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  • Articles  (58)
  • 81R50  (58)
  • 1990-1994  (58)
  • Mathematics  (58)
  • Electrical Engineering, Measurement and Control Technology
  • Energy, Environment Protection, Nuclear Power Engineering
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  • Articles  (58)
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  • Mathematics  (58)
  • Electrical Engineering, Measurement and Control Technology
  • Energy, Environment Protection, Nuclear Power Engineering
  • Physics  (58)
  • 1
    Electronic Resource
    Electronic Resource
    On affine yangians (1994)
    Springer
    Letters in mathematical physics 32 (1994), S. 269-274 
    Details
    ISSN: 1573-0530
    Keywords: 17B37 ; 81R50 ; 82B23
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract A Yangian $$Y(\widetilde{sl}(2))$$ , a deformation of the universal enveloping algebra of the two-dimensional loop algebra sl(2) ⊗C [t −1,t;u], is constructed. This deformation is an analogue of a Yangian which was constructed by V. Drinfeld for any simple Lie algebra. The PBW theorem for $$Y(\widetilde{sl}(2))$$ is proved and some representations are constructed. Like usual Yangians, $$Y(\widetilde{sl}(2))$$ possesses a one-dimensional group of auto- morphisms and at zero level - a two-dimensional group of automorphisms. This observation allows one to conjecture that the representation theory of $$Y(\widetilde{sl}(2))$$ should give rise to new solutions of QYBE. Yangians of other affine algebras can be constructed similarly and they enjoy similar properties.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00761137
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  • 2
    Electronic Resource
    Electronic Resource
    Topological representations of U q (sl2(ℂ)) on the torus and the mapping class group (1994)
    Springer
    Letters in mathematical physics 30 (1994), S. 71-85 
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    ISSN: 1573-0530
    Keywords: 81R50 ; 81Txx
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We compute the mapping class group action on cycles on the configuration space of the torus with one puncture, with coefficients in a local system arising in conformal field theory. This action commutes with the topological action of the quantum group U q (sl2(ℂ)), and is given in vertex form.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00761424
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  • 3
    Electronic Resource
    Electronic Resource
    The quantum de Rham complexes associated with SL h (2) (1994)
    Springer
    Letters in mathematical physics 30 (1994), S. 87-98 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 16S80 ; 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Quantum de Rham complexes on the quantum plane and the quantum group itself are constructed for the nonstandard deformation of Fun(SL(2)). It is shown that in contrast to the standardq-deformation of SL(2), the above complexes are unique for SL h (2). Also, as a byproduct, a new deformation of the two-dimensional Heisenberg algebra is obtained which can be used to construct models ofh-deformed quantum mechanics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939696
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  • 4
    Electronic Resource
    Electronic Resource
    Differential calculus on the inhomogeneous quantum groups IGL q (n) (1994)
    Springer
    Letters in mathematical physics 30 (1994), S. 233-239 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 22E70 ; 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We obtain the inhomogeneousq-groups IGL q (n) via a projection from GL q (n + 1). The bicovariant differential calculus of IGL q (n) is constructed, and the corresponding quantum Lie algebra is given explicitly.
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    URL: http://dx.doi.org/10.1007/BF00805855
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  • 5
    Electronic Resource
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    Free boson representation of $$U_q (\widehat{sl}_3 )$$ (1994)
    Springer
    Letters in mathematical physics 30 (1994), S. 207-216 
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    ISSN: 1573-0530
    Keywords: 81R50 ; 16W30 ; 17B37
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract A representation of the quantum affine algebra $$U_q (\widehat{sl}_3 )$$ of an arbitrary levelk is constructed in the Fock module of eight boson fields. This realization reduces the Wakimoto representation in theq → 1 limit. The analogues of the screening currents are also obtained. They commute with the action of $$U_q (\widehat{sl}_3 )$$ modulo total differences of some fields.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00805853
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  • 6
    Electronic Resource
    Electronic Resource
    On aq-deformation of the discrete Painlevé I equation andq-orthogonal polynomials (1994)
    Springer
    Letters in mathematical physics 30 (1994), S. 327-336 
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    ISSN: 1573-0530
    Keywords: 17B37 ; 33D45 ; 39A70 ; 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We present aq-analog of the discrete Painlevé I equation, and a special realization of it in terms ofq-orthogonal polynomials.
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    URL: http://dx.doi.org/10.1007/BF00751068
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  • 7
    Electronic Resource
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    Metric on quantum spaces (1994)
    Springer
    Letters in mathematical physics 31 (1994), S. 35-39 
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    ISSN: 1573-0530
    Keywords: 46L87 ; 81R50 ; 17B37
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We introduce the analogue of the metric tensor in the case ofq-deformed differential calculus. We analyse the consequences of the existence of the metric, showing that this enforces severe restrictions on the parameters of the theory. We discuss in detail examples of the Manin plane and theq-deformation of SU(2). Finally we touch the topic of relations with Connes' approach.
