Kronecker product decomposition for Poincaré-like groups I. A general constructive method

For Poincaré-like groupsW=T×G, i.e. such Type I-groups which are semidirect products of a regularity embedded abelian normal subgroupT with a locally compact groupG, the problem of the Kronecker product decomposition has been solved, in the framework of induced representations theory, up to a reduction of little group representations. For practical applications some computational hints are given.The autors would like to thank Prof. H.Boseck, Dr. J.Niederle and Dr. J.Tolar for helpful and interesting discussions. One of us (Th. G.) would also like to thank Dr. J.Tolar for his kind hospitality during a stay at the Czech Technical University, Prague.     

On Majorana Representations of A6 and A7

The Majorana representations of groups were introduced in Ivanov (The Monster Group and Majorana Involutions, 2009) by axiomatising some properties of the 2A-axial vectors of the 196 884-dimensional Monster algebra, inspired by the sensational classification of such representations for the dihedral groups achieved by Sakuma (Int Math Res Notes, 2007). This classification took place in the heart of the theory of Vertex Operator Algebras and expanded earlier results by Miyamoto (J Alg 268:653–671, 2003). Every subgroup G of the Monster which is generated by its intersection with the conjugacy class of 2A-involutions possesses the (possibly unfaithful) Majorana representation obtained by restricting to G the action of the Monster on its algebra. This representation of G is said to be based on an embedding of G in the Monster. So far the Majorana representations have been classified for the groups G isomorphic to the symmetric group S ₄ of degree 4 (Ivanov et al. in J Alg 324:2432–2463, 2010), the alternating group A ₅ of degree 5 (Ivanov AA, Seress á in Majorana Representations of A ₅, 2010), and the general linear group GL ₃(2) in dimension 3 over the field of two elements (Ivanov AA, Shpectorov S in Majorana Representations of L ₃(2), 2010). All these representations are based on embeddings in the Monster of either the group G itself or of its direct product with a cyclic group of order 2. The dimensions and shapes of these representations are given in the following table:     

Icosahedral symmetry adaptation of |JM〉 bases

The behaviour of the orbital and spin harmonic kets |JM〉 under the icosahedral double group I? is described by means of the irreducible icosahedral representations in a D₂ quantization. Real and complex forms are discussed. The results are listed in practical subduction tables, both for integer (J = 1 → 6) and half-integer (J = 1/2 → 11/21 harmonics.     

Schur duality in the toroidal setting

The classical Frobenius-Schur duality gives a correspondence between finite dimensional representations of the symmetric and the linear groups. The goal of the present paper is to extend this construction to the quantum toroidal setup with only elementary (algebraic) methods. This work can be seen as a continuation of [J, D1 and C2] (see also [C-P and G-R-V]) where the cases of the quantum groups U _q (sl(n)), Y(sl(n)) (the Yangian) and U _q (sl(n)) are given. In the toroidal setting the two algebras involved are deformations of Cherednik's double affine Hecke algebra introduced in [C1] and of the quantum toroidal group as given in [G-K-V]. Indeed, one should keep in mind the geometrical construction in [G-R-V] and [G-K-V] in terms of equivariant K-theory of some flag manifolds. A similar K-theoretic construction of Cherednik's algebra has motivated the present work. At last, we would like to lay emphasis on the fact that, contrary to [J, D1 and C2], the representations involved in our duality are infinite dimensional. Of course, in the classical case, i.e.,q=1, a similar duality holds between the toroidal Lie algebra and the toroidal version of the symmetric group. The authors would like to thank V. Ginzburg for a useful remark on a preceding version of this paper. Communicated by M. Jimbo     

Simple construction ofSU(3) representations using theSU(2) projection technique

SU(3) representations for theSU(3) SU(2) chain are constructed in a new way usingSU(2) projection operators. TheSU(3) D-functions, presented in new parametrizations, are expressed in terms ofSU(2) D-functions and 6j-symbols.Valuable discussions with Dr. J. Niederle are gratefully acknowledged.     

Canonical Representations of Sp(1,n) Associated with Representations of Sp(1)

Canonical representations of Sp(1,n) associated with finite dimensional irreducible representations of Sp(1) are defined using vector-valued Berezin kernels. Their decomposition into irreducible representations is determined by decomposing the corresponding reproducing distributions in terms of positive definite trace spherical functions on Sp(1,n). The canonical representatons are also identified with the restriction to Sp(1,n) of certain maximal degenerate representations of SL(n+1,H). Received: 1 May 1998 / Accepted: 18 November 1998     

Quantum Algebra U q(gl(3)) and Nonlinear Optics

Indecomposable representations are investigated for the U _q(gl(3)) quantum algebra. The matrix elements are explicitly determined for the elementary representations, and the extremal vectors which characterize invariant subspaces are given in explicit form. Quotient spaces are used to derive other representations from the elementary representations, including the finite-dimensional irreducible representations and infinite-dimensional representations which are bounded above. Applications to nonlinear-optical phenomena are discussed.     

