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

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

Reflection equations and q-Minkowski space algebras

We express the defining relations of the $q$-deformed Minkowski space algebra as well as that of the corresponding derivatives and differentials in the form of reflection equations. This formulation encompasses the covariance properties with respect the quantum Lorentz group action in a straightforward way.Comment: 10 page

Peripheric Extended Twists

The properties of the set L of extended jordanian twists are studied. It is shown that the boundaries of L contain twists whose characteristics differ considerably from those of internal points. The extension multipliers of these "peripheric" twists are factorizable. This leads to simplifications in the twisted algebra relations and helps to find the explicit form for coproducts. The peripheric twisted algebra U(sl(4)) is obtained to illustrate the construction. It is shown that the corresponding deformation U_{P}(sl(4)) cannot be connected with the Drinfeld--Jimbo one by a smooth limit procedure. All the carrier algebras for the extended and the peripheric extended twists are proved to be Frobenius.Comment: 16 pages, LaTeX 209. Some misprints have been corrected and new Comments adde

Quantization of the N=2 Supersymmetric KdV Hierarchy

We continue the study of the quantization of supersymmetric integrable KdV hierarchies. We consider the N=2 KdV model based on the $sl^{(1)}(2|1)$ affine algebra but with a new algebraic construction for the L-operator, different from the standard Drinfeld-Sokolov reduction. We construct the quantum monodromy matrix satisfying a special version of the reflection equation and show that in the classical limit, this object gives the monodromy matrix of N=2 supersymmetric KdV system. We also show that at both the classical and the quantum levels, the trace of the monodromy matrix (transfer matrix) is invariant under two supersymmetry transformations and the zero mode of the associated U(1) current.Comment: LaTeX2e, 12 page

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

Extended and Reshetikhin Twists for sl(3)

The properties of the set {L} of extended jordanian twists for algebra sl(3) are studied. Starting from the simplest algebraic construction --- the peripheric Hopf algebra U_ P'(0,1)(sl(3)) --- we construct explicitly the complete family of extended twisted algebras {U_ E(\theta)(sl(3))} corresponding to the set of 4-dimensional Frobenius subalgebras {L(\theta)} in sl(3). It is proved that the extended twisted algebras with different values of the parameter \theta are connected by a special kind of Reshetikhin twist. We study the relations between the family {U_E(\theta)(sl(3))} and the one-dimensional set {U_DJR(\lambda)(sl(3))} produced by the standard Reshetikhin twist from the Drinfeld--Jimbo quantization U_DJ(sl(3)). These sets of deformations are in one-to-one correspondence: each element of {U_E(\theta)(sl(3))} can be obtained by a limiting procedure from the unique point in the set {U_DJR(\lambda)(sl(3))}.Comment: 14 pages, LaTeX 20

q-Supersymmetric Generalization of von Neumann's Theorem

Assuming that there exist operators which form an irreducible representation of the q-superoscillator algebra, it is proved that any two such representations are equivalent, related by a uniquely determined superunitary transformation. This provides with a q-supersymmetric generalization of the well-known uniqueness theorem of von Neumann for any finite number of degrees of freedom.Comment: 10 pages, Latex, HU-TFT-93-2

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.