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  • Articles  (55)
  • 16W30  (55)
  • 1990-1994  (55)
  • Mathematics  (55)
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  • Articles  (55)
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  • 1
    Electronic Resource
    Electronic Resource
    Quantum homogeneous spaces, duality and quantum 2-spheres (1994)
    Springer
    Geometriae dedicata 52 (1994), S. 291-315 
    Details
    ISSN: 1572-9168
    Keywords: 16W30 ; 17B37 ; 22E99 ; quantum groups ; quantum homogeneous spaces ; coaction ; dual Hopf algebras ; two-sided coideals ; infinitesimal invariance ; quantum SU(2) ; sl(2) quantized universal enveloping algebra ; quantum 2-spheres
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For a quantum groupG the notion of quantum homogeneousG-space is defined. Two methods to construct such spaces are discussed. The first one makes use of quantum subgroups, the second more general one is based upon the notion of infinitesimal invariance with respect to certain two-sided coideals in the Hopf algebra dual to the Hopf algebra ofG. These methods are applied to the quantum group SU(2). As two-sided coideals we take the subspaces spanned by twisted primitive elements in the sl(2) quantized universal enveloping algebra. A one-parameter series of mutually non-isomorphic quantum 2-spheres is obtained, together with the spectral decomposition of the corresponding right regular representation of quantum SU(2). The link with the quantum spheres defined by Podleś is established.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01278478
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  • 2
    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 
    Details
    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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  • 3
    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 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00805855
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  • 4
    Electronic Resource
    Electronic Resource
    Free boson representation of $$U_q (\widehat{sl}_3 )$$ (1994)
    Springer
    Letters in mathematical physics 30 (1994), S. 207-216 
    Details
    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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  • 5
    Electronic Resource
    Electronic Resource
    Quantized affine Lie algebras and diagonalization of braid generators (1994)
    Springer
    Letters in mathematical physics 30 (1994), S. 267-277 
    Details
    ISSN: 1573-0530
    Keywords: 81R10 ; 17B37 ; 16W30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Let U q $$(\hat{\mathcal{G}})$$ be a quantized affine Lie algebra. It is proven that the universal R-matrix R of U q $$(\hat{\mathcal{G}})$$ satisfies the celebrated conjugation relationR + =TR withT the usual twist map. As applications, the braid generator is shown to be diagonalizable on arbitrary tensor product modules of integrable irreducible highest weight U q $$(\hat{\mathcal{G}})$$ -module and a spectral decomposition formula for the braid generator is obtained which is the generalization of Reshetikhin and Gould forms to the present affine case. Casimir invariants are constructed and their eigenvalues computed by means of the spectral decomposition formula. As a by-product, an interesting identity is found.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00751063
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  • 6
    Electronic Resource
    Electronic Resource
    Casimir invariants for quantized affine Lie algebras (1994)
    Springer
    Letters in mathematical physics 31 (1994), S. 77-83 
    Details
    ISSN: 1573-0530
    Keywords: 81R10 ; 17B37 ; 16W30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Casimir invariants for quantized affine Lie algebras are constructed and their eigenvalues computed in any irreducible highest-weight representation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00751173
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  • 7
    Electronic Resource
    Electronic Resource
    Quantum affine algebras and universalR-matrix with spectral parameter (1994)
    Springer
    Letters in mathematical physics 31 (1994), S. 101-110 
    Details
    ISSN: 1573-0530
    Keywords: 81R10 ; 17B37 ; 16W30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Using the previously obtained universalR-matrix for the quantized nontwisted affine Lie algebras U q (A 1 (1) ) and U q (A 2 (1) ), we determine the explicitly spectral dependent universalR-matrix for the corresponding quantum Lie algebras U q (A 1) and U q (A 2). As applications, we reproduce the well known results in the fundamental representations and we also derive an extremely explicit formula of the spectral-dependentR-matrix for the adjoint representation of U q (A 2), the simplest nontrivial case when the tensor product decomposition of the representation with itself has nontrivial multiplicity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00750144
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  • 8
    Electronic Resource
    Electronic Resource
    Quantization of U q [so(2n + 1)] with deformed para-Fermi operators (1994)
    Springer
    Letters in mathematical physics 31 (1994), S. 151-157 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00750149
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  • 9
    Electronic Resource
    Electronic Resource
    Existence of a Lie bialgebra structure on every Lie algebra (1994)
    Springer
    Letters in mathematical physics 31 (1994), S. 225-231 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 17Bxx
