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  • 1
    Electronic Resource
    Electronic Resource
    On some nonunitary representations of the Poincaré group and their use for the construction of free quantum fields (1989)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 30 (1989), S. 2692-2707 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In this paper a class of nonunitary infinite dimensional Hilbert space representations of a semidirect product is investigated. The equivalence of this category with the category of finite dimensional representations of the stability subgroups is shown. This theory is applied to the Poincaré group and to the construction of free quantum fields. In an appendix a method is introduced for building an infinite family of finite dimensional indecomposable representations of the noncompact Euclidean group in two dimensions. Such representations are used for carrying out the analysis of the massless fields.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.528502
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  • 2
    Electronic Resource
    Electronic Resource
    Hilbert space representations of the Poincaré group for the Landau gauge (1991)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 32 (1991), S. 1076-1090 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The theory of the electromagnetic quantum field in the Landau gauge has always lacked a treatment based on a Hilbert space of C4 -valued functions (the one particle space) on which the Poincaré group acts via a continuous representation realized by bounded operators. In the present paper the gap will be filled and it will be shown, through the mathematical technique of jets on the forward light cone in momentum space, how the theory presented extends the previous treatments of the problem. In order to show that the massless vector field theory in the Landau gauge can be obtained, the equation of motion and the two-point function will be derived in the present model according to the Wightman axioms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.529333
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  • 3
    Electronic Resource
    Electronic Resource
    Representation theory approach to the polynomial solutions of q-difference equations: Uq(sl(3)) and beyond (1994)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 35 (1994), S. 6058-6075 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: A new approach to the theory of polynomial solutions of q-difference equations is proposed. The approach is based on the representation theory of simple Lie algebras G and their q-deformations and is presented here for Uq(sl(n)). First a q-difference realization of Uq(sl(n)) in terms of n(n−1)/2 commuting variables and depending on n−1 complex representation parameters, ri, is constructed. From this realization lowest weight modules (LWM) are obtained which are studied in detail for the case n=3 (the well-known n=2 case is also recovered). All reducible LWM are found and the polynomial bases of their invariant irreducible subrepresentations are explicitly given. This also gives a classification of the quasi-exactly solvable operators in the present setting. The invariant subspaces are obtained as solutions of certain invariant q-difference equations, i.e., these are kernels of invariant q-difference operators, which are also explicitly given. Such operators were not used until now in the theory of polynomial solutions. Finally, the states in all subrepresentations are depicted graphically via the so-called Newton diagrams.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.530728
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  • 4
    Electronic Resource
    Electronic Resource
    Poincare group representations on spinor spaces and grassmann algebras
    Amsterdam : Elsevier
    Nuclear Physics B (Proceedings Supplements) 6 (1989), S. 121-123 
    Details
    ISSN: 0920-5632
    Source: Elsevier Journal Backfiles on ScienceDirect 1907 - 2002
    Topics: Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1016/0920-5632(89)90414-3
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  • 5
    Electronic Resource
    Electronic Resource
    Exceptional groups and elementary particle structures
    Amsterdam : Elsevier
    Physica A: Statistical Mechanics and its Applications 114 (1982), S. 257-270 
    Details
    ISSN: 0378-4371
    Source: Elsevier Journal Backfiles on ScienceDirect 1907 - 2002
    Topics: Physics
    Type of Medium: Electronic Resource
    URL: http://linkinghub.elsevier.com/retrieve/pii/0378-4371(82)90294-1
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  • 6
    Electronic Resource
    Electronic Resource
    Three graded exceptional algebras and symmetric spaces (1986)
    Springer
    The European physical journal 33 (1986), S. 47-65 
    Details
    ISSN: 1434-6052
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The exceptional algebras of typeE 7 are studied from the point of view of their three graded structure. The connection between three-grading and the Jordan Pair structure of such Lie algebras is analyzed. The Jordan Pair content is in turn related to the symmetric spaces and . Coset spaces of this type have been recently suggested as possible scalar manifolds in supergravity. We develop a way of representingE 7 in a matrix form, which makes the Jordan Pair content of theE 7 real and complex forms quite transparent and shows whether such forms admit a three graded structure.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01410452
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  • 7
    Electronic Resource
    Electronic Resource
    The concept of a quantum semisimple group (1991)
    Springer
    Letters in mathematical physics 21 (1991), S. 287-292 
    Details
    ISSN: 1573-0530
    Keywords: 81D99
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We explore the possibility of introducing the concept of a quantum semisimple group by exhibiting a class of deformations of classical groups whose Dynkin diagrams are disconnected at the classical level, but become connected at the quantum level. The possibility of applications to the quantization of Lorentz and Poincaré groups are mentioned.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00398326
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  • 8
    Electronic Resource
    Electronic Resource
    Universal deformations of simple Lie algebras (1992)
    Springer
    Letters in mathematical physics 24 (1992), S. 63-71 
    Details
    ISSN: 1573-0530
    Keywords: 81S99 ; 57T015
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract This Letter examines the question of the structure of the Hopf algebra deformations of the universal enveloping algebras of the simple Lie algebras. Deformations of a complex algebra A are viewed as algebras defined over formal power series rings that specialize to A when the parameters go to 0. Only the case of U(sl(2,C)) is treated but the methods are general. Under the Ansatz that the two Borel subalgebras are deformed as Hopf algebras but possibly differently, we construct a universal two-parameter deformation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00430004
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  • 9
    Electronic Resource
    Electronic Resource
    Universal deformations of reductive Lie algebras (1992)
    Springer
    Letters in mathematical physics 26 (1992), S. 53-65 
    Details
    ISSN: 1573-0530
    Keywords: 81S99
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We construct multiparameter quantizations of reductive Lie algebras which have the property of universality within a certain class of deformations. The universal deformations can be defined so that the algebra structure on each simple component is the same as that of the standard one-parameter quantization, the remaining parameters being relegated to the coalgebra structure. We discuss an example in which only the latter parameters appear, as a special case of deformations of a semisimple algebra whose simple components remain classical. Deformations are defined as algebras over power series rings and it is essential to require them to be torsion free to secure the universality. The Poincaré-Birkhoff-Witt theorem and the torsion freeness are established for the universal deformation on the basis of results on the representation theory of the deformed algebras.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00420518
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  • 10
    Electronic Resource
    Electronic Resource
    The Jordan Pair content of the magic square and the geometry of the scalars in N=2 supergravity (1985)
    Springer
    Letters in mathematical physics 9 (1985), S. 255-261 
    Details
    ISSN: 1573-0530
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The close connection between Jordan and Lie algebras makes these Jordan structures of interest to physicists. The Freudenthal-Tits Magic Square, which exemplifies this connection, has recently entered into constructing supergravity. We show how Jordan pairs-which are, from several points of view, a most natural Jordan structure-are imbedded in the Magic Square. We compare our approach with that of Gürsey and show show the Hermitian symmetric spaces parametrized by the scalars of N=2, d=4 supergravity theories are related either to Jordan pairs or to geometries of projective dimension two, whose elements belong to a Jordan pair.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00402838
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