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  • 1
    Electronic Resource
    Electronic Resource
    q-epsilon tensor for quantum and braided spaces (1995)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 1991-2007 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The machinery of braided geometry introduced previously is used now to construct the ε "totally antisymmetric tensor'' on a general braided vector space determined by R-matrices. This includes natural q-Euclidean and q-Minkowski spaces. The formalism is completely covariant under the corresponding quantum group such as SOq(4) or SOq(1,3). The Hodge * operator and differentials are also constructed in this approach. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531098
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  • 2
    Electronic Resource
    Electronic Resource
    *-structures on braided spaces (1995)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 4436-4449 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: *-structures on quantum and braided spaces of the type defined via an R-matrix are studied. These include q-Minkowski and q-Euclidean spaces as additive braided groups. The duality between the *-braided groups of vectors and covectors is proved and some first applications to braided geometry are made. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530900
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  • 3
    Electronic Resource
    Electronic Resource
    On the addition of quantum matrices (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 2617-2632 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: An addition law is introduced for the usual quantum matrices A(R) by means of a coaddition Δ(underbar)t=t⊗1+1⊗t. It supplements the usual comultiplication Δt=t⊗t and together they obey a codistributivity condition. The coaddition does not form a usual Hopf algebra but a braided one. The same remarks apply for rectangular m×n quantum matrices. As an application, left-invariant vector fields are constructed on A(R) and other quantum spaces. They close in the form of a braided Lie algebra. As another application, the wave functions in the lattice approximation of Kac–Moody algebras and other lattice fields can be added and functionally differentiated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530527
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  • 4
    Electronic Resource
    Electronic Resource
    Free braided differential calculus, braided binomial theorem, and the braided exponential map (1993)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 34 (1993), S. 4843-4856 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Braided differential operators ∂i are obtained by differentiating the addition law on the braided covector spaces introduced previously (such as the braided addition law on the quantum plane). These are affiliated to a Yang–Baxter matrix R. The quantum eigenfunctions expR(x||v) of the ∂i (braided-plane waves) are introduced in the free case where the position components xi are totally noncommuting. A braided R-binomial theorem and a braided Taylor theorem expR(a||∂)f(x)=f(a+x) are proven. These various results precisely generalize to a generic R-matrix (and hence to n dimensions) the well-known properties of the usual one-dimensional q-differential and q-exponential. As a related application, it is shown that the q-Heisenberg algebra px−qxp=1 is a braided semidirect product C[x]×C[ p] of the braided line acting on itself (a braided Weyl algebra) and similarly for its generalization to an arbitrary R- matrix.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530326
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  • 5
    Electronic Resource
    Electronic Resource
    q-Euclidean space and quantum Wick rotation by twisting (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 5025-5034 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The quantum matrix algebra R21x1x2=x2x1R is studied and proposed for the standard 2×2 case as the coordinates of q-deformed Euclidean space. The algebra in this simplest case is isomorphic to the usual quantum matrices Mq(2) but in a form which is naturally covariant under the Euclidean rotations SUq(2)⊗SUq(2). A quantum Wick rotation is introduced that twists this system precisely into the approach to q-Minkowski space based on braided matrices and their associated spinorial q-Lorentz group.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530828
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  • 6
    Electronic Resource
    Electronic Resource
    Random walk and the heat equation on superspace and anyspace (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 3753-3760 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Random walks are used to study diffusion on anyspace. Anyspace is characterized by coordinate ξ with ξN=0 and statistics ξξ'=e2πi/Nξ'ξ between independent copies. Anyonic integration and anyonic Dirac δ functions are introduced, and reduced to familiar results for supersymmetry when N=2. These ingredients are then used to formulate and solve the resulting anyonic diffusion equation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530868
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  • 7
    Electronic Resource
    Electronic Resource
    The braided Heisenberg group (1993)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 34 (1993), S. 3588-3606 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The braided groups and braided matrices B(R) for the solution R of the Yang–Baxter equation associated to the quantum Heisenberg group are computed. It is also shown that a particular extension of the quantum Heisenberg group is dual to the Heisenberg universal enveloping algebra Uq(h), and this result is used to derive an action of Uq(h) on the braided groups. The various covariance properties are then demonstrated using the braided Heisenberg group as an explicit example. In addition, the braided Heisenberg group is found to be self-dual. Finally, a physical application to a system of n braided harmonic oscillators is discussed. An isomorphism is found between the n-fold braided and unbraided tensor products, and the usual "free'' time evolution is shown to be equivalent to an action of a primitive generator of Uq(h) on the braided tensor product.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530047
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  • 8
    Electronic Resource
    Electronic Resource
    Quantum and braided linear algebra (1993)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 34 (1993), S. 1176-1196 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Quantum matrices A(R) are known for every R matrix obeying the quantum Yang–Baxter equations. It is also known that these act on "vectors'' given by the corresponding Zamalodchikov algebra. This interpretation is developed in detail, distinguishing between two forms of this algebra, V(R) (vectors) and V*(R) (covectors). A(R)→V(R21)⊗V*(R) is an algebra homomorphism (i.e., quantum matrices are realized by the tensor product of a quantum vector with a quantum covector), while the inner product of a quantum covector with a quantum vector transforms as a scaler. It is shown that if V(R) and V*(R) are endowed with the necessary braid statistics Ψ then their braided tensor-product V(R)⊗(underbar)V*(R) is a realization of the braided matrices B(R) introduced previously, while their inner product leads to an invariant quantum trace. Introducing braid statistics in this way leads to a fully covariant quantum (braided) linear algebra. The braided groups obtained from B(R) act on themselves by conjugation in a way impossible for the quantum groups obtained from A(R).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530193
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  • 9
    Electronic Resource
    Electronic Resource
    Fermionic q-Fock space and braided geometry (1997)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 38 (1997), S. 4845-4853 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We write the fermionic q-Fock space representation of Uq(sl(circumflex)n) as an infinite extended braided tensor product of finite-dimensional fermionic Uq(sln)-quantum planes or exterior algebras. Using braided geometrical techniques developed for such quantum exterior algebras, we provide a new R-matrix approach to the Kashiwara–Miwa–Stern action of the Heisenberg algebra on the q-fermionic Fock space, obtaining the action in detail for the lowest nontrivial case [b2, b−2]=2((1−q−4n)/(1−q−4)). © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532128
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  • 10
    Electronic Resource
    Electronic Resource
    The quantum double as quantum mechanics
    Amsterdam : Elsevier
    Journal of Geometry and Physics 13 (1994), S. 169-202 
    Details
    ISSN: 0393-0440
    Keywords: Duality ; Lorentz metric ; Mackey quantization ; Non-commutative geometry ; Quantum double ; Quantum groups ; [Mathematical Subject Codes] 81R50
    Source: Elsevier Journal Backfiles on ScienceDirect 1907 - 2002
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://linkinghub.elsevier.com/retrieve/pii/0393-0440(94)90026-4
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