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We prove that every non-Abelian, finite-dimensional Lie algebra admits an exact bialgebra structure.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00761714
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  • 10
    Electronic Resource
    Electronic Resource
    SU q (2) covariant $$\hat R$$ -matrices for reducible representations (1994)
    Springer
    Letters in mathematical physics 31 (1994), S. 279-288 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 81R05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We consider SU q (2) covariant $$\hat R$$ -matrices for the reducible3 ⊕1 representation. There are three solutions to the Yang-Baxter equation. They coincide with the previously known $$\hat R$$ -matrices for SO q (3) and SO q (3, 1). Also, they are the three $$\hat R$$ -matrices which can be constructed by using four different SU q (2) doublets. Only two of the three $$\hat R$$ -matrices allow a differential structure on the reducible four-dimensional quantum space.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00762790
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  • 11
    Electronic Resource
    Electronic Resource
    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 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00739419
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  • 12
    Electronic Resource
    Electronic Resource
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00750660
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  • 13
    Electronic Resource
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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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00761142
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  • 14
    Electronic Resource
    Electronic Resource
    Diagonalization of the braid generator on unitary irreps of quantum supergroups (1994)
    Springer
    Letters in mathematical physics 32 (1994), S. 231-240 
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    ISSN: 1573-0530
    Keywords: 81R10 ; 17B37 ; 16W30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract It is shown that the braid generator σ associated with the universalR-matrix is diagonalizable on all unitary representations of quantum supergroups. An example is considered using U q (gl(2|1)) and a family of eight-dimensional typical representations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00750665
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  • 15
    Electronic Resource
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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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  • 16
    Electronic Resource
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    Simple WZW superselection sectors (1993)
    Springer
    Letters in mathematical physics 28 (1993), S. 31-41 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 17B37 ; 81T05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The superselection structure of simple currents of chiral Wess-Zumino-Witten theories, at arbitrary valuek ∈ ℕ of the corresponding affine Lie algebra, is described in terms of explicit localizable automorphisms of the affine algebra. These automorphisms are induced by certain Dynkin diagram automorphisms; under composition, they form an Abelian group isomorphic to the center of the relevant simply connected simple Lie group and, hence, reproduce the WZW fusion rules.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00739564
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  • 17
    Electronic Resource
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    The double of a Jacobian quasi-bialgebra (1993)
    Springer
    Letters in mathematical physics 28 (1993), S. 13-29 
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    ISSN: 1573-0530
    Keywords: 17B37 ; 16W30 ; 17B70
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We show that the construction of the double of a Lie bialgebra can be extended to the case where a vector space is only equipped with the structure of a Jacobian quasi-bialgebra (also called a Lie quasi-bialgebra). In this case, the double is itself a Jacobian quasi-bialgebra and it is quasi-triangular. The more general case of the double of a proto-Lie bialgebra is also discussed. In the first section, the notions of exact, strictly exact, quasi-triangular and triangular Jacobian quasi-bialgebras are defined and their equivalence classes under twisting are studied.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00739563
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  • 18
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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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  • 19
    Electronic Resource
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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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  • 20
    Electronic Resource
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    Baxterization of ther-matrix for the adjoint representation of U q [D(2, 1;α)] (1993)
    Springer
    Letters in mathematical physics 27 (1993), S. 51-58 
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    ISSN: 1573-0530
    Keywords: 81R10 ; 17B37 ; 16W30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The Baxterization procedure is applied to the Braid group representation arising from the adjoint representation of the quantum supergroup U q [D(2, 1;α)]. This yields a two-parameter family of solutions to the (parameter dependent) Yang-Baxter equation of interest in supersymmetric lattice models.
