Analyticity properties of trilinear SL(2, C) invariant forms in elementary representation parameters

Trilinear invariant forms over spaces transforming under the so-called elementary representations of SL (2, C) (obtained from the principal series by analytic continuation in the representation parameters) are studied with regard to their analyticity properties in the representation parameters. The method is based on a natural one-one correspondence between the invariant forms and invariant separately homogeneous distributions (called kernels of the forms) in three complex two-dimensional non-zero vectors. There exists a family Ψ of kernels of forms with analytic dependence on the representation parameters (Ψ being unique up to a family of complex multiples dependent on the parameters). Also associated kernels obtained by differentiating Ψ in the parameters are studied.     

Matrix Kernels for Measures on Partitions

We consider the problem of computation of the correlation functions for the z-measures with the deformation (Jack) parameters 2 or 1/2. Such measures on partitions are originated from the representation theory of the infinite symmetric group, and in many ways are similar to the ensembles of Random Matrix Theory of β=4 or β=1 symmetry types. For a certain class of such measures we show that correlation functions can be represented as Pfaffians including 2×2 matrix valued kernels, and compute these kernels explicitly. We also give contour integral representations for correlation kernels of closely connected measures on partitions. Supported by US-Israel Binational Science Foundation (BSF) Grant No. 2006333.     

On invariant and covariant Schwartz distributions in the case of a compact linear group

A proof is given for the representations of invariant and covariant (Schwartz) distributions onR ⁿ , which are often used in theoretical physics. We express invariant distributions as distributions of standard polynomial invariants and decompose covariant distributions in standard polynomial covariants. Our consideration is restricted to compact groups acting linearly onR ⁿ . The representation for invariant distributions is obtained provided the standard invariants form an algebraically independent generating set in the ring of invariant polynomials. As for the standard covariants we assume that in the class of covariant polynomials they provide a unique decomposition into a sum of the standard covariants multiplied with invariant polynomials.     

Noether-Symmetry Analysis using Alternative Lagrangian Representations

We have sought to work with an approach to Noether symmetry analysis which uses the properties of infinitesimal point transformations in the space-time (q, t) variable to establish the association between symmetries and conservation laws of a dynamical system. In this approach symmetries are expressed in the form of generators. We have studied the variational or Noether symmetries of two uncoupled Harmonic oscillators and two such oscillators coupled by an interaction. Both these systems can have alternative Lagrangian representations. We have studied in detail how the association between symmetries and conservation laws changes as one alters the analytic or Lagrangian representation. This analysis is carried out with a view to explicitly demonstrate that the correlation between symmetry transformation and corresponding invariant quantity depends crucially on the choice of the analytic representation. PACS 45.20.Jj, 45.20.df, 45.20.dh     

Industrial application of impulsive high pressures in sheet forming with parameter optimization

Abstract

Impulsive high pressures for sheet forming are widely used in the modem industry. Applied aspects of the impulsive forming problem, including both control external stress and heat influence parameters are stated in the present research. The problem is reproduced in the vector representation in the multidimensional space of optimizing functions. In the case of a limited number of measurements the task leads to well known scalar representations with similar physical characteristics'.     

The Cohomology of the Variational Bicomplex Invariant under the Symmetry Algebra of the Potential Kadomtsev-Petviashvili Equation

Abstract

The variational bicomplex of forms invariant under the symmetry algebra of the potential Kadomtsev-Petviashvili equation is described and the cohomology of the associated Euler-Lagrange complex is computed. The results are applied to a characterization problem of the Kadomtsev-Petviashvili equation by its symmetry algebra originally posed by David, Levi, and Winternitz.     

Irreducible highest weight representations of quantum groupsU q (gl(n,C))

Explicit recurrence formulas of canonical realization (boson representation) for quantum enveloping algebrasU _q (gl(n, C)) are given. Using them, irreducible highest weight representations ofU _q (gl(n, C)) are obtained as restriction of representation of Fock space to invariant subspace generated by vacuum as a cyclic vector.     

Finite-dimensional representations of the euclidean group in the plane

In this article two theorems are given which permit, together with the concept of a representation vector diagram, to classify all (linear) finite-dimensional representations of the algebra and group E ₂ which are induced by a master representation on the place of the universal enveloping algebra of the algebra E ₂. Apart from a classification of the finite-dimensional representations, the two theorems make it possible to obtain the matrix elements of these representations for both, algebra and group, in explicit form. The material contained in this letter forms part of an analysis of indecomposable (finite- and infinite-dimensional) representations of the algebra and group E ₂ which is contained in Reference [1]. No proofs will be given in this letter. We refer instead to [1].     

Representations of Hermitian Kernels¶by Means of Krein Spaces.¶II. Invariant Kernels

In this paper we study hermitian kernels invariant under the action of a semigroup with involution. We characterize those hermitian kernels that realize the given action by bounded operators on a Kre?n space. This is motivated by the GNS representation of *-algebras associated to hermitian functionals, the dilation theory of hermitian maps on C ^*-algebras, as well as others. We explain the key role played by the technique of induced Kre?n spaces and a lifting property associated to them. Received: 27 March 2000/ Accepted: 5 September 2000     

Mappin class group actions on quantum doubles

We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non-semisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT's is circumvented. We find compact formulae for theS ^±1-matrices using the canonical, non-degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford's relations between the canonical forms and the moduli of integrals. We analyze the projectiveSL(2, Z)-action on the center ofU _q(sl₂) forq anl=2m+1^st root of unity. It appears that the 3m+1-dimensional representation decomposes into anm+1-dimensional finite representation and a2m-dimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation ofSL(2, Z) and the finite,m-dimensional representation, obtained from the truncated TQFT of the semisimplified representation category ofU _q(sl₂).     

r-Matrix for the Restricted KdV Flows with the Neumann Constraints

Abstract

Under the Neumann constraints, each equation of the KdV hierarchy is decomposed into two finite dimensional systems, including the well-known Neumann model. Like in the case of the Bargmann constraint, the explicit Lax representations are deduced from the adjoint representation of the auxiliary spectral problem. It is shown that the Lax operator satisfies the r-matrix relation in the Dirac bracket. Thus, the integrabilities of these resulting systems with the Neumann constraints are obtained.     