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    URL: http://dx.doi.org/10.1007/BF00751169
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  • 8
    Electronic Resource
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    Quantization of U q [so(2n + 1)] with deformed para-Fermi operators (1994)
    Springer
    Letters in mathematical physics 31 (1994), S. 151-157 
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    ISSN: 1573-0530
    Keywords: 81R50 ; 16W30 ; 17B37
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The observation thatn pairs of para-Fermi (pF) operators generate the universal enveloping algebra of the orthogonal Lie algebra so(2n + 1) is used in order to define deformed pF operators. It is shown that these operators are an alternative to the Chevalley generators. With this background U q [so(2n + 1)] and its ‘Cartan-Weyl’ generators are written down entirely in terms of deformed para-Fermi operators.
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    URL: http://dx.doi.org/10.1007/BF00750149
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  • 9
    Electronic Resource
    Electronic Resource
    R-matrix method for Heisenberg quantum groups (1994)
    Springer
    Letters in mathematical physics 31 (1994), S. 159-166 
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    ISSN: 1573-0530
    Keywords: 17B37 ; 81R50 ; 70H05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract TheR-matrix method is systematically applied to get several Heisenberg quantum groups depending on two or three parameters. It turns out that the associatedR-matrices have to verify a weaker form of the QYBE. Only for particular cases of quantum groups, we can imposeR to be a solution of the QYBE. The corresponding quantum Heisenberg Lie algebras are obtained by duality.
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    URL: http://dx.doi.org/10.1007/BF00750150
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  • 10
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    Quantum Heisenberg groups and sklyanin algebras (1994)
    Springer
    Letters in mathematical physics 31 (1994), S. 167-177 
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    ISSN: 1573-0530
    Keywords: Primary: 17B37 ; Secondary: 16W30 ; 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract New bialgebra structures on the Heisenberg-Lie algebra and their quantizations are introduced. Some of these quantizations give rise to new multiplications in homogeneous coordinate rings of Abelian varieties, via the well-known identification of theta functions with suitable matrix coefficients of the Stone-von Neumann representations.
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    URL: http://dx.doi.org/10.1007/BF00761709
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  • 11
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    Classification of three-dimensional covariant differential calculi on Podles' quantum spheres and on related spaces (1994)
    Springer
    Letters in mathematical physics 32 (1994), S. 25-36 
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    ISSN: 1573-0530
    Keywords: 17B37 ; 46L87 ; 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Covariant differential calculi on the quantum space $$\mathfrak{X}_{q\lambda \rho }$$ for the quantum group SL q (2) are classified. Our main assumptions are thatq is not a root of unity and that the differentials de j of the generators of $$\mathfrak{X}_{q\lambda \rho }$$ form a free right module basis for the first-order forms. Our result says, in particular, that apart from the two casesc =c(3), ∞ there exists a unique differential calculus with the above properties on the space $$\mathfrak{X}_{q\lambda \rho }$$ which corresponds to Podles' quantum sphereS qc /2 .
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    URL: http://dx.doi.org/10.1007/BF00761121
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  • 12
    Electronic Resource
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    Representation theory of quantized poincaré algebra. tensor operators and their applications to one-particle systems (1994)
    Springer
    Letters in mathematical physics 32 (1994), S. 85-101 
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    ISSN: 1573-0530
    Keywords: 81R50 ; 16W30 ; 17B37
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract A representation theory of the quantized Poincaré (κ-Poincaré) algebra (QPA) is developed. We show that the representations of this algebra are closely connected with the representations of the nondeformed Poincaré algebra. A theory of tensor operators for QPA is considered in detail. Necessary and sufficient conditions are found in order for scalars to be invariants. Covariant components of the four-momenta and the Pauli-Lubanski vector are explicitly constructed. These results are used for the construction of someq-relativistic equations. The Wigner-Eckart theorem for QPA is proven.
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    URL: http://dx.doi.org/10.1007/BF00739419
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  • 13
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    A simple construction of ellipticR-matrices (1994)
    Springer
    Letters in mathematical physics 32 (1994), S. 167-171 
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    ISSN: 1573-0530
    Keywords: 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We show that Belavin's solutions of the quantum Yang-Baxter equation can be obtained by restricting an infiniteR-matrix to suitable finite-dimensional subspaces. This infiniteR-matrix is a modified version of the Shibukawa-UenoR-matrix acting on functions of two variables.
