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  • 1
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    Harmonic Analysis : on the n-Dimensional Lorentz Group and Its Application to Conformal Quantum Field Theory (1977)
    Berlin ; Heidelberg : Springer
    Details
    ISBN: 9783540081500
    URL: http://www.springerlink.com/openurl.asp?genre=book&isbn=978-3-540-08150-0
    Language: English
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  • 2
    Electronic Resource
    Electronic Resource
    Duality for the matrix quantum group GLp,q(2,C) (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 3419-3430 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: It is shown that the algebra Up,q dual to GLp,q(2,C) is isomorphic to U(pq)1/2(sl(2,C)) ⊗ Z as a commutation algebra, where Z is a subalgebra central in Up,q. The subalgebra Z is a Hopf subalgebra of Up,q, while the commutation subalgebra U(pq)1/2(sl(2,C)) is not a Hopf subalgebra.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529890
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  • 3
    Electronic Resource
    Electronic Resource
    Normalized Uq(sl(3)) Gel'fand–(Weyl)–Zetlin basis and new summation formulas for q-hypergeometric functions (2000)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 41 (2000), S. 7752-7768 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: An explicit realization of the normalized Gel'fand–(Weyl)–Zetlin (GWZ) basis for Uq(sl(3)) in terms of polynomial functions in three variables (real or complex) is given. The construction uses two different realizations of the Uq(sl(3)) unnormalized GWZ basis which were given previously, and whose transformation properties were not known. It turns out that finding these properties enables us to find the (GWZ-dependent) proportionality constant between these two realizations. The scalar product is also fixed by this in both unnormalized realizations, and then, by normalization, the normalized GWZ states are obtained. As by-products new summation formulas are obtained which seem new also for q=1. The main new formula is a double sum which is given in terms of the proportionality constant mentioned above. This double sum can be written as single sum over a q−3F2 hypergeometric function, or as a q-hypergeometric function of two variables. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1310360
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  • 4
    Electronic Resource
    Electronic Resource
    Representation theory approach to the polynomial solutions of q-difference equations: Uq(sl(3)) and beyond (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 6058-6075 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A new approach to the theory of polynomial solutions of q-difference equations is proposed. The approach is based on the representation theory of simple Lie algebras G and their q-deformations and is presented here for Uq(sl(n)). First a q-difference realization of Uq(sl(n)) in terms of n(n−1)/2 commuting variables and depending on n−1 complex representation parameters, ri, is constructed. From this realization lowest weight modules (LWM) are obtained which are studied in detail for the case n=3 (the well-known n=2 case is also recovered). All reducible LWM are found and the polynomial bases of their invariant irreducible subrepresentations are explicitly given. This also gives a classification of the quasi-exactly solvable operators in the present setting. The invariant subspaces are obtained as solutions of certain invariant q-difference equations, i.e., these are kernels of invariant q-difference operators, which are also explicitly given. Such operators were not used until now in the theory of polynomial solutions. Finally, the states in all subrepresentations are depicted graphically via the so-called Newton diagrams.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530728
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  • 5
    Electronic Resource
    Electronic Resource
    Erratum: Structural analysis and elementary representations of SL(4, R) and GL(4, R) and their covering groups [J. Math. Phys. 27, 883 (1986)] (1986)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 27 (1986), S. 1927-1927 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527062
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  • 6
    Electronic Resource
    Electronic Resource
    Elementary representations and intertwining operators for SU(2, 2). I (1985)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 26 (1985), S. 235-251 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The structure of the group SU(2, 2) and of its Lie algebra is studied in detail. The results will be applied in subsequent parts devoted to the explicit construction of elementary representations of SU(2, 2) induced from different parabolic subgroups and of the intertwining operators between these representations. A summary of some results of Parts II and III is given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.526653
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  • 7
    Electronic Resource
    Electronic Resource
