Generalized Virasoro algebra: left-symmetry and related algebraic and hydrodynamic properties

Motivated by the work of Kupershmidt (J. Nonlin. Math. Phys. 6 (1998), 222 –245) we discuss the occurrence of left symmetry in a generalized Virasoro algebra. The multiplication rule is defined, which is necessary and sufficient for this algebra to be quasi-associative. Its link to geometry and nonlinear systems of hydrodynamic type is also recalled. Further, the criteria of skew-symmetry, derivation and Jacobi identity making this algebra into a Lie algebra are derived. The coboundary operators are defined and discussed. We deduce the hereditary operator and its generalization to the corresponding 3–ary bracket. Further, we derive the so-called ρ–compatibility equation and perform a phase-space extension. Finally, concrete relevant particular cases are investigated.     

交换超算符方法的李代数研究

本文讨论了交换超算符方法的理论基础,结果表明由交换超算符所定义的算符集合g是一个李代数,交换超算符的定义就是李代数中内导子的定义,由此得出一些交换超算符间的代数关系。证明了g中所有算符诱导的超算符集合也是一个李代数,指出了与g对应的是由复盖群派生的,有内积定义的李群,而角动量超算符是由矢量场的内禀角动量和单位算符的直积所生成。结论是交换超算符方法的理论基础是李代数。     

The Operator Formulation of Classical Mechanics and Semiclassical Limit

The algebra of polynomials in operators that represent generalized coordinate and momentum and depend on the Planck constant is defined. The Planck constant is treated as the parameter taking values between zero and some nonvanishing h ₀. For the later of these two extreme values, introduced operator algebra becomes equivalent to the algebra of observables of quantum mechanical system defined in the standard manner by operators in the Hilbert space. For the vanishing Planck constant, the generalized algebra gives the operator formulation of classical mechanics since it is equivalent to the algebra of variables of classical mechanical system defined, as usually, by functions over the phase space. In this way, the semiclassical limit of kinematical part of quantum mechanics is established through the generalized operator framework.     

Darboux related quantum integrable systems on a constant curvature surface

We consider integrable deformations of the Laplace–Beltrami operator on a constant curvature surface, obtained through the action of first-order Darboux transformations. Darboux transformations are related to the symmetries of the underlying geometric space and lead to separable potentials which are related to the KdV equation. Eigenfunctions of the corresponding operators are related to highest weight representations of the symmetry algebra of the underlying space.     

VECTOR LADDER OPERATORS FOR THE CENTRAL POTENTIALS

A new class of nonlinear Lie algebra has been found, which is generated naturally by the Hamiltonian operator, the square of the angular momentum operator and the ladder operator for the central potentials. According to the theory of nonlinear Lie algebra, without using the factorization method, we obtained the vector ladder operators for the three-dimensional isotropic harmonic oscillator and hydrogen atom. The radial components of these operators, which are independent of the quantum numbers, are just the radial ladder operators for the same potentials.     

A Spinor Approach to the SU(2) Clebsch-Gordan Coefficients *

We review the irreducible representation of an angular momentum vector operator constructed in terms of spinor algebra. We generalize the idea of spinor approach to study the coupling of the eigenstates of two independent angular momentum vector operators. Utilizing the spinor algebra, we are able to develop a simple way for calculating the SU(2) Clebsch-Gordan (CG) coefficients. The explicit expression for the SU(2) CG coefficients is worked out, and some simple physical examples are presented to illustrate the spinor approach.     

Hubbard model,conserved quantities,and computer algebra

The constants of motion of the half-filled four-point Hubbard model with cyclic boundary conditions are given in Wannier and Bloch representation. The total number operator and total spin operator are conserved and spin-reversal symmetry exists. In Wannier representation we have additionally the C_4v symmetry and in Bloch representation we have the total momentum operator which is conserved. The anticommutation relations for Fermi operators with spin are implemented using computer algebra. Using computer algebra, all the constants of motion are given. The one-dimensional Hubbard model admits a Lax representation. From the Lax pair we find a new constant of motion.     