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    URL: http://dx.doi.org/10.1007/BF00739425
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  • 14
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    Reflection equations andq-Minkowski space algebras (1994)
    Springer
    Letters in mathematical physics 32 (1994), S. 173-182 
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    ISSN: 1573-0530
    Keywords: 81P05 ; 81R50 ; 16W30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We express the defining relations of theq-deformed Minkowski space algebra as well as that of the corresponding derivatives and differentials in the form of reflection equations. This formulation encompasses the covariance properties with respect to the quantum Lorentz group action in a straightforward way. Different equivalences ofq-Minkowski algebras are pointed out.
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    URL: http://dx.doi.org/10.1007/BF00750660
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  • 15
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    TrigonometricR matrices related to ‘dilute’ Birman-Wenzl-Murakami algebra (1994)
    Springer
    Letters in mathematical physics 32 (1994), S. 183-187 
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    ISSN: 1573-0530
    Keywords: 20F36 ; 81R50 ; 82B23
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Explicit expressions for three series ofR matrices which are related to a ‘dilute’ generalisation of the Birman-Wenzl-Murakami algebra are presented. Of those, one series is equivalent to the quantumR matrices of theD n+1 (2) generalised Toda systems, whereas the remaining two series appear to be new.
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    URL: http://dx.doi.org/10.1007/BF00750661
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  • 16
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    A poisson—lie structure on the diffeomorphism group of a circle (1994)
    Springer
    Letters in mathematical physics 32 (1994), S. 307-313 
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    ISSN: 1573-0530
    Keywords: 17B66 ; 22E65 ; 58D05 ; 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Starting from the Gelfand-Fuks-Virasoro cocycle on the Lie algebraX(S 1) of the vector fields on the circleS 1 and applying the standard procedure described by Drinfel'd in a finite dimension, we obtain a classicalr-matrix (i.e. an elementr ∈ X(S 1) ∧X(S 1) satisfying the classical Yang-Baxter equation), a Lie bialgebra structure onX(S 1), and a sort of Poisson-Lie structure on the group $$\widetilde{Diff}(S^1 )$$ of diffeomorphisms. Quantizations of such Lie bialgebra structures may lead to ‘quantum diffeomorphism groups’.
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    URL: http://dx.doi.org/10.1007/BF00761141
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  • 17
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    CQG algebras: A direct algebraic approach to compact quantum groups (1994)
    Springer
    Letters in mathematical physics 32 (1994), S. 315-330 
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    ISSN: 1573-0530
    Keywords: 81R50 ; 16W30 ; 22C05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The purely algebraic notion of CQG algebra (algebra of functions on a compact quantum group) is defined. In a straightforward algebraic manner, the Peter-Weyl theorem for CQG algebras and the existence of a unique positive definite Haar functional on any CQG algebra are established. It is shown that a CQG algebra can be naturally completed to aC *-algebra. The relations between our approach and several other approaches to compact quantum groups are discussed.
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    URL: http://dx.doi.org/10.1007/BF00761142
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  • 18
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    Left regular representation and contraction of sl q (2) to e q (2) (1994)
    Springer
    Letters in mathematical physics 32 (1994), S. 249-258 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 17B37 ; 81R10 ; 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The left regular representation of the quantum algebras sl q (2) and e q (2) are discussed and shown to be related by contraction. The reducibility is studied andq-difference intertwining operators are constructed.
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    URL: http://dx.doi.org/10.1007/BF00750667
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  • 19
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    Stochastic differential calculus, the Moyal *-product, and noncommutative geometry (1993)
    Springer
    Letters in mathematical physics 28 (1993), S. 123-137 
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    ISSN: 1573-0530
    Keywords: 17B37 ; 58A10 ; 58G32 ; 81R50 ; 81S20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract A reformulation of the Itô calculus of stochastic differentials is presented in terms of a differential calculus in the sense of noncommutative geometry (with an exterior derivative operator d satisfying d2 = 0 and the Leibniz rule). In this calculus, differentials do not commute with functions. The relation between both types of differential calculi is mediated by a generalized Moyal *-product. In contrast to the Itô calculus, the new framework naturally incorporates analogues of higher-order differential forms. A first step is made towards an understanding of their stochastic meaning.
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    URL: http://dx.doi.org/10.1007/BF00750305
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  • 20
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    A quantum-group-like structure on noncommutative 2-tori (1993)
    Springer
    Letters in mathematical physics 28 (1993), S. 251-255 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 46L87 ; 58B30 ; 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We show that on noncommutative 2-tori, there are natural structures which have analogous formal properties as Hopf algebra structures, but where the comultiplication has values in a deformation of the tensor product.
