86 research outputs found

    Hilbert Space Representation of an Algebra of Observables for q-Deformed Relativistic Quantum Mechanics

    Full text link
    Using a representation of the q-deformed Lorentz algebra as differential operators on quantum Minkowski space, we define an algebra of observables for a q-deformed relativistic quantum mechanics with spin zero. We construct a Hilbert space representation of this algebra in which the square of the mass p2 p^2 is diagonal.Comment: 13 pages, LMU-TPW 94-

    Heisenberg double as braided commutative Yetter-Drinfel'd module algebra over Drinfel'd double in multiplier Hopf algebra case

    Full text link
    Based on a pairing of two regular multiplier Hopf algebras AA and BB, Heisenberg double H\mathscr{H} is the smash product A#BA \# B with respect to the left regular action of BB on AA. Let D=A⋈B\mathscr{D}=A\bowtie B be the Drinfel'd double, then Heisenberg double H\mathscr{H} is a Yetter-Drinfel'd D\mathscr{D}-module algebra, and it is also braided commutative by the braiding of Yetter-Drinfel'd module, which generalizes the results in [10] to some infinite dimensional cases.Comment: 18 pages. arXiv admin note: text overlap with arXiv:math/0404029 by other author

    U-duality (sub-)groups and their topology

    Full text link
    We discuss some consequences of the fact that symmetry groups appearing in compactified (super-)gravity may be non-simply connected. The possibility to add fermions to a theory results in a simple criterion to decide whether a 3-dimensional coset sigma model can be interpreted as a dimensional reduction of a higher dimensional theory. Similar criteria exist for higher dimensional sigma models, though less decisive. Careful examination of the topology of symmetry groups rules out certain proposals for M-theory symmetries, which are not ruled out at the level of the algebra's. We conclude with an observation on the relation between the ``generalized holonomy'' proposal, and the actual symmetry groups resulting from E_10 and E_11 conjectures.Comment: LaTeX, 8 pages, 2 tables, 1 figure, uses IOP-style files. Contributed to the proceedings of the RTN-workshop ``The quantum structure of space-time and the geometrical nature of the fundamental interactions,'', Copenhagen, Denmark, september 200

    Bicovariant Quantum Algebras and Quantum Lie Algebras

    Get PDF
    A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given by elements of the pure braid group. These operators --- the `reflection matrix' Y≡L+SL−Y \equiv L^+ SL^- being a special case --- generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation for YY in SOq(N)SO_q(N).Comment: 38 page

    Characteristic Relations for Quantum Matrices

    Full text link
    General algebraic properties of the algebras of vector fields over quantum linear groups GLq(N)GL_q(N) and SLq(N)SL_q(N) are studied. These quantum algebras appears to be quite similar to the classical matrix algebra. In particular, quantum analogues of the characteristic polynomial and characteristic identity are obtained for them. The qq-analogues of the Newton relations connecting two different generating sets of central elements of these algebras (the determinant-like and the trace-like ones) are derived. This allows one to express the qq-determinant of quantized vector fields in terms of their qq-traces.Comment: 11 pages, latex, an important reference [16] added

    Diagrammar and metamorphosis of coset symmetries in dimensionally reduced type IIB supergravity

    Full text link
    Studying the reduction of type IIB supergravity from ten to three space-time dimensions we describe the metamorphosis of Dynkin diagram for gravity line "caterpillar" into a type IIB supergravity "dragonfly" that is triggered by inclusion of scalars and antisymmetric tensor fields. The final diagram corresponds to type IIB string theory E8 global symmetry group which is the subgroup of the conjectured E11 hidden symmetry group. Application of the results for getting the type IIA/IIB T-duality rules and for searching for type IIB vacua solutions is considered.Comment: 9 pp, 7 figs, LATEX; to be published in JETP Let

    The topology of U-duality (sub-)groups

    Full text link
    We discuss the topology of the symmetry groups appearing in compactified (super-)gravity, and discuss two applications. First, we demonstrate that for 3 dimensional sigma models on a symmetric space G/H with G non-compact and H the maximal compact subgroup of G, the possibility of oxidation to a higher dimensional theory can immediately be deduced from the topology of H. Second, by comparing the actual symmetry groups appearing in maximal supergravities with the subgroups of SL(32,R) and Spin(32), we argue that these groups cannot serve as a local symmetry group for M-theory in a formulation of de Wit-Nicolai type.Comment: 18 pages, LaTeX, 1 figure, 2 table

    More on quantum groups from the the quantization point of view

    Full text link
    Star products on the classical double group of a simple Lie group and on corresponding symplectic grupoids are given so that the quantum double and the "quantized tangent bundle" are obtained in the deformation description. "Complex" quantum groups and bicovariant quantum Lie algebras are discused from this point of view. Further we discuss the quantization of the Poisson structure on symmetric algebra S(g)S(g) leading to the quantized enveloping algebra Uh(g)U_{h}(g) as an example of biquantization in the sense of Turaev. Description of Uh(g)U_{h}(g) in terms of the generators of the bicovariant differential calculus on F(Gq)F(G_q) is very convenient for this purpose. Finally we interpret in the deformation framework some well known properties of compact quantum groups as simple consequences of corresponding properties of classical compact Lie groups. An analogue of the classical Kirillov's universal character formula is given for the unitary irreducible representation in the compact case.Comment: 18 page
    • 

    corecore