Anyons,deformed oscillator algebras and projectors

We initiate an algebraic approach to the many-anyon problem based on deformed oscillator algebras. The formalism utilizes a generalization of the deformed Heisenberg algebras underlying the operator solution of the Calogero problem. We define a many-body Hamiltonian and an angular momentum operator which are relevant for a linearized analysis in the statistical parameter ν. There exists a unique ground state and, in spite of the presence of defect lines, the anyonic weight lattices are completely connected by the application of the oscillators of the algebra. This is achieved by supplementing the oscillator algebra with a certain projector algebra.     

Symmetries and exact solutions of some integrable Haldane-Shastry-like spin chains

Here we construct some integrable Haldane-Shastry (HS) like spin chains, which exhibit multi-parameter deformed or non-standard variants of Y(gl_m) Yangian symmetry. By projecting the eigenstates of Dunkl operators in a suitable way, we also derive a class of exact eigenfunctions for such spin chains and subsequently conjecture that these exact eigenfunctions would lead to the highest weight states (HWS) associated with multi-parameter deformed or non-standard variants of Y(gl_M) Yangian algebra. By using this conjecture, and acting descendent operator on the HWS associated with a non-standard Y(gl₂) Yangian algebra, we are able to find out the complete set of eigenvalues and eigenfunctions for the related HS-like chain. It turns out that some additional energy levels, which are forbidden due to a selection rule in the case of SU(2) HS model, interestingly appear in the spectrum of the above mentioned HS-like spin chain with non-standard Y(gl²) symmetry.     

A symmetry-based approach to the effective Hamiltonian for symmetric top molecules in degenerate vibronic states

The effective Hamiltonian for a symmetric top molecule in a degenerate vibronic state is obtained. Included in this Hamiltonian are the rotational, spin-rotational, spin-orbit coupling and electronic spin-spin interactions. The terms of the Hamiltonian are expressed as the product of molecular ‘constants,’ rotational angular momentum operators, and symmetry operators. A formalism is derived, and tables included, to determine whether or not a symmetry operator vanishes for a given vibronic state of a particular molecular symmetry. In this way, one can easily obtain all the non-vanishing Hamiltonian terms for a particular application.     

The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra

The superintegrability, wavefunctions and overlap coefficients of the Dunkl oscillator model in the plane were considered in the first part. Here finite-dimensional representations of the symmetry algebra of the system, called the Schwinger–Dunkl algebra sd(2), are investigated. The algebra sd(2) has six generators, including two involutions and a central element, and can be seen as a deformation of the Lie algebra \({\mathfrak{u}(2)}\) . Two of the symmetry generators, J ₃ and J ₂, are respectively associated to the separation of variables in Cartesian and polar coordinates. Using the parabosonic creation/annihilation operators, two bases for the representations of sd(2), the Cartesian and circular bases, are constructed. In the Cartesian basis, the operator J ₃ is diagonal and the operator J ₂ acts in a tridiagonal fashion. In the circular basis, the operator J ₂ is block upper-triangular with all blocks 2 × 2 and the operator J ₃ acts in a tridiagonal fashion. The expansion coefficients between the two bases are given by the Krawtchouk polynomials. In the general case, the eigenvectors of J ₂ in the circular basis are generated by the Heun polynomials, and their components are expressed in terms of the para-Krawtchouk polynomials. In the fully isotropic case, the eigenvectors of J ₂ are generated by little ?1 Jacobi or ordinary Jacobi polynomials. The basis in which the operator J ₂ is diagonal is considered. In this basis, the defining relations of the Schwinger–Dunkl algebra imply that J ₃ acts in a block tridiagonal fashion with all blocks 2 × 2. The matrix elements of J ₃ in this basis are given explicitly.     

A representation of angular momentum (SU(2)) algebra

This paper seeks to construct a representation of the algebra of angular momentum (SU(2) algebra) in terms of the operator relations corresponding to Gentile statistics in which one quantum state can be occupied by n particles. First, we present an operator realization of Gentile statistics. Then, we propose a representation of angular momenta. The result shows that there exist certain underlying connections between the operator realization of the Gentile statistics and the angular momentum (SU(2)) algebra.     