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    URL: http://dx.doi.org/10.1007/BF00761493
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  • 21
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    Representations of the coordinate ring of GL q (3) (1993)
    Springer
    Letters in mathematical physics 28 (1993), S. 207-217 
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    ISSN: 1573-0530
    Keywords: 17B37 ; 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract It is shown that finite-dimensional irreducible representations of the quantum matrix algebraM q (3) (the coordinate ring of GL q (3)) exist only whenq is a root of unity (q p = 1). The dimensions of these representations can only be one of the following values:p 3,p 3/2,p 3/4, orp 3/8. The topology of the space of states ranges between two extremes, from a three-dimensional torusS 1 ×S 1 ×S 1 (which may be thought of as a generalization of the cyclic representation) to a three-dimensional cube [0, 1] × [0, 1] × [0, 1].
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    URL: http://dx.doi.org/10.1007/BF00745152
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  • 22
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    On solutions to the braid equation identified by Woronowicz (1993)
    Springer
    Letters in mathematical physics 27 (1993), S. 13-17 
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    ISSN: 1573-0530
    Keywords: 57T05 ; 16W30 ; 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Solutions of the braid equation recently identified by Woronowicz are shown to be derived from subrepresentations of the regular representations of a quantum double Hopf algebra.
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    URL: http://dx.doi.org/10.1007/BF00739584
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  • 23
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    On PBW bases for Yangians (1993)
    Springer
    Letters in mathematical physics 27 (1993), S. 37-42 
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    ISSN: 1573-0530
    Keywords: 17B37 ; 81R50 ; 82B23
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract PBW bases for Yangians are constructed.
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    URL: http://dx.doi.org/10.1007/BF00739587
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  • 24
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    Boson realizations of quantum algebras and some physical spaces with quantum algebra symmetry (1993)
    Springer
    Letters in mathematical physics 27 (1993), S. 273-277 
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    ISSN: 1573-0530
    Keywords: 81R50 ; 17B37
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The generators ofq-boson algebra are expressed in terms of those of boson algebra, and the relations among the representations of a quantum algebra onq-Fock space, on Fock space, and on coherent state space are discussed in a general way. Two examples are also given to present concrete physical spaces with quantum algebra symmetry. Finally, a new homomorphic mapping from a Lie algebra to boson algebra is presented.
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    URL: http://dx.doi.org/10.1007/BF00777374
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  • 25
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    Remarks on bicovariant differential calculi and exterior Hopf algebras (1993)
    Springer
    Letters in mathematical physics 27 (1993), S. 287-300 
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    ISSN: 1573-0530
    Keywords: 81R50 ; 16W30 ; 58B30 ; 46L87 ; 05A30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We show that every bicovariant differential calculus over the quantum groupA defines a bialgebra structure on its exterior algebra. Conversely, every exterior bialgebra ofA defines bicovariant bimodule overA. We also study a quasitriangular structure on exterior Hopf algebras in some detail.
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  • 26
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    On spectral properties ofq-oscillator operators (1993)
    Springer
    Letters in mathematical physics 29 (1993), S. 13-18 
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    ISSN: 1573-0530
    Keywords: 17B37 ; 33D25 ; 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The property of self-adjointness of the operatorQ =a + +a - in three types ofq-oscillator algebras is considered. Spectral measures and generalized eigenfunctions ofQ are found in the cases when this operator is bounded. Generalized eigenvectors are expressed in terms ofq-Hermite polynomials. If the operatorQ is unbounded, then its closure $$\bar Q$$ is not self-adjoint. However, in this case, $$\bar Q$$ admits self-adjoint extensions. Deficiency subspaces are one-dimensional. These subspaces are explicitly found.
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    URL: http://dx.doi.org/10.1007/BF00760854
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  • 27
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    Difference schrödinger operators with linear and exponential discrete spectra (1993)
    Springer
    Letters in mathematical physics 29 (1993), S. 63-73 
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    ISSN: 1573-0530
    Keywords: 33D45 ; 39A70 ; 17B37 ; 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Using the factorization method, we construct finite-difference Schrödinger operators (Jacobi matrices) whose discrete spectra are composed from independent arithmetic, or geometric series. Such systems originate from the periodic, orq-periodic closure of a chain of corresponding Darboux transformations. The Charlier, Krawtchouk, Meixner orthogonal polynomials, theirq-analogs, and some other classical polynomials appear as the simplest examples forN = 1 andN = 2 (N is the period of closure). A natural generalization involves discrete versions of the Painlevé transcendents.
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    URL: http://dx.doi.org/10.1007/BF00760860
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  • 28
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    Fusion structures from quantum groups: II. Why truncation is necessary (1993)
    Springer
    Letters in mathematical physics 29 (1993), S. 83-90 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 17B37 ; 81R50 ; 81T05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract It is shown that a finite, reflection positive, and nontruncated fusion structure on an arbitrary Hopf algebraℋ is trivial in the sense thatq-traces coincide with ordinary traces andq-dimensions coincide with ordinary dimensions. Thus, nontruncated fusion structures are ruled out to describe the fusion rules of quantum field theories with noninteger statistical dimensions and a finite number of superselection sectors.
