q-Difference Realization of Uq(sl(M|N)) and Its Application to Free Boson Realization of Uq $$(\widehat{{\text{sl}}}(2|1))$$

We present a q-difference realization of the quantum superalgebra U_q(sl(M|N)), which includes Grassmann even and odd coordinates and their derivatives. Based on this result, we obtain a free boson realization of the quantum affine superalgebra U_q of an arbitrary level k.     

Scasimir Operator,Scentre and Representations of Uq(osp(1|2))

Recently, Borchers has shown that in a theory of local observables, certain unitary and antiunitary operators, which are obtained from an elementary construction suggested by Bisognano and Wichmann, have the same commutation relations with translation operators as Lorentz boosts and P₁CT operators would have, respectively. It is concluded from this that as soon as the operators considered implement any symmetry, this symmetry can be fixed up to at most some translation. As a symmetry, any unitary or antiunitary operator is admitted under whose adjoint action any algebra of local observables is mapped onto an algebra which can be localized somewhere in Minkowski space.     

On Highest Weight Modules Over Elliptic Quantum Groups

The purpose of this Letter is to define and construct highest weight modules for Felder's elliptic quantum groups. This is done by using exchange matrices for intertwining operators between modules over quantum affine algebras.     

A Remark on the FRTS Realization and Drinfeld Realization of Quantum Affine Superalgebra U q (o $${\hat s} $$ sp(1,2))

In this Letter, we present the hidden symmetry behind the Faddeev–Reshetikhin–Takhtajan–Semenov–Tian–Shansky (FRTS) realization of quantum affine superalgebras U(o sp(1,2)) and add the q-Serre relation to the Drinfeld realization of U(o sp(1,2)), derived from the FRTS realization.     

Bethe Vectors of the osp(1|2) Gaudin Model

The eigenvectors of the osp(1|2) invariant Gaudin Hamiltonians are found using explicitly constructed creation operators. Commutation relations between the creation operators and the generators of the loop superalgebra are calculated. The coordinate representation of the Bethe states is presented. The relation between the Bethe vectors and solutions to the Knizhnik–Zamolodchikov equation yields the norm of the eigenvectors.     

Universal R-matrix of the quantum superalgebra osp(2 | 1)

A quantum analogue of the simplest superalgebra osp(2 | 1) and its finite-dimensional, irreducible representations are found. The corresponding constant solution to the Yang-Baxter equation is constructed and is used to formulate the Hopf superalgebra of functions on the quantum supergroup OSp(2 | 1).     

Two-Parameter Deformations of SU(2) and SU (1,l) Algebras

The algebras SU(2) and SU(1,1) are promoted to two-parameter quantum universal enveloping algebras (QUEA) by a doubleparameter deformation in this paper. The discrete unitary irreducible representations and their deformed coherent states are studied. The deformed generators are given by a Jordan-Schwinger realization and a Bargmann-Fock representation. It is also interesting that the two-parameter deformed coherent states are found to relate to the oneparameter deformed ones by a simple scaling transformation and this can be used to derive the completeness relation of the former.     

关于双参数量子代数SUqs(2)和双参数形变谐振子的若干结果

构造了双参数形变量子代数SU_qs(2)的Holstein-Primakoff实现和Nodvik实现,并给出了量子代数SU_qs(2)和双参数形变谐振子的形变映射.     

Structure constants for the osp(1|2) current algebra

We study the free field realization of the two-dimensional osp(1|2) current algebra. We consider the case in which the level of the affine osp(1|2) symmetry is a positive integer. Using the Coulomb gas technique we obtain integral representations for the conformal blocks of the model. In particular, from the behaviour of the four-point function, we extract the structure constants for the product of two arbitrary primary operators of the theory. From this result we derive the fusion rules of the osp(1|2) conformal field theory and we explore the connections between the osp(1|2) affine symmetry and the N = 1 superconformal field theories.     

Projective indecomposable modules over the small quantum osp(1|2)

In this paper, we construct the finite dimensional Hopf superalgebra u _q (osp(1|2)) arising from U _q(osp(1|2)) when q is a root of unity and describe the projective objects and the irreducible morphisms in a category of Z-graded u _q (osp(l|2))-modules.     

SUPERSYMMETRIC SU(1|2) GAUDIN MODEL

In this paper, we propose a supersymmetric SU(1|2) Gaudin model and have derived its eigenvalues. We also present the well-defined eigenstates through the quasi-classical limit of the eigenstates in the supersymmetric t-J model.     

osp(1|4)Toda模型解的构造

将Leznov–Saveliev代数分析和Drinfeld–Sokolov构造这种方法推广到超对称情形,并运用这种方法给出osp(1|4)Toda模型的解,从而将这种方法推广到二秩情况.     

THE FIRST COHOMOLOGY OF THE SUPERCONFORMAL ALGEBRA K(1|4)

The infinitesimal deformations of the embedding of the Lie superalgebra of contact vector fields on the supercircle S^1|4 into the Poisson superalgebra of symbols of pseudodifferential operators on S^1|2 are explicitly calculated.     

Some Tensor Product Representations of Uq(s $${\hat l}{\kern 1pt} $$ 2) at Roots of Unity

When q is a root of unity, a triangular decomposition of U _q(s ₂) is given and irreducibility conditions concerning some tensor product representations of U _q(s ₂) are presented. Their connection with physics is also pointed out.     

Quasi-Hopf Deformations of Quantum Groups

The search for elliptic quantum groups leads to a modified quantum Yang–Baxter relation and to a special class of quasi-triangular quasi-Hopf algebras. This Letter calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac–Moody algebras are more rigid than their loop algebra quotients and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with sl(n) are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang–Baxter relation and are calculated more or less explicitly. The modified classical Yang–Baxter relation is obtained and the elliptic solutions are worked out explicitly.The same method is used to construct the Universal R-matrices associated with Felder's quantization of the Knizhnik–Zamolodchikov–Bernard equation, to throw some light on the quasi-Hopf structure of conformal field theory and (perhaps) the Calogero–Moser models.     

Integrable Deformations of the (2+1)-Dimensional Heisenberg Ferromagnetic Model

Based on the covariant prolongation structure technique,we construct the integrable higher-order deformations of the (2+1)-dimensional Heisenberg ferromagnet model and obtain their su(2)×R(λ) prolongation structures.By associating these deformed multidimensional Heisenberg ferromagnet models with the moving space curve in Euclidean space and using the Hasimoto function,we derive their geometrical equivalent counterparts,i.e.,higher-order (2+1)-dimensional nonlinear Schrödinger equations.