    Structural analysis and elementary representations of SL(4, R) and GL(4, R) and their covering groups (1986)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 27 (1986), S. 883-899 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The structure of the groups SL(4,R) and GL(4,R) and their universal cover− ing groups SL(4,R) and GL(4,R), respectively, and Lie algebras sl(4,R) and gl(4,R), respectively, are studied. The parabolic subgroups and subalgebras are identified and the cuspidal parabolic subgroups singled out. The Iwasawa and Bruhat decompositions are given explicitly. All elementary representations (ER) of SL(4,R) are explicitly given in two equivalent realizations. Using the preceding detailed structural analysis the SL(4,R) constructions are used for the explicit realization of all ER of SL(4,R), GL(4,R), and GL(4,R). The results shall be applied (among other things) elsewhere for the construction of all irreducible representations of the above groups.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.527162
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  • 8
    Electronic Resource
    Electronic Resource
    Irregular Uq(sl(3)) representations at roots of unity via Gel'fand–(Weyl)–Zetlin basis (1997)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 38 (1997), S. 2631-2651 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The Gel'fand–(Weyl)–Zetlin (GWZ) description of the Uq(sl(3)) irregular irreps at roots of unity is explicitly given. Those are irreps fixed by the same parameters as the unitary irreducible representations (UIRs) of SU(3) yet having dimensions smaller than their classical counterparts, the reason being that to obtain an irregular irrep one has to make factorization of an additional submodule. This description is made geometrically transparent by an arrangement of the standard SU(3) GWZ basis in a hexagonal pyramid, which is valid for any q and seems new even for q=1. The pyramid has as a base the standard hexagon which gives the weight space of the UIRs of SU(3) in the plane of third component of isospin Iz and hypercharge Y, while third dimension of this pyramid is related to the isospin I. Algebraically this arrangement is related to a one-to-one correspondence between the abstract GWZ states and monomials in the algebra of raising generators Uq(G+); however, those monomials are not in the standard Poincaré–Birkhoff–Witt (PBW) basis of Uq(G+). The additional factorization corresponds to taking away an upper part of the pyramid, itself being a hexagonal pyramid representing another SU(3) irrep of smaller dimension, which for roots of unity becomes the submodule to be factored out. The technical tool in this factorization is the explicit coincidence of two polynomials: one giving the singular vectors of the Verma modules, the other used in the algebraic description of the pyramid. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532001
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  • 9
    Electronic Resource
    Electronic Resource
    q-difference intertwining operators for a Lorentz quantum algebra (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 971-985 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Representations πˆr,r¯ of a Lorentz quantum algebra U are constructed. They are labeled by two complex numbers r,r¯ and act in the space of formal power series of two noncommuting variables η,η¯. These variables are built from elements of the matrix Lorentz quantum group L which is dual to U. The conditions for reducibility of πˆr,r¯ are given. q-difference intertwining operators in η, η¯, which realize partial equivalences of the representations πˆr,r¯ , are constructed explicitly. The whole construction is a generalization of a known procedure for q=1. The case when q is a root of unity is also considered in detail.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530624
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  • 10
    Electronic Resource
    Electronic Resource
    Polynomial realization of the Uq(sl(3)) Gel'fand–(Weyl)–Zetlin basis (1997)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 38 (1997), S. 3750-3767 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: An explicit realization of the U≡Uq(sl(3)) Gel'fand–(Weyl)–Zetlin (GWZ) basis as polynomial functions in three variables (real or complex) is given. This realization is obtained in two complementary ways. First, a known correspondence is used between the abstract GWZ basis and explicit polynomials in the quantum subgroup U+ of the raising generators. Then an explicit construction is used of arbitrary lowest weight (holomorphic) representations of U in terms of three variables on which the generators of U are realized as q-difference operators. The application of the GWZ corresponding polynomials in this realization of the lowest weight vector (the function 1) produces the first realization of this GWZ basis. Another realization of the GWZ polynomial basis is found by the explicit diagonalization of the operators of isospin Î2, third component of isospin Îz, and hypercharge Y(circumflex), in the same realization as q-difference operators. The result is that the eigenvectors can be written in terms of q-hypergeometric polynomials in the three variables. Finally an explicit scalar product is constructed by adapting the Shapovalov form to this setting. The orthogonality of the GWZ polynomials with respect to this scalar product is proven using both realizations. This provides a polynomial construction for the orthonormal GWZ basis. The results here are for generic q, leaving the root of unity case for a following paper. It seems that the results are new also in the classical situation (q=1). © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532074
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