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    URL: http://dx.doi.org/10.1007/BF00749724
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  • 29
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    On quantum multi-hamiltonian systems (1993)
    Springer
    Letters in mathematical physics 29 (1993), S. 197-203 
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    ISSN: 1573-0530
    Keywords: 81R50 ; 81B05 ; 17B37 ; 70H05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract On studying various examples with one degree of freedom, we discuss the possibility of extending the classical concept of multi-Hamiltonian systems to the quantum domain. We show that an adaptation is possible in each case generally leading to distinguish classes of systems characterized globally by their observable algebras.
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    URL: http://dx.doi.org/10.1007/BF00761107
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  • 30
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    A borel-weil-bott approach to representations of sl q (2, ℂ) (1993)
    Springer
    Letters in mathematical physics 29 (1993), S. 215-217 
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    ISSN: 1573-0530
    Keywords: 81R50 ; 16W30
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract We use a quite concrete and simple realization of sl q (2, ℂ) involving finite difference operators. We interpret them as derivations (in the noncommutative sense) on a suitable graded algebra, which gives rise to the ‘noncommutative’ scheme ℙ1 II ℙ1* as the counterpart of the standard ℙ1 = Sl(2, ℂ)/B.
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  • 31
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    Duality for a Lorentz quantum group (1993)
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    Letters in mathematical physics 29 (1993), S. 259-269 
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    Keywords: 17B37 ; 16W30 ; 81R50
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    Notes: Abstract We give the algebra ℒ q /* dual to the matrix Lorentz quantum group ℒ q of Podles-Woronowicz, and Watamuraet al. As a commutation algebra, it has the classical form ℒ q /* ≅ U q (sl(2, ℂ)) ⊗ U q (sl(2, ℂ)). However, this splitting is not preserved by the coalgebra structure which we also give. For the derivation, we use a generalization of the approach of Sudbery, viz. tangent vectors at the identity.
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    On a possible algebra morphism of U q [osp(1/2n)] onto the deformed oscillator algebra W q (n) (1993)
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    Letters in mathematical physics 28 (1993), S. 187-193 
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    Keywords: 81R50 ; 16W30 ; 17B37
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    Notes: Abstract We formulate a conjecture stating that the algebra ofn pairs of deformed Bose creation and annihilation operators is a factor algebra of U q [osp(1/2n)], considered as a Hopf algebra, and prove it for then = 2 case. To this end, we show that for any value ofq, U q [osp(1/4)] can be viewed as a superalgebra freely generated by two pairsB 1 ± ,B 2 ± of deformed para-Bose operators. We write down all Hopf algebra relations, an analogue of the Cartan-Weyl basis, the ‘commutation’ relations between the generators and a basis in U q [osp(1/2n)] entirely in terms ofB 1 ± ,B 2 ± .
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    Self-similar potentials and theq-oscillator algebra at roots of unity (1993)
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    Letters in mathematical physics 28 (1993), S. 59-74 
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    Keywords: 34L40 ; 17B37 ; 33D80 ; 81R50
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    Notes: Abstract Properties of the simplest class of self-similar potentials are analyzed. Wave functions of the corresponding Schrödinger equation provide bases of representations of theq-deformed Heisenberg-Weyl algebra. When the parameterq is a root of unity, the functional form of the potentials can be found explicitly. The generalq 3 = 1 and the particularq 4 = 1 potentials are given by the equi-anharmonic and (pseudo) lemniscatic Weierstrass functions, respectively.
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    The quantum bialgebra associated with the eight-vertexR-matrix (1993)
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    Letters in mathematical physics 28 (1993), S. 139-141 
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    Keywords: 16W30 ; 81R50
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    Notes: Abstract The quantum bialgebra related to the Baxter's eight-vertexR-matrix is found as a quantum deformation of the Lie algebra of sl(2)-valued automorphic functions on a complex torus.
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    A superalgebra morphism of U q [osp(1/2N)] onto the deformed oscillator superalgebra W q (N) (1993)
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    Letters in mathematical physics 28 (1993), S. 321-328 
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    Keywords: 81R50 ; 16W30 ; 17B37
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    Notes: Abstract We prove that the deformed oscillator superalgebra W q (n) (which in the Fock representation is generated essentially byn pairs ofq-bosons) is a factor algebra of the quantized universal enveloping algebra U q [osp(1/2n)]. We write down aq-analog of the Cartan-Weyl basis for the deformed osp(1/2n) and also give an oscillator realization of all Cartan-Weyl generators.