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    URL: http://dx.doi.org/10.1007/BF00739589
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  • 21
    Electronic Resource
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    On the quantum group and quantum algebra approach toq-special functions (1993)
    Springer
    Letters in mathematical physics 27 (1993), S. 179-190 
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    ISSN: 1573-0530
    Keywords: 22E70 ; 17B37 ; 16W30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Two interpretations ofq-special functions based on quantum groups and algebras have been presented in the literature. The connection between these approaches is explained using as an example the case whereU q (sl(2)) is the basic structure.
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    URL: http://dx.doi.org/10.1007/BF00739576
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  • 22
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    Inhomogeneous quantum groups and their quantized universal enveloping algebras (1993)
    Springer
    Letters in mathematical physics 27 (1993), S. 217-222 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 81R05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The quantum group IGL q (N), the inhomogenization of GL q (N), is formulated with $$\hat R$$ -matrices. Theq-deformed universal enveloping algebra is constructed as the algebra of regular functionals in this formulation and contains the partial derivatives of the covariant differential calculus on the quantum space.
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    URL: http://dx.doi.org/10.1007/BF00739579
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  • 23
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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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    URL: http://dx.doi.org/10.1007/BF00777376
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  • 24
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    On universalR-matrix for quantized nontwisted rank 3 affine KM algebras (1993)
    Springer
    Letters in mathematical physics 29 (1993), S. 19-31 
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    ISSN: 1573-0530
    Keywords: 81R10 ; 17B37 ; 16W30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Explicit formulas of the universalR-matrix are given for all quantized nontwisted rank 3 affine KM algebras U q (A 2 (1) ), U q (C 2 (1) ) and U q (G 2 (1) ).
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    URL: http://dx.doi.org/10.1007/BF00760855
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  • 25
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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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  • 26
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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
    Topics: Mathematics , Physics
    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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    URL: http://dx.doi.org/10.1007/BF00761109
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  • 27
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    Duality for a Lorentz quantum group (1993)
    Springer
    Letters in mathematical physics 29 (1993), S. 259-269 
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    ISSN: 1573-0530
    Keywords: 17B37 ; 16W30 ; 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    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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    URL: http://dx.doi.org/10.1007/BF00750960
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  • 28
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    On a possible algebra morphism of U q [osp(1/2n)] onto the deformed oscillator algebra W q (n) (1993)
    Springer
    Letters in mathematical physics 28 (1993), S. 187-193 
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    ISSN: 1573-0530
    Keywords: 81R50 ; 16W30 ; 17B37
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    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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    URL: http://dx.doi.org/10.1007/BF00745150
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  • 29
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    The quantum bialgebra associated with the eight-vertexR-matrix (1993)
    Springer
    Letters in mathematical physics 28 (1993), S. 139-141 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    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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    URL: http://dx.doi.org/10.1007/BF00750306
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  • 30
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    A superalgebra morphism of U q [osp(1/2N)] onto the deformed oscillator superalgebra W q (N) (1993)
    Springer
    Letters in mathematical physics 28 (1993), S. 321-328 
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    ISSN: 1573-0530
    Keywords: 81R50 ; 16W30 ; 17B37
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    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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    URL: http://dx.doi.org/10.1007/BF00761500
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  • 31
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    Primitive ideals of C q [SL(3)] (1993)
    Springer
    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
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    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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    Differential operators on quantum spaces for GL q (n) and SO q (n) (1992)
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    Letters in mathematical physics 24 (1992), S. 245-255 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 81R05
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    Notes: Abstract We prove that the rings of q-differential operators on quantum planes of the GL q (n) and SO q (n) types are isomorphic to the rings of classical differential operators. Also, we construct decompositions of the rings of q-differential operators into tensor products of the rings of q-differential operators with less variables.
