Irreducible representations of deformed oscillator algebra and q-special functions

Different generators of a deformed oscillator algebra give rise to one-parameter families of $q$-exponential functions and $q$-Hermite polynomials related by generating functions. Connections of the Stieltjes and Hamburger classical moment problems with the corresponding resolution of unity for the $q$-coherent states and with 'coordinate' operators - Jacobi matrices, are also pointed out.Comment: Contribution to the workshop IWCQIS-96 (JINR, Dubna

Quantum resolution of the nonlinear super-Schrodinger equation

We introduce a Z_2-graded version of the nonlinear Schrodinger equation that includes one fermion and one boson at the same time. This equation is shown to possess a supersymmetry which proves to be itself part of a super-Yangian symmetry based on gl(1|1). The solution exhibits a super version form of the classical Rosales solution. Then, we second quantize these results, and give a Lax pair formulation (based on gl(2|1)) for the model.Comment: 20 pages, no figur

Deformation of orthosymplectic Lie superalgebra osp(1|2)

Triangular deformation of the orthosymplectic Lie superalgebra osp(1|4) is defined by chains of twists. Corresponding classical r-matrix is obtained by a contraction procedure from the trigonometric r-matrix. The carrier space of the constant r-matrix is the Borel subalgebra.Comment: LaTeX, 8 page

Jordanian deformation of the open XXX-spin chain

The general solution to the reflection equation associated with the jordanian deformation of the SL(2) invariant Yang R-matrix is found. The same K-matrix is obtained by the special scaling limit of the XXZ-model with general boundary conditions. The Hamiltonian with the boundary terms is explicitly derived according to the Sklyanin formalism. We discuss the structure of the spectrum of the deformed XXX-model and its dependence on the boundary conditions.Comment: 13 pages; typos correcte

Twist Deformation of the rank one Lie Superalgebra

The Drinfeld twist is applyed to deforme the rank one orthosymplectic Lie superalgebra $osp(1|2)$. The twist element is the same as for the $sl(2)$ Lie algebra due to the embedding of the $sl(2)$ into the superalgebra $osp(1|2)$. The R-matrix has the direct sum structure in the irreducible representations of $osp(1|2)$. The dual quantum group is defined using the FRT-formalism. It includes the Jordanian quantum group $SL_\xi(2)$ as subalgebra and Grassmann generators as well.Comment: LaTeX, 9 page

Algebraic Bethe ansatz for the gl(1∣|2) generalized model II: the three gradings

The algebraic Bethe ansatz can be performed rather abstractly for whole classes of models sharing the same $R$-matrix, the only prerequisite being the existence of an appropriate pseudo vacuum state. Here we perform the algebraic Bethe ansatz for all models with $9 \times 9$, rational, gl(1$|$2)-invariant $R$-matrix and all three possibilities of choosing the grading. Our Bethe ansatz solution applies, for instance, to the supersymmetric t-J model, the supersymmetric $U$ model and a number of interesting impurity models. It may be extended to obtain the quantum transfer matrix spectrum for this class of models. The properties of a specific model enter the Bethe ansatz solution (i.e. the expression for the transfer matrix eigenvalue and the Bethe ansatz equations) through the three pseudo vacuum eigenvalues of the diagonal elements of the monodromy matrix which in this context are called the parameters of the model.Comment: paragraph added in section 3, reference added, version to appear in J.Phys.

Separation of variables in the quantum integrable models related to the Yangian Y[sl(3)]

There being no precise definition of the quantum integrability, the separability of variables can serve as its practical substitute. For any quantum integrable model generated by the Yangian Y[sl(3)] the canonical coordinates and the conjugated operators are constructed which satisfy the ``quantum characteristic equation'' (quantum counterpart of the spectral algebraic curve for the L operator). The coordinates constructed provide a local separation of variables. The conditions are enlisted which are necessary for the global separation of variables to take place.Comment: 15 page