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  • 1
    Electronic Resource
    Electronic Resource
    A short survey of noncommutative geometry (2000)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 41 (2000), S. 3832-3866 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the Riemann–Hilbert problem. We discuss at length two issues. The first is the relevance of the paradigm of geometric space, based on spectral considerations, which is central in the theory. As a simple illustration of the spectral formulation of geometry in the ordinary commutative case, we give a polynomial equation for geometries on the four-sphere with fixed volume. The equation involves an idempotent e, playing the role of the instanton, and the Dirac operator D. It is of the form 〈(e−〈fraction SHAPE="CASE"〉12)[D,e]4〉=γ5 and determines both the sphere and all its metrics with fixed volume form. The expectation 〈x〉 is the projection on the commutant of the algebra of 4 by 4 matrices. We also show, using the noncommutative analog of the Polyakov action, how to obtain the noncommutative metric (in spectral form) on the noncommutative tori from the formal naive metric. We conclude with some questions related to string theory. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.533329
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  • 2
    Electronic Resource
    Electronic Resource
    Noncommutative geometry and reality (1995)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 6194-6231 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We introduce the notion of real structure in our spectral geometry. This notion is motivated by Atiyah's KR-theory and by Tomita's involution J. It allows us to remove two unpleasant features of the "Connes–Lott'' description of the standard model, namely, the use of bivector potentials and the asymmetry in the Poincaré duality and in the unimodularity condition. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531241
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  • 3
    Electronic Resource
    Electronic Resource
    Renormalization in Quantum Field Theory and the Riemann–Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem (2000)
    Springer
    Communications in mathematical physics 210 (2000), S. 249-273 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract: This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann–Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra which is commutative as an algebra. It is the dual Hopf algebra of the enveloping algebra of a Lie algebra whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of . We shall then show that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop where C is a small circle of complex dimensions around the integer dimension D of space-time. Our main result is that the renormalized theory is just the evaluation at z=D of the holomorphic part γ+ of the Birkhoff decomposition of γ. We begin to analyse the group G and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. The analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002200050779
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  • 4
    Electronic Resource
    Electronic Resource
    Caractère multiplicatif d'un module de Fredholm (1988)
    Springer
    K-Theory 2 (1988), S. 431-463 
    Details
    ISSN: 1573-0514
    Keywords: Cyclic homology ; Fredholm modules ; loop groups ; C *-algebras ; group of diffeomorphisms of the circle
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Résumé L'objectif essential de cet article est de définir un accouplement $$Ell^p (A) \times K_{p + 1} (A) \to \mathbb{C}^* $$ où K(A) désigne la K-théorie algébrique de A et où Ell p(A) est le groupe engendré par les modules de Fredholm de dimension p. Nous relions cet accouplement au déterminant de Fredholm et aux extensions centrales des groupes de lacets et du groupe des difféomorphismes du cercle.
    Notes: Abstract The main object of this paper is to define a pairing $$Ell^p (A) \times K_{p + 1} (A) \to \mathbb{C}^* $$ , where K(A) is the algebraic K-theory of A and Ell p(A) is the group generated by Fredholm modules of dimension p. We relate this pairing to Fredholm determinants and central extensions of loop groups and the group of diffeomorphisms of the circle.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00533391
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  • 5
    Electronic Resource
    Electronic Resource
    Matrix Vieta Theorem Revisited (1997)
    Springer
    Letters in mathematical physics 39 (1997), S. 349-353 
    Details
    ISSN: 1573-0530
    Keywords: Vieta theorem ; matrix equations ; symmetric functions.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We give another proof of the noncommutative analog of the Vieta theorem. This proof gives a little bit stronger statement and leads to some generalizations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1007373114601
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  • 6
    Electronic Resource
    Electronic Resource
    Closed star products and cyclic cohomology (1992)
