On Characters and Superdimensions of Some Infinite-Dimensional Irreducible Representations of osp(m|n)

Physics of Atomic Nuclei - Chiral spinors and self dual tensors of the Lie superalgebra osp(m|n) are infinite-dimensional representations belonging to the class of representations with Dynkin...     

Two Types of Infinite-Dimensional Representations of the Quantum Algebra OSPq(l, 4)

In this letter, we discuss the quantum superalgebra OSP_q(l,4). With the help of the SU_q(2)×U(1) basis, we construct two types of representations for the quantum superalgebra, called the-1/2-representation and the 1/2-representation. Both of them are infinitedimensional.     

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.     

osp(N, 2) AKNS equations

The osp(N, 2) extension of the AKNS scheme is reconsidered. It leads to a general class of integrable nonlinear evolution equations for 2+N(N–1)/2 commuting and 2N anticommuting fields. by reduction, various osp(N, 2) versions of the Korteweg-de Vries equation can be obtained. One of these is shown to be bi-Hamiltonian and its second Hamiltonian structure corresponds to the classical limit of the so(N) superconformal algebra. The nonlinear Schrödinger and modified Korteweg-de Vries reductions are also briefly discussed.Work supported by NSerc and FCAR.     

Finite- and Infinite-Dimensional Representations of the Lorentz Group

Finite- and infinite-dimensional representations of the Lorentz group are discussed and various topics in which this group is currently in use are mentioned. The infinitesimal approach of finding representations is reviewed and all finite-dimensional spinor representations of the Lorentz group are obtained. Infinite-dimensional representations are then discussed, including the principal, complementary, and complete series of representations. A generalized Fourier transformation is introduced which enables one to use the global approach to representation theory so as to express infinite-dimensional representations in terms of matrices. This method is shown to lead to a generalization of the spinor form of finite-dimensional representation to the infinite-dimensional case. However, whereas the usual spinor representations are nonunitary, the obtained new form describes both unitary and non-unitary representations, depending on the choice of certain parameters appearing in the representation formula.     

Representation of osp (2, 2) q (2) andS-matrix of super sine-Gordon theory

We investigate the representations of the osp (2, 2) _q ⁽²⁾ algebra, which leads to theS-matrix of super sine-Gordon theory. TheS-matrix has been derived from supersymmetric conformal field theory with some assumptions. We show that the conjecturedS-matrix can be derived from the representation theory using a correspondence between the representations of osp (1, 2) _q and those of sl(2) _q .     

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).     

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.     

Quantum Deformation of the Lie Superalgebra spl(2, 1)

In this paper we obtain a q-deformation of theLie superalgebra spl(2, 1), which is in correspondencewith its one-parameter quantum group. This proceduresuggests a two-parameter deformation of the Lie superalgebra which leads to the two-parameterquantum group.     

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.     

Quantum Group Structure of Lie Superalgebra osp (1/2)

We quantize the enveloping Lie superalgebraU(osp(1/2)) in the nonstandard simple root system withtwo odd simple roots. The quantum supergroup structureassociated with the nonstandard simple root system is also given.     

On Two-Parameter Deformations of osp(1|2)(1)

An elliptic two-parameter deformation of the (universal enveloping superalgebra of) affine Lie superalgebra osp(1|2)⁽¹⁾ is proposed in terms of free boson realization. This deformed superalgebra is shown to fit in the framework of infinite Hopf family of superalgebras, a generalization of the infinite Hopf family of algebras proposed earlier by the authors. The trigonometric and rational degenerations are briefly discussed.     

Representation theory of osp(1,2) q

We investigate the representations of the osp(1, 2) _q algebra. We derive all the finite-dimensional irreducible representations, whenq is not a root of unity. We also discuss the connection between those of osp(1, 2) _q and sl(2) _q .     

Induced Representations and Invariant Integral Operators for SU (2, 2)

The elementary representations (ER) of SU (2, 2) induced from the three non-trivial parabolic subgroups P₀, P₁, P₂ are explicitly constructed in two equivalent realizations. We exhibit a one-to-one correspondence between the P₀ and P₂ induced representations. The corresponding representations are equivalent. We also exhibit a two-to-one correspondence between the P₁ ERs and a subset of the P₀ ERs; however, the corresponding representations are only partially equivalent. The Knapp-Stein integral intertwining operators are explicitly given for all representations in consideration.     

A New q-Deformed Boson Realization of the Quantum Algebra Cq(l) and Its Representations

A new q-deformed boson realization of the quantum algebra C_q(l) is given and some of its finite dimensional representations are explicitly presented when q is a root of unity.     

A remarkable connection between the representations of the Lie superalgebras osp(1, 2n) and the Lie algebras o(2n+1)

We show that there is a one-to-one correspondence between the graded representations of osp(1, 2n) and the non-spinorial representations of o(2n+1). The Clebsch-Gordan series for osp(1, 2n) reduce to the corresponding series for o(2n+1) and the properly defined Casimir operators of order at least up to four have the same eigenvalues.Supported by the Deutsche Forschungsgemeinschaft     

Star products for SL(2)

The deformation program (the use of star products in harmonic analysis) leads to the definition of an adapted Fourier transform, unitary transformation between spaces of square integrable functions of the group G and on the dual of its Lie algebra, describing the unitary dual of G and its Plancherel transform. This Letter is an application of this program to the universal covering group of SL(2).