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  • 1
    Electronic Resource
    Electronic Resource
    Braided geometry of the conformal algebra (1996)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 37 (1996), S. 6495-6509 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We show that the action of the special conformal transformations of the usual (undeformed) conformal group is the q→1 scaling limit of the braided adjoint action or R-commutator of q-Minkowski space on itself. We also describe the q-deformed conformal algebra in R-matrix form and its quasi-* structure. © 1996 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531749
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  • 2
    Electronic Resource
    Electronic Resource
    Erratum: Liouville theorem for the Yang–Mills self-duality equations [J. Math. Phys. 29, 2303 (1988)] (1989)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 30 (1989), S. 1652-1652 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528249
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  • 3
    Electronic Resource
    Electronic Resource
    Quantum groups and noncommutative geometry (2000)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 41 (2000), S. 3892-3942 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalization of symmetry groups for certain integrable systems, and on the other as part of a generalization of geometry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in the form of the first theoretical predictions for Planck-scale physics via ongoing astronomical measurements. Noncommutativity of space–time, in particular, amounts to a postulated new force or physical effect called cogravity. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.533331
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  • 4
    Electronic Resource
    Electronic Resource
    Quantum geometry of field extensions (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 2311-2323 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We introduce a new kind of topological gauge configuration or "soliton" associated to the extension C&supuline;R of the real numbers to the complex ones. These configurations describe zero-curvature gauge fields and nontrivial cohomology over the real line, but with a quantum choice of differential calculus. In general, the quantum differential 1-forms on the line with coordinate algebra k[x] are in correspondence with field extensions of k, and the quantum cohomology detects the nontriviality of the extension. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532866
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  • 5
    Electronic Resource
    Electronic Resource
    Nonstandard quantum groups and superization (1995)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 7081-7097 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We obtain the universal R-matrix of the nonstandard quantum group associated to the Alexander–Conway knot polynomial. We show further that this nonstandard quantum group is related to the super-quantum group Uq gl(1||1) by a general process of superization, which we describe. We also study a twisted variant of this nonstandard quantum group and obtain, as a result, a twisted version of Uq gl(1||1) as a q supersymmetry of the exterior differential calculus of any quantum plane of Hecke type, acting by mixing the bosonic xi coordinates and the forms dxi. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531344
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  • 6
    Electronic Resource
    Electronic Resource
    Algebraic q-integration and Fourier theory on quantum and braided spaces (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 6802-6837 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: An algebraic theory of integration on quantum planes and other braided spaces is introduced. In the one-dimensional case a novel picture of the Jackson q- integral as indefinite integration on the braided group of functions in one variable x is obtained. Here x is treated with braid statistics q rather than the usual bosonic or Grassmann ones. It is shown that the definite integral ∫x∞−x∞ can also be evaluated algebraically as multiples of the integral of a q-Gaussian, with x remaining as a bosonic scaling variable associated with the q-deformation. Further composing the algebraic integration with a representation then leads to ordinary numbers for the integral. Integration is also used to develop a full theory of q-Fourier transformation F The braided addition Δx=x⊗1+1⊗x and braided-antipode S is used to define a convolution product, and prove a convolution theorem. It is also proven that F2=S. The analogous results are proven on any braided group, including integration and Fourier transformation on quantum planes associated to general R matrices, including q-Euclidean and q-Minkowski spaces. © 1994 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530644
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  • 7
    Electronic Resource
    Electronic Resource
    Braided momentum in the q-Poincaré group (1993)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 34 (1993), S. 2045-2058 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The q-Poincaré group of M. Schlieker et al. [Z. Phys. C 53, 79 (1992)] is shown to have the structure of a semidirect product and coproduct B× SOq(1,3) where B is a braided-quantum group structure on the q-Minkowski space of four-momentum with braided-coproduct Δ(underbar)p=p⊗1+1⊗p. Here the necessary B is not a usual kind of quantum group, but one with braid statistics. Similar braided vectors and covectors V(R'), V*(R') exist for a general R-matrix. The abstract structure of the q-Lorentz group is also studied.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530154
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  • 8
    Electronic Resource
    Electronic Resource
    Liouville theorem for the Yang–Mills self-duality equations (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 2303-2310 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: It is shown that under certain conditions one may associatively matrix-multiply Lie-algebra-valued matrices with a componentwise Lie bracket. Using this, a simple algebraic constraint on a Lie-algebra-valued antisymmetric n×n matrix F, which in n=4 is essentially self-duality or anti-self-duality, is described. Somewhat in analogy with Liouville's theorem for the Cauchy–Riemann equations in n=2, it is shown that, for n〉4, the constraint implies that the Lie subalgebra generated by the matrix elements {Fμν } decomposes into copies of S(underbar)O(underbar)(n) plus a few degenerate cases. The result may be relevant to the structure of the quantum chromodynamic vacuum.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528204
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  • 9
    Electronic Resource
    Electronic Resource
    C-statistical quantum groups and Weyl algebras (1992)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 3431-3444 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: C-statistical quantum groups and algebras are obtained by a process of transmutation. They are like super-quantum groups and algebras but with statistics given now by a complex number. Among the examples, it is shown that the Weyl algebra of canonical quantization may be viewed equivalently as a C-statistical plane. Related examples are the "noncommutative torus'' and a double loop-variable quantization of photons. These results lead toward a reformulation of ordinary quantum mechanics as a classical theory with C-statistics. In this reformulation, the role of the ±1 factors of super statistics is played by the free-particle time evolution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529891
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  • 10
    Electronic Resource
    Electronic Resource
    Examples of braided groups and braided matrices (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 3246-3253 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Matrix braided groups are developed as an analog of the "coordinate functions'' on a group or supergroup. The ±1 in the super case is replaced by braid statistics. There are braided group analogs of all the classical simple Lie groups as well as braided matrix groups and braided matrices B(R) for every regular solution R of the quantum Yang–Baxter equations. A direct verification of B(R) is provided and some of the simplest examples are computed in detail.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529485
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