The group with Grassmann structureUOSP(1.2)

The finite-dimensional representations of the Lie superalgebraosp(1.2) and the group with Grassmann structureOSP(1.2) have been studied. The explicit expression of the projection operator of the superalgebraosp(1.2) has been found. The operator permits an arbitrary finite-dimensional representation to be expanded in the components multiple to the irreducible ones. The Clebsch-Gordan coefficients for the tensor product of two arbitrary irreducible representations have been obtained. The matrix elements of the irreducible representations of the groupUOSP(1.2) [the analoque of the compact form of the groupOSP(1.2)] are studied. The explicit form of these matrix elements, the differential equations satisfied by them, and the integral of their product have been found.     

SU3羣不可约表示直乘的分解

本文利用SU₃羣无穷小算子的对易关系,求出了SU₃羣的所有不可约U表示,并导出了SU₃羣的约化系数满足的方程。作为例子,我们计算了SU₃羣的(01)×(10),(11)×(10),(11)×(11),(30)×(11)的约化系数。     

The Simultaneous Coupling Scheme for SU(3)

We present a simple method for calculating the Clebsch-Gordan coefficients for the tensor product of two unitary irreducible representations of SU(3). The calculation given here is an application and extension of the simultaneous coupling scheme proposed in 2010 for the rotation group SU(2).     

The Clebsch-Gordan coefficients of SU(4)

A set of recurrence relations connecting the matrix elements of finite transformation belonging to the same irreducible representation of SU(4) is used to obtain a wide class of matrix elements. An expression for the Clebsch-Gordan coefficient is obtained by integrating the product of three matrix elements belonging to three different irreducible representations of the group. The symmetry properties of the matrix elements and the Clebsch-Gordan coefficients are discussed.The author is grateful to Professors S.Datta Majumdar and G.Bandyopadhyay of the Department of Physics, I.I.T., Kharagpur, for many helpful discussions. This work was supported by the C.S.I.R., Government of India.     

Notes on quaternionic group representations

We study quaternionic group representations of finite groups systematically and obtain some basic tools of the theory, such as orthogonality relations and the Clebsch-Gordan series for reducible representations. We also derive all irreducible inequivalentQ-representations of a groupG, classifying them according to a suitable generalization of the Wigner-Frobenius-Schur classification.     

Finite dimensional representations of the quantum Lorentz group

All finite dimensional irreducible representations of the quantum Lorentz group SL _q (2,) are described explicitly and it is proved all finite dimensional representations of SL _q (2,) are completely reducible. The conjecture of Podle and Woronowicz will be answered affirmatively.     

SU3群的多项式基底及其Clebsch-Gordan系数的明显表达式

利用对称性与Young图在多项式空间上构成了SU_n群的不可约表示及积表示的完全基底,从而不但区分了非单纯权也区分了积表示的非简单可约性,并在此基础上用构成不变量法得到SU₃群的Clebsch-Gordan系数的明显表达式     

The jordanian deformation of Su(2) and clebsch-gordan coefficients

Representation theory for the Jordanian quantum algebraU _h (sl(2)) is developed using a nonlinear relation between its generators and those of sl(2). Closed form expressions are given for the action of the generators ofU _h (sl(2)) on the basis vectors of finite dimensional irreducible representations. In the tensor product of two such representations, a new basis is constructed on which the generators ofU _h (sl(2)) have a simple action. Using this basis, a general formula is obtained for the Clebsch-Gordan coefficients ofU _h (sl(2)). Some remarkable properties of these Clebsch-Gordan coefficients are derived. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.     

Irreducible tensors for the groupSU 3

The explicit determination of the matrix elements of theSU ₃ irreducible tensors is carried out by a purely algebraic method. These expressions may be used to compute the Clebsch-Gordan coefficients by orthogonalisation. For the special case of (0,q) tensors simple formulas are derived.In partial fulfillment for the requierements of the doctoral degree at the Institute of Physics of the Academy.     

Circular matrices,duality theorems and the reality of Clebsch-Gordan coefficients

Schur's lemma and the properties of circular matrices are used to establish a necessary and sufficient condition for the finite-dimensional irreducible matrix representations of an arbitrary groupG to admit real coupling (or Clebsch-Gordan) coefficients. The Pontryagin-Van Kampen and Tannaka-Krein duality theorems are found to be of considerable value in implementing the condition, which requires that complex conjugation effects an automorphism on the group of all matrices having the same reduction of their tensor products as the matrix representations ofG. This result is noted to be relevant to a generalization of the Frobenius-Schur invariant.     

Representations of the cyclically symmetric q-deformed algebra U q(so3)

An algebra homomorphism from the nonstandard q-deformed (cyclically symmetric) algebra U _q(so₃) to the extension Û _q(sl₂) of the Hopf algebra U _q(sl₂) is constructed. Not all irreducible representations (IR) of U _q(sl₂) can be extended to representations of Û _q(sl₂). Composing the homomorphism with irreducible representations of Û _q(sl₂) we obtain representations of U _q(so₃). Not all of these representations of U _q(so₃) are irreducible. Reducible representations of U _q(so₃) are decomposed into irreducible components. In this way we obtain all IR of U _q(so₃) when q is not a root of unity. A part of these representations turn into IR of the Lie algebra so₃ when q 1.     