Realization of the three-dimensional quantum Euclidean space by differential operators

The three-dimensional quantum Euclidean space is an example of a non-commutative space that is obtained from Euclidean space by q-deformation. Simultaneously, angular momentum is deformed to , it acts on the q-Euclidean space that becomes a -module algebra this way. In this paper it is shown, that this algebra can be realized by differential operators acting on functions on . On a factorspace of a scalar product can be defined that leads to a Hilbert space, such that the action of the differential operators is defined on a dense set in this Hilbert space and algebraically self-adjoint becomes self-adjoint for the linear operator in the Hilbert space. The self-adjoint coordinates have discrete eigenvalues, the spectrum can be considered as a q-lattice. Received: 27 June 2000 / Published online: 9 August 2000     

Vertex Operators in 4D Quantum Gravity Formulated as CFT

We study vertex operators in 4D conformal field theory derived from quantized gravity, whose dynamics is governed by the Wess-Zumino action by Riegert and the Weyl action. Conformal symmetry is equal to diffeomorphism symmetry in the ultraviolet limit, which mixes positive-metric and negative-metric modes of the gravitational field and thus these modes cannot be treated separately in physical operators. In this paper, we construct gravitational vertex operators such as the Ricci scalar, defined as space-time volume integrals of them are invariant under conformal transformations. Short distance singularities of these operator products are computed and it is shown that their coefficients have physically correct signs. Furthermore, we show that conformal algebra holds even in the system perturbed by the cosmological constant vertex operator as in the case of the Liouville theory shown by Curtright and Thorn.     

Geometry of differential operators of second order,the algebra of densities,and groupoids

In our previous works, we introduced, for each (super)manifold, a commutative algebra of densities. It is endowed with a natural invariant scalar product. In this paper, we study geometry of differential operators of second order on this algebra. In the more conventional language they correspond to certain operator pencils. We consider the self-adjoint operators and analyze the operator pencils that pass through a given operator acting on densities of a particular weight. There are ‘singular values’ for pencil parameters. They are related with interesting geometric picture. In particular, we obtain operators that depend on certain equivalence classes of connections (instead of connections as such). We study the corresponding groupoids. From this point of view we analyze two examples: the canonical Laplacian on an odd symplectic supermanifold appearing in Batalin–Vilkovisky geometry and the Sturm–Liouville operator on the line, related with classical constructions of projective geometry. We also consider the canonical second order semi-density arising on odd symplectic supermanifolds, which has some similarity with mean curvature of surfaces in Riemannian geometry.     

Semiunitary Transformation and Isospectral Hamiltonians in Arbitrary Dimensional Space

By means of the complex Clifford algebra, a new realization of multi-dimensional semiunitary transformation is put forward and then applied to studying the isospectrality of nonrelativistic Hamiltonians of multi-dimensional quantum mechanical systems, in which the generalized Pauli coupling interaction and spin-orbit coupling interaction appear naturally. Moreover, it is shown that the semiunitary operators, together with the Hamiltonian of quantum mechanical system, satisfy the polynomially-deformed angular momentum algebra.     

Quantum mechanics of a constrained particle on an ellipsoid: Bein formalism and Geometric momentum

In this work we apply the Dirac method in order to obtain the classical relations for a particle on an ellipsoid. We also determine the quantum mechanical form of these relations by using Dirac quantization. Then by considering the canonical commutation relations between the position and momentum operators in terms of curved coordinates, we try to propose the suitable representations for momentum operator that satisfy the obtained commutators between position and momentum in Euclidean space. We see that our representations for momentum operators are the same as geometric one.     

通用的角动量阶梯算符

利用最新发展的非线性代数理论,给出了一般角动量阶梯算符所应满足的代数方程,并具体构造出了这些算符,所构造的北算符能对所有角动量本征态的解量子数和磁量子数起升降作用,具有很好的通用性。