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    Quantum groups and diagonalization of the braid generator (1992)
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    Letters in mathematical physics 24 (1992), S. 183-196 
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    ISSN: 1573-0530
    Keywords: 81R10 ; 17B37 ; 16W30
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    Notes: Abstract It is shown that the braid generator is diagonalizable on arbitrary tensor product modules V ⊗ V, V an irreducible module for a quantum group. A generalization of the Reshetikhin form for the braid generator is thereby obtained in the general case. As an application, a general closed formula is determined for link polynomials.
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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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    ISSN: 1573-0530
    Keywords: 17A70 ; 16W30 ; 17B37 ; 81R50
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    Topics: Mathematics , Physics
    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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    Construction of a new quantum double and new solutions to the Yang-Baxter equation (1992)
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    Letters in mathematical physics 24 (1992), S. 197-203 
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    ISSN: 1573-0530
    Keywords: 81R10 ; 17B37 ; 16W30
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    Topics: Mathematics , Physics
    Notes: Abstract A new quantum double is established from a new Hopf algebra and a new kind of quantum R-matrix is obtained.
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    Real forms of U q(g) (1992)
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    Letters in mathematical physics 24 (1992), S. 49-58 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 57T05 ; 17B37
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    Topics: Mathematics , Physics
    Notes: Abstract In this Letter we consider the real forms of quantum groups associated to generalized Cartan matrices. There are two main results. The first is a description of the Hopf algebra automorphisms and of the Hopf *-algebra structures of the quantum group. This immediately yields a precise description of the real forms. The second result establishes a correspondence of these real forms when the quantum group is associated to a complex simple Lie algebra with objects associated to the real forms of the classical object.
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    The uniqueness theorem for the universal R-matrix (1992)
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    Letters in mathematical physics 24 (1992), S. 231-244 
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    ISSN: 1573-0530
    Keywords: 81R10 ; 17B37 ; 16W30
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    Topics: Mathematics , Physics
    Notes: Abstract Up to now, the universal R-matrix for quantized Kac-Moody algebras Here the name \lsKac-Moody algebras\rs includes all semisimple finite-dimensional Lie algebras and all infinite-dimensional affine Kac-Moody algebras. is believed to be uniquely determined (for some ansatz) by properties of a quasi-cocommutativity and a quasi-triangularity. We prove here that the universal R-matrix (for the same ansatz) is uniquely determined by the property of the quasi-cocommutativity only. Thus, the quasi-triangular property (and the Yang-Baxter equation!) for the universal R-matrix is a consequence of the linear equation of the quasi-cocommutativity. The proof is based on properties of singular vectors in the tensor product of the Verma modules and the structure of extremal projector for quantized algebras. Explicit expressions of the universal R-matrix for quantized algebras U q (A inf1 sup(1) ) and U q (A inf2 sup(2) ) are given.
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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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    ISSN: 1573-0530
    Keywords: 16W30 ; 81R50 ; 17B37
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    Topics: Mathematics , Physics
    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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    ISSN: 1573-0530
    Keywords: 16W30 ; 17B37 ; 81R50
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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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    Rational forms for twistings of enveloping algebras of simple Lie algebras (1992)
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    Letters in mathematical physics 25 (1992), S. 111-120 
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    ISSN: 1573-0530
    Keywords: 17B20 ; 16S35 ; 16W30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We give ‘rational’ forms for twistings of classical enveloping algebras. We also remark a link with the ‘generalized formalism’ of Gurevich, Manin, and Cartier.
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    Reality in the differential calculus on q-Euclidean spaces (1992)
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    Letters in mathematical physics 25 (1992), S. 121-130 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 81R05
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    Topics: Mathematics , Physics
    Notes: Abstract The nonlinear reality structure of the derivatives and the differentials for the Euclidean q-spaces are found. A real Laplacian is constructed and reality properties of the exterior derivative are given.