Algebraic investigations in landau model of structural phase transitions

First we introduce the basic notions of the theory of permutation representations: stabilizers, orbits, stable subsets and strata. Then we consider the relation between permutation and linear representations which lead to some formulae connecting subduction coefficients, Clebsch-Gordan coefficients, and dimensions of stability spaces. This relation also leads to the concept of suborbits. Epikernels, the subgroups which are stabilizers of vectors of irreducible subspaces (either on the complex or on the real field) — are, studied and several theorems about them are proved. Further we consider the relation between epikernels, stability spaces and strata for subspaces irreducible on the real field as compared with subspaces irreducible on the complex field. Finally, the exomorphism is defined with use of permutation representations. The vectors of irreducible subspaces and corresponding epikernels (their stabilizers) for real ireps (representations irreducible on the real) of the classical crystal point groups are given in the appendix.     

An Operator Method for the Addition of Two Angular Momenta by Symmetrizing the Product of n Spin Functions

A general expression of Clebsch-Gordan coefficients is derived by means of de Broglie-like méthode de fusion: the second angular momentum is added to the first in n steps of half each, the maximum symmetric composition of the n spin functions being taken into account explicitly. The procedure is operatorial. Group-theoretically speaking the irreducible representations of the direct product of two irreducible representations of the rotation group are realized via symmetric group S_n with the aid of projection operators (equivalent to Young's symmetrizers). The decomposition of the representations of S_n therein is performed inductively by utilizing the chain of subgroups: S_n ⊃ S_n−1 … ⊃ S₂ ⊃ S₁.     

Irreducible representations of the quantum analogue of SU(2)

We give the complete set of irreducible representations of U(SU(2))_q when q is a mth root of unity. In particular, we show that their dimensions are less or equal to m. Some of them are not highest-weight representations.     

Positive-energy irreps of the quantum anti de Sitter algebra

We obtain positive-energy irreducible representations of theq-deformed anti de Sitter algebraU _q (so(3, 2)) by deformation of the classical ones. When the deformation parameterq isN-th root of unity, all these irreducible representations become unitary and finite-dimensional. Generically, their dimensions are smaller than those of the corresponding finite-dimensional non-unitary representations ofso(3, 2). We discuss in detail the singleton representations, i.e. the Di and Rac. WhenN is odd, the Di has dimension 1/2(N ²–1) and the Rac has dimension 1/2(N ²+1), while ifN is even, both the Di and Rac have dimension 1/2N ². These dimensions are classical only forN=3 when the Di and Rac are deformations of the two fundamental non-unitary representations ofso(3, 2).Presented at the 4th Colloquium Quantum groups and integrable systems, Prague, 22–24 June 1995.On leave from Bulgarian Acad. Sci., Institute of Nuclear Research and Nuclear Energy, 72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria.On leave from Pennsylvania State University (Fulbright scholar).     

Nonstandard GLh(n) quantum groups and contraction of covariant q-bosonic algebras

GL_h(n) × GL_h(m)-covariant h-bosonic algebras are built by contracting the GL_q(n) × GL_q(m)-covariant q-bosonic algebras considered by the present author some years ago. Their defining relations are written in terms of the corresponding R _h-matrices. Whenever n = 2, and m = 1 or 2, it is proved by using U_h(sl(2)) Clebsch-Gordan coefficients that they can also be expressed in terms of coupled commutators in a way entirely similar to the classical case. Some U_h(sl(2)) rank-(1/2) irreducible tensor operators, recently constructed by Aizawa in terms of standard bosonic operators, are shown to provide a realization of the h-bosonic algebra corresponding to n = 2 and m = 1.     

Multiplet classification of the reducible elementary representations of real semisimple Lie groups: the SO e (p,q) example

A multiplet classification of the reducible elementary representations (=generalized principal series representations) is proposed as a useful intermediate step in the important problem of determining the irreducible composition factors of the elementary representations and their multiplicities and, thus, of the distribution characters of all irreducible representations of real semisimple Lie groups. As an example, the results for the elementary representations of SO _e (p, q) induced from its minimal parabolic subgroup are given. Further research is also indicated. Presented at the 1st National Congress of Bulgarian Physicists, Sofia, September 1983.     

The cyclic representations of the quantum superalgebra U q osp(2, 1) withq a root of unity

By generalizing De Concini and Kac's cyclic representation theory of quantum groups at roots of unity, the cyclic representations of the quantum superalgebra U _q osp(2, 1) are constructed in three classes: irreducible representations with single multiplicities, irreducible representations with the multiplicities larger than one, and indecomposable representations.This work is supported in part by the National Sciene Foundation in China.     

The GL q (2) gauge theory

We present some partial results on theq-deformation of the GL(2) Yang-Mills theory. In particular, the irreducible representations needed to describe the complete set of physical states are obtained by a simple procedure.     

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).