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    Primitive ideals of C q [SL(3)] (1993)
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    Communications in mathematical physics 156 (1993), S. 581-605 
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    ISSN: 1432-0916
    Keywords: Quantum group ; primitive ideal ; Poisson Lie group ; symplectic leaves ; 17B37 ; 16W30 ; 16S80 ; 16S30 ; 58F06 ; 81R50
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    Notes: Abstract The primitive ideals of the Hopf algebraC q [SL(3)] are classified. In particular it is shown that the orbits in PrimC q [SL(3)] under the action of the representation groupH ≅C *×C * are parameterized naturally byW×W, whereW is the associated Weyl group. It is shown that there is a natural one-to-one correspondence between primitive ideals ofC q [SL(3)] and symplectic leaves of the associated Poisson algebraic groupSL(3,C).
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    Quantized universal enveloping superalgebra of type Q and a super-extension of the Hecke algebra (1992)
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    Letters in mathematical physics 24 (1992), S. 93-102 
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    Keywords: 17A70 ; 17B35 ; 81R50
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    Notes: Abstract The quantized universal enveloping algebra U q(q(n)) of the ‘strange’ Lie superalgebra q(n) and a super-analogue HC q (N) of the Hecke algebra H q (N) are constructed. These objects are in a duality similar to the known duality between U q (gl(n)) and H q (N).
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    Serre-type relations for special linear Lie superalgebras (1992)
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    Letters in mathematical physics 24 (1992), S. 173-181 
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    Keywords: 17A70 ; 16W30 ; 17B37 ; 81R50
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    Notes: Abstract It is pointed out that, for m, n ≥2, the naive Serre presentation corresponding to the simplest Cartan matrix of sl(m, n) does not define the Lie superalgebra sl(m, n) but a larger algebra s(m, n) of which sl(m, n) is a nontrivial quotient. The supplementary relations for the generators are found and the definition of the q-deformed universal enveloping algebra of sl(m, n) is modified accordingly.
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    On the representations of quantum oscillator algebra (1992)
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    Letters in mathematical physics 24 (1992), S. 147-153 
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    Keywords: 81R50 ; 22E70
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    Notes: Abstract It is shown that for q〈1, the quantum oscillator algebra has a supplementary family of representations inequivalent to the usual q-Fock representation, with no counterpart at the limit q=1. They are used to build representations of SU q (1,1) and E(2) in Schwinger's way.
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    Star-products and phase space realizations of quantum groups (1992)
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    Letters in mathematical physics 25 (1992), S. 51-59 
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    Keywords: 17B37 ; 81R50 ; 81S30
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    Notes: Abstract It is shown for a family of *-products (i.e. different ordering rules) that, under a strong invariance condition, the functions of the quadratic preferred observables (which generate the Cartan subalgebra in phase space realization of Lie algebras) take only the linear or exponential form. An exception occurs for the case of a symmetric ordering *-product where trigonometric functions and two special polynomials can also be included. As an example, the ‘quantized algebra’ of the oscillator Lie algebra is argued.
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    A natural and rigid model of quantum groups (1992)
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    Letters in mathematical physics 25 (1992), S. 75-84 
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    Keywords: 16W30 ; 81R50 ; 17B37
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    Notes: Abstract We introduce a natural (Fréchet-Hopf) algebra A containing all generic Jimbo algebras U t (sl(2)) (as dense subalgebras). The Hopf structures on A extend (in a continuous way) the Hopf structures of generic U t (sl(2)). The Universal R-matrices converge in A $$\hat \otimes $$ A. Using the (topological) dual of A, we recover the formalism of functions of noncommutative arguments. In addition, we show that all these Hopf structures on A are isomorphic (as bialgebras), and rigid in the category of bialgebras.
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    A *-product on SL(2) and the corresponding nonstandard $$(\mathfrak{s}\mathfrak{l}({\text{2}}))$$ (1992)
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    Letters in mathematical physics 25 (1992), S. 85-88 
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    Notes: Abstract We obtain Zakrzewski's deformation of Fun SL(2) through the construction of a *-product on SL(2). We then give the deformation of $$(\mathfrak{s}\mathfrak{l}({\text{2}}))$$ dual to this, as well as a Poincaré basis for both algebras.
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    Finite-dimensional representations of U q (osp(1/2n)) and its connection with quantum so(2n+1) (1992)
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    Letters in mathematical physics 25 (1992), S. 317-325 
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    Keywords: 17B37 ; 81R50 ; 17A70
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    Notes: Abstract It is shown that the quantum supergroup U q (osp(1/2n)) is essentially isomorphic to the quantum group U -q (so(2n+1)) restricted to tensorial representations. This renders it straightforward to classify all the finite-dimensional irreducible representations of U q (osp(1/2n)) at generic q. In particular, it is proved that at generic q, every-dimensional irrep of this quantum supergroup is a deformation of an osp(1/2n) irrep, and all the finite-dimensional representations are completely reducible.
