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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Communications in mathematical physics 172 (1995), S. 571-622 
    ISSN: 1432-0916
    Keywords: 17B67 ; 17B81 ; 81R10 ; 83E30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract An attempt is made to understand the root spaces of Kac Moody algebras of hyperbolic type, and in particularE 10, in terms of a DDF construction appropriate to a subcritical compactified bosonic string. While the level-one root spaces can be completely characterized in terms of transversal DDF states (the level-zero elements just span the affine subalgebra), longitudinal DDF states are shown to appear beyond level one. In contrast to previous treatments of such algebras, we find it necessary to make use of a rational extension of the self-dual root lattice as an auxiliary device, and to admit non-summable operators (in the sense of the vertex algebra formalism). We demonstrate the utility of the method by completely analyzing a non-trivial level-two root space, obtaining an explicit and comparatively simple representation for it. We also emphasize the occurrence of several Virasoro algebras, whose interrelation is expected to be crucial for a better understanding of the complete structure of the Kac Moody algebra.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract: We study the embedding of Kac–Moody algebras into Borcherds (or generalized Kac–Moody) algebras which can be explicitly realized as Lie algebras of physical states of some completely compactified bosonic string. The extra “missing states” can be decomposed into irreducible highest or lowest weight “missing modules” w.r.t. the relevant Kac–Moody subalgebra; the corresponding lowest weights are associated with imaginary simple roots whose multiplicities can be simply understood in terms of certain polarization states of the associated string. We analyse in detail two examples where the momentum lattice of the string is given by the unique even unimodular Lorentzian lattice or , respectively. The former leads to the Borcherds algebra , which we call “gnome Lie algebra”, with maximal Kac--Moody subalgebra A 1. By the use of the denominator formula a complete set of imaginary simple roots can be exhibited, while the DDF construction provides an explicit Lie algebra basis in terms of purely longitudinal states of the compactified string in two dimensions. The second example is the Borcherds algebra , whose maximal Kac–Moody subalgebra is the hyperbolic algebra E 10. The imaginary simple roots at level 1, which give rise to irreducible lowest weight modules for E 10, can be completely characterized; furthermore, our explicit analysis of two non-trivial level-2 root spaces leads us to conjecture that these are in fact the only imaginary simple roots for .
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 38 (1997), S. 4435-4450 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: An affine vertex operator construction at an arbitrary level is presented which is based on a completely compactified chiral bosonic string whose momentum lattice is taken to be the (Minkowskian) affine weight lattice. This construction is manifestly physical in the sense of string theory, i.e., the vertex operators are functions of Del Giudice–Di Vecchia–Fubini (DFF) "oscillators" and the Lorentz generators, both of which commute with the Virasoro constraints. We therefore obtain explicit representations of affine highest weight modules in terms of physical (DDF) string states. This opens new perspectives on the representation theory of affine Kac–Moody algebras, especially in view of the simultaneous treatment of infinitely many affine highest weight representations of arbitrary level within a single state space as required for the study of hyperbolic Kac–Moody algebras. A novel interpretation of the affine Weyl group as the "dimensional null reduction" of the corresponding hyperbolic Weyl group is given, which follows upon re-expression of the affine Weyl translations as Lorentz boosts. © 1997 American Institute of Physics.
    Type of Medium: Electronic Resource
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