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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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    ISSN: 1573-0530
    Keywords: 81R50 ; 16W30 ; 58B30 ; 57M25 ; 05A30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    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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    The hopf algebra of vector fields on complex quantum groups (1992)
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    Letters in mathematical physics 26 (1992), S. 91-96 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 81R05
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    Topics: Mathematics , Physics
    Notes: Abstract We derive the equivalence of the complex quantum enveloping algebra and the algebra of complex quantum vector fields for the Lie algebra types A n , B n , C n , and D n by factorizing the vector fields uniquely into a triangular and a unitary part and identifying them with the corresponding elements of the algebra of regular functionals.
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    Two variable link polynomials from quantum supergroups (1992)
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    Letters in mathematical physics 26 (1992), S. 187-198 
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    Keywords: 81R10 ; 17B37 ; 16W30
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    Topics: Mathematics , Physics
    Notes: Abstract New two variable link polynomials are constructed corresponding to a one-parameter family of representations of the quantum supergroup U q [gl(2 | 1)]. Their connection with the Kauffman polynomials is also investigated.
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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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    ISSN: 1573-0530
    Keywords: 16E40 ; 16W30 ; 17B37 ; 81R50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    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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    On the defining relations of quantum superalgebras (1991)
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    Letters in mathematical physics 23 (1991), S. 127-131 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 16S80 ; 22E70
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract In defining quantum superalgebras, extra relations need to be added to the Serre-like relations. They are obtained for sl q (m, n) and osp q (m, 2n) usingq-oscillator representations.
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    Six generatorq-deformed Lorentz algebra (1991)
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    Letters in mathematical physics 23 (1991), S. 233-240 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 81R05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The six generator deformation of the Lorentz algebra is presented. The Hopf algebra structure and the reality conditions are found. The chiral decomposition of SL(2, C) is generalized to theq-case. Casimir operators for theq-Lorentz algebra are given.
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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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    ISSN: 1573-0530
    Keywords: 81R50 ; 17B37 ; 16W30 ; 81R30 ; 81S05 ; 81T40 ; 20F36
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    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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    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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    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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    Topics: Mathematics , Physics
    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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    Braided groups and algebraic quantum field theories (1991)
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    Letters in mathematical physics 22 (1991), S. 167-175 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 18D10 ; 20N99 ; 81T05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We introduce the notion of a braided group. This is analogous to a supergroup with Bose-Fermi statistics ±1 replaced by braid statistics. We show that every algebraic quantum field theory in two dimensions leads to a braided group of internal symmetries. Every quantum group can be viewed as a braided group.
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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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    An algebra related to the fusion rules of Wess-Zumino-Witten models (1991)
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    Letters in mathematical physics 22 (1991), S. 291-296 
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    ISSN: 1573-0530
    Keywords: 16W30
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    Topics: Mathematics , Physics
    Notes: Abstract We introduce a new family of examples of ‘quantum groups’. Its irreducible representations are indexed by level l dominant integral weights of affine Lie algebra sī(n), and their branching rules with respect to the ‘truncated tensor product’ $$\bar\otimes $$ is given by the fusion rules of SU(n) l -Wess-Zumino-Witten model in conformal field theory.
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    Fusion rule engineering (1991)
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    Letters in mathematical physics 23 (1991), S. 11-18 
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    ISSN: 1573-0530
    Keywords: 16W30 ; 16Dxx ; 81T
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    Topics: Mathematics , Physics
    Notes: Abstract The determination of the fusion rules of WZW theories constitutes an important step in the program of classifying and solving conformal field theories. A drawback of the existing algorithms for such calculations is that, even at a modestly large number of primary fields, they require an enormous amount of computer time and memory. However, by breeding together two of these algorithms in an appropriate manner, it is possible to speed up the calculations and, therefore, greatly extend the accessible range of theories.
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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
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    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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