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    Quantum group-symmetric fock spaces with bargmann-fock representation (1992)
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    Letters in mathematical physics 26 (1992), S. 1-12 
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    Notes: Abstract SUq(n) symmetric algebras of creation and annihilation operators are represented on a Hilbert space of deformed holomorphic functions. The scalar product on this space is given by an algebraically defined integral. In all calculations the bosonic and the fermionic case are treated simultaneously.
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    A class of bicovariant differential calculi on hopf algebras (1992)
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    Letters in mathematical physics 26 (1992), S. 67-78 
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    Keywords: 81R50 ; 16W30 ; 58B30 ; 57M25 ; 05A30
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    Notes: Abstract We introduce a large class of bicovariant differential calculi on any quantum group A, associated to Ad-invariant elements. For example, the deformed trace element on SLq (2) recovers Woronowicz's 4D ± calculus. More generally, we obtain a class of differential calculi on each quantum group A(R), based on the theory of the corresponding braided groups B(R). Here R is any regular solution of the QYBE.
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    Fundamental representations of quantum groups at roots of 1 (1992)
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    Letters in mathematical physics 26 (1992), S. 133-146 
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    ISSN: 1573-0530
    Keywords: 17B37 ; 81R50
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    Notes: Abstract To every finite-dimensional irreducible representation V of the quantum group Uε(g) where ε is a primitive lth root of unity (l odd) and g is a finite-dimensional complex simple Lie algebra, de Concini, Kac and Procesi have associated a conjugacy class C V in the adjoint group G of g. We describe explicitly, when g is of type A n , B n , C n , or D n , the representations associated to the conjugacy classes of minimal positive dimension. We call such representations fundamental and prove that, for any conjugacy class, there is an associated representation which is contained in a tensor product of fundamental representations.
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    Free particle in q-deformed configuration space (1992)
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    Letters in mathematical physics 26 (1992), S. 245-258 
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    ISSN: 1573-0530
    Keywords: 46E20 ; 81Q05 ; 81R20 ; 81R50
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    Notes: Abstract The SO q (N)-invariant Schrödinger equation for the free particle is formulated in polar coordinates as a partial differential equation in noncommutative geometry. For each value of the total angular momentum, a Hilbert space of radial functions is constructed as the space of normalizable functions respective to the q-integral. The spectrum of the Hamiltonian is found to be discrete.
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    Cohomology and associated deformations for not necessarily co-associative bialgebras (1992)
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    Letters in mathematical physics 26 (1992), S. 277-283 
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    Keywords: 16E40 ; 16W30 ; 17B37 ; 81R50
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    Notes: Abstract In this Letter, a cohomology and an associated theory of deformations for (not necessarily co-associative) bialgebras are studied. The cohomology was introduced in a previous paper (Lett. Math. Phys. 25, 75–84 (1992)). This theory has several advantages, especially in calculating cohomology spaces and in its adaptability to deformations of quasi-co-associative (qca) bialgebras and even quasi-triangular qca bialgebras.
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    The Hamiltonian systems on the Poisson structure of the quasi-classical limit of GL q (∞) (1991)
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    Letters in mathematical physics 23 (1991), S. 121-126 
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    Keywords: 81R50
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    Notes: Abstract We consider the Hamiltonian systems on the Poisson structure of GL(∞) which is introduced from the quantum group GL q (∞) by the so-called quasi-classical limit of GL q (∞). Furthermore, we show that the Toda lattice hierarchy is a Hamiltonian system of this structure.
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    Braid group action on theq-Weyl algebra (1991)
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    Letters in mathematical physics 23 (1991), S. 151-158 
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    Keywords: 17B37 ; 20F36 ; 81R50
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    Notes: Abstract We provide a braid group action on theq-deformed Weyl algebraW q (n). The restriction of this action to the representations ofU q (A n−1 ) andU q (C n ) inW q (n) is seen to agree with the braid group action introduced by Lusztig on these quantum algebras.
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    On the universalR-matrix for Uq(241-1241-1241-1) (1991)
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    Letters in mathematical physics 23 (1991), S. 241-250 
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    Keywords: 81R50 ; 17B37 ; 16W30 ; 81R30 ; 81S05 ; 81T40 ; 20F36
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    Notes: Abstract A compact form for the universalR-matrix of U q (sl n ) is derived and illustrated by simple applications.
