By Helmer Aslaksen, Eng-Chye Tan, Chen-bo Zhu (auth.), Bruno Gruber (eds.)

ISBN-10: 1461357837

ISBN-13: 9781461357834

ISBN-10: 1461519152

ISBN-13: 9781461519157

Invariant conception of Matrices; *H. Aslaksen, et al.* Symmetries of easy debris Revisited; *A.O. Barut.* Perturbative SU(1,1); *H.**Beker.* A twin constitution for the Quantal Rotation staff, SU(2); *L.C.**Biedenharn, M.A. Lohe.* a few issues within the Quantization of Relativistic Grassmann based interplay structures; *A. Del Sol Mesa, R.P.**Martinez y Romero.* q-Difference Intertwining Operators for Uq(sI(4)) and q-Conformal Invariant Equations; *V.K. Dobrev.* A Quantum Mechanical Evolution Equation for combined States from Symmetry and Kinematics; *H.D.**Doebner, J.D. Hennig.* Quantum Mechanical Motions over the gang Manifolds and similar Potentials; *I.H. Duru.* Quantum Violation of vulnerable Equivalence crucial within the Brans-Dicke thought; *Y. Fujii.* Quantum Unitary and Pseudounitary teams and Generalized Hadron Mass family; *A.M. Gavrilik.* Linear Coxeter teams; *J. Getino.* Diffeomorphism teams, Quasiinvariant Measures, and endless Quantum platforms; *G.A. Goldin, U. Moschella.* Algebraic Shells and the Interacting Boson version of the Nucleus; *B. Gruber.* fresh advancements within the software of Vector Coherent States; *K.T. Hecht.* Algebraic conception of the Threebody challenge; *F. Iachello.* 18 extra articles. Index.

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It is very useful for the further development of this idea to have an explicit realization of the corresponding (6 - j) operators as linear operators on a well-defined set of vector spaces which carry irreps of the dual algebra. That such a realization exists [4] is in itself rather surprising. Consider the unitary group U(3). Then the complete set of (integral) unitary irreps can be labelled by the Young frames denoted by [m] == [m13m23m33], where the mi3 are (positive, negative or zero) integers such that m13 ~ m23 ~ m33.

We will denote the coupling effected by the (6-j) operators by the symbol D. 3). 9), identify the coupling 0 by (6-j) coefficients to be the analog-for the dual Hopf algebra-of the coupling x by the wee coefficients for the SU(2) Hopf algebra. It follows that we can determine both the algebra and the co-algebra structure of the dual Hopf algebra from the (6- j) coupling coefficients. 3 The Dual Algebra We have stated (in Section 2) that the coupling law for the group (the WCG coefficients) determines the Lie algebra structure.

2 These operators are of two types - differential and integral. For the canonical construction of the integral invariant operators (which we shall not consider) we refer to [2]. As stated we are interested in the invariant differential operators. , [1], however, most of these rely on constructions which are not yet available for quantum groups. Here we shall apply a procedure [5] which is rather algebraic and can be generalized almost straightforwardly to quantum groups. 3. 2. Here we shall sketch the procedure of [5] illustrating the general notions with the conformal group SU(2,2).

### Symmetries in Science VIII by Helmer Aslaksen, Eng-Chye Tan, Chen-bo Zhu (auth.), Bruno Gruber (eds.)

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