The order parameter for the superconducting phases of UPt3

I review the principal theories that have been proposed for the superconducting phases of UPt₃. The detailed H-T phase diagram places constraints on any theory for the multiple superconducting phases. Much attention has been given to the Ginzberg-Landau region of the phase diagram where the phase boundaries of three phases appear to meet at a tetracritical point. It has been argued that the existence of a tetracritical point for all field orientations eliminates the two-dimensional (2D) orbital representations coupled to a symmetry-breaking field (SBF) as a viable theory of these phases and favours either a theory based on two primary order parameters belonging to different irreducible representations that are accidentally degenerate, as described by Chen and Garg 1993, or a spin-triplet, orbital one-dimensional representation with non spin-orbit coupling in the pairing channel, as described by Machida and Ozaki 1991. I comment on the limitations of the models proposed so far for the superconducting phases of UPt₃. I also find that a theory in which the order parameter belongs to an orbital 2D representation coupled to a SBF is a viable model for the phases of UPt₃, based on the existing body of experimental data. Specifically, I show that the existing phase diagram (including an apparent tetracritical point for all field orientations), the anisotropy of the upper critical field over the full temperature range, the correlation between superconductivity and basal plane antiferromagnetism and the low-temperature power laws in the transport and thermodynamic properties can be explained qualitatively, and in many respects quantitatively, by an odd-parity E_2u order parameter with a pair spin projection of zero along the ?c axis. The coupling of an antiferromagnetic moment to the superconducting order parameter acts as a SBF which is responsible for the apparent tetracritical point, in addition to the zero-field double transition. The new results presented here for the E_2u representation are based on an analysis of the material parameters calculated within the Bardeen-Cooper-Schrieffer theory for the 2D representations, and a refinement of the SBF model given by Hess et al. (1989). I also discuss possible experiments to test the symmetry of the order parameter.     

Gelfand-Tzetlin coordinates for the unitary supergroup

The Gelfand-Tzetlin method provides explicit coordinates on the parameter space of the unitary groupU(k) which make direct evaluations of group integrals possible. It is closely related to the Gelfand construction of finite-dimensional irreducible representations. We generalize the Gelfand-Tzetlin method to the unitary supergroupU(k ₁/k₂). The coordinates on the parameter space for supergroup integrals and the invariant Haar measure are evaluated. As an example, the supersymmetric Harish-Chandra-Itzykson-Zuber integral is calculated. A generalized Gelfand pattern containing anticommuting variables is introduced which determines the representation.This article was processed by the author using the Latex style filepljour1 from Springer-Verlag.     

Group-Theoretical Approach to Extended Conformal Supersymmetry: Function Space Realizations and Invariant Differential Operators

We give function space realizations of all representations of the conformal superalgebra su(2,2/N) and of the supergroup SU(2, 2 /N) induced from irreducible finite-dimensional Lorentz and SU(N) representations realized without spin and isospin indices. We use the lowest weight module structure of our su(2,2/N) representations to present a general procedure (adapted from the semisimple Lie algebra case) for the canonical construction of invariant differential operators closely related to the reducible (indecomposable) representations. All conformal supercovariant derivatives are obtained in this way. Examples of higher order invariant differential operators are given.     

∗-Representations of the Quantum Algebra U q(sl(3))

Abstract

Studied in this paper are real forms of the quantum algebra U _q(sl(3)). Integrable operator representations of ?-algebras are defined. Irreducible representations are classified up to a unitary equivalence.     

(p,q)-Analog of Two-Dimensional Conformal Field Theory. The Ward Identities and Correlation Functions

Abstract

A (p, q)-analog of two-dimensional conformally invariant field theory based on the quantum algebra U_pq (su(1, 1)) is proposed. The representation of the algebra U_pq (su(1, 1)) on the space of quasi-primary fields is given. The (p, q)-deformed Ward identities of conformal field theory are defined. The two- and three-point correlation functions of quasi-primary fields are calculated.     

Fock representations and BRST cohomology inSL(2) current algebra

We investigate the structure of the Fock modules overA ₁ ⁽¹⁾ introduced by Wakimoto. We show that irreducible highest weight modules arise as degree zero cohomology groups in a BRST-like complex of Fock modules. Chiral primary fields are constructed as BRST invariant operators acting on Fock modules. As a result, we obtain a free field representation of correlation functions of theSU(2) WZW model on the plane and on the torus. We also consider representations of fractional level arising in Polyakov's 2D quantum gravity. Finally, we give a geometrical, Borel-Weil-like interpretation of the Wakimoto construction.     

Extended Higher-Spin Superalgebras and Their Realizations in Terms of Quantum Operators

The realization of the N = 1 higher-spin superalgebra, proposed earlier by E. S. FRADKIN and the author, is found in terms of bosonic quantum operators. The extended higher-spin super-algebras, generalizing ordinary extended supersymmetry with arbitrary N > 1, are constructed by adding fermion quantum operators. Automorphisms, real forms, subalgebras, contractions and invariant forms of these infinite-dimensional superalgebras are studied. The formulation of the higher-spin superalgebras is described in terms of symbols of operators by BEREZIN. We hope that this formulation will provide in future the powerful tool for constructing the complete solution of the higher-spin problem, the problem of introducing a consistent gravitational interaction for massless higher-spin fields (S > 2).