Partial wave analysis of dilatation-invariant relativisticS-matrix

Free particle states andS-matrix are defined on carrier spaces of representations of the Weyl groupW (Poincaré group extended by dilatations). By using Clebsch-Gordan coefficients ofW the extension of the relativistic partial wave analysis of the two-bodyS-matrix is obtained. Scattering of particles with arbitrary masses and spins is treated and a comparison with relativistic case is discussed.One of the authors (J. N.) would like to thank Professor M.Flato for a stimulating discussion.     

Soliton equations and fundamental representations of A 2l (2)

Fundamental representations of the Euclidean Lie algebra A _2l ⁽²⁾ is constructed by decomposing the vertex representations of gI(∞). For l=1 the multiplicities of highest weights are determined. Soliton equations associated with each of these representations are also discussed.     

On the representations of deformed Galilei group

The projective representations of k-Galilei group G _k are found by contracting the relevant representations of –Poincare group. The projective multiplier is found. It is shown that it is not possible to replace the projective representations of G _k by vector representations of some its extension.     

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.     

Nonclassical type representations of the q-deformed algebra U′q(son)

The nonstandard q-deformation U_q(so_n) of the universal enveloping algebra U(so _n ) has irreducible finite dimensional representations which are a q-deformation of the well-known irreducible finite dimensional representations of U(so _n ). But U_q(so_n) also has irreducible finite dimensional representations which have no classical analogue. The aim of this paper is to give these representations which are called nonclassical type representations. They are given by explicit formulas for operators of the representations corresponding to the generators of U_q(so_n).     

The exponential map for representations ofU p,q (gl(2))

For the quantum groupGL _p,q (2) and the corresponding quantum algebraU _p,q (gl(2)) Fronsdal and Galindo [Lett. Math. Phys.27 (1993) 59] explicitly constructed the so-called universalT-matrix. In a previous paper [J. Phys. A28 (1995) 2819] we showed how this universalT-matrix can be used to exponentiate representations from the quantum algebra to get representations (left comodules) for the quantum group. Here, further properties of the universalT-matrix are illustrated. In particular, it is shown how to obtain comodules of the quantum algebra by exponentiating modules of the quantum group. Also the relation with the universalR-matrix is discussed.Presented at the 4th International Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.     

Deformations of representations

A connection between deformation of Lie group representations and deformations of associated Lie algebra representations is established. Applications are given to the theory of analytic continuation of K-finite quasi-simple representations of semi-simple Lie groups. A construction process of all TCI representations of SL(2,R) is obtained.     

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.     

Spectra,eigenvectors and overlap functions for representation operators ofq-deformed algebras

Operators of representations corresponding to symmetric elements of theq-deformed algebrasU _q (su_1,1),U _q (so_2,1),U _q (so_3,1),U _q (so _n ) and representable by Jacobi matrices are studied. Closures of unbounded symmetric operators of representations of the algebrasU _q (su_1,1) andU _q (so_2,1) are not selfadjoint operators. For representations of the discrete series their deficiency indices are (1,1). Bounded symmetric operators of these representations are trace class operators or have continuous simple spectra. Eigenvectors of some operators of representations are evaluated explicitly. Coefficients of transition to eigenvectors (overlap coefficients) are given in terms ofq-orthogonal polynomials. It is shown how results on eigenvectors and overlap coefficients can be used for obtaining new results in representation theory ofq-deformed algebras.     

A Kohno–Drinfeld Theorem for the Monodromy of Cyclotomic KZ Connections

We compute explicitly the monodromy representations of “cyclotomic” analogs of the Knizhnik–Zamolodchikov differential system. These are representations of the type B braid group B_n¹{B_n^1} . We show how the representations of the braid group B _n obtained using quantum groups and universal R-matrices may be enhanced to representations of B_n¹{B_n^1} using dynamical twists. Then, we show how these “algebraic” representations may be identified with the above “analytic” monodromy representations.     

On fine gradings and their symmetries

In this paper fine gradings ofgl(n, C) associated with the Pauli matrices inn dimensions are studied with the subsequent graded contractions ofsl(n, C) in view. It is shown that, ifn≥3 is a prime, the discrete symmetries of the gradings involve the specialn-dimensional representations ofSL(2,F _n), whereF _n is the finite field of ordern. These symmetries may be used to simplify the system of contraction equations. Presented by J. Tolar at the DI-CRM Workshop held in Prague, 18–21 June 2000.     

Localization of Unitary Braid Group Representations

Governed by locality, we explore a connection between unitary braid group representations associated to a unitary R-matrix and to a simple object in a unitary braided fusion category. Unitary R-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary R-matrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q = e ^πi/6 specialization of the unitary Jones representation of the braid groups can be localized by a unitary 9 × 9 R-matrix. Actually this Jones representation is the first one in a family of theories (SO(N), 2) for an odd prime N > 1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science.     

The representations of the oscillator group

Using the Mackey theory of induced representations all the unitary continuous irreducible representations of the 4-dimensional Lie groupG generated by the canonical variables and a positive definite quadratic hamiltonian are found. These are shown to be in a one to one correspondence with the orbits underG in the dual spaceG to the Lie algebraG ofG, and the representations are obtained from the orbits by inducing from one-dimensional representations provided complex subalgebras are admitted. Thus a construction analogous to that ofKirillov andBernat gives all the representations of this group.The research reported in this document has been sponsored in part by the Air Force Office of Scientific Research OAR through the European Office Aerospace Research, United States Air Force.