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    Quantum E(2) group and its Pontryagin dual (1991)
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    Letters in mathematical physics 23 (1991), S. 251-263 
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    Keywords: 81R50 ; 22D25 ; 22D35
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    Notes: Abstract The quantum deformation of the group of motions of the plane and its Pontryagin dual are described in detail. It is shown that the Pontryagin dual is a quantum deformation of the group of transformations of the plane generated by translations and dilations. An explicit expression for the unitary bicharacter describing the Pontryagin duality is found. The Heisenberg commutation relations are written down.
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    Singular vectors of quantum group representations for straight Lie algebra roots (1991)
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    Letters in mathematical physics 22 (1991), S. 251-266 
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    Keywords: 17B37 ; 81R50 ; 16W30
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    Notes: Abstract We give explicit formulae for singular vectors of Verma modules over Uq(G), where G is any complex simple Lie algebra. The vectors we present correspond exhaustively to a class of positive roots of G which we call straight roots. In some special cases, we give singular vectors corresponding to arbitrary positive roots. For our vectors we use a special basis of Uq(G -), where G - is the negative roots subalgebra of G, which was introducted in our earlier work in the case q=1. This basis seems more economical than the Poincaré-Birkhoff-Witt type of basis used by Malikov, Feigin, and Fuchs for the construction of singular vectors of Verma modules in the case q=1. Furthermore, this basis turns out to be part of a general basis recently introduced for other reasons by Lusztig for Uq(ℬ-), where ℬ- is a Borel subalgebra of G.
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    SO q (n+1,n−1) as a real form of SO q (2n,ℂ) (1991)
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    Letters in mathematical physics 23 (1991), S. 45-49 
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    Keywords: 81R50 ; 81R10
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    Notes: Abstract Quantum pseudo-orthogonal groups SO q (n+1,n−1) are defined as real forms of quantum orthogonal groups SO q (n+1,n−1) by means of a suitable antilinear involution. In particular, the casen=2 gives a quantized Lorentz group.
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    Differential calculi on quantum vector spaces with Hecke-type relations (1991)
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    Letters in mathematical physics 23 (1991), S. 133-141 
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    Keywords: 81R50 ; 16O80 ; 16W30 ; 17B37
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    Notes: Abstract From a vector spaceV equipped with a Yang-Baxter operatorR one may form the r-symmetric algebraS R V=TV/〈v ⊗w−R(v ⊗w)〉, which is a quantum vector space in the sense of Manin, and the associated quantum matrix algebraM R V=T(End(V))/〈f ⊗g−R(f ⊗g)R -1〉. In the case whenR satisfies a Hecke-type identityR 2=(1−q)R+q, we construct a differential calculus Ω R V forS R V which agrees with that constructed by Pusz, Woronowicz, Wess, and Zumino whenR is essentially theR-matrix of GL q (n). Elements of Ω R V may be regarded as differential forms on the quantum vector spaceS R V. We show that Ω R V isM R V-covariant in the sense that there is a coaction Φ*: Ω R V →M R V ⊗ Ω R V with Φ*d=(1 ⊗ d)Φ* extending the natural coaction Φ:S R V →M R V ⊗S R V.
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    On a possible origin of quantum groups (1991)
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    Letters in mathematical physics 22 (1991), S. 155-160 
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    Keywords: 17B37 ; 81R50 ; 81S30
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    Notes: Abstract A Poisson bracket structure having the commutation relations of the quantum group SL q (2) is quantized by means of the Moyal star-product on C ∞(ℝ2), showing that quantum groups are not exactly quantizations, but require a quantization (with another parameter) in the background. The resulting associative algebra is a strongly invariant nonlinear star-product realization of the q-algebra U q (sl(2)). The principle of strong invariance (the requirement that the star-commutator is star-expressed, up to a phase, by the same function as its classical limit) implies essentially the uniqueness of the commutation relations of U q (sl(2)).
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    Orbifolds, Hopf algebras, and the Moonshine (1991)
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    Letters in mathematical physics 22 (1991), S. 187-194 
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    Keywords: 81R50 ; 16W30
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    Notes: Abstract We describe the modular properties and fusion rules of holomorphic orbifold models by Hopf algebraic techniques, using the representation theory of the orbifold quantum group. We apply this theory to the construction of generalized Thompson series, and discuss its connections with Moonshine.
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    Solutions of the braid equation related to a Hopf algebra (1991)
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    Letters in mathematical physics 23 (1991), S. 143-145 
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    Keywords: 57T05 ; 16W30 ; 81R50
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    Notes: Abstract Two solutionsT andT′ of the braid equation acting onA ⊗A (whereA is a Hopf algebra) are described. IfA is a cocommutative, thenT=σ. IfA is commutative, thenT′=σ (σ denotes the flip: σ(a ⊗b) =b ⊗a for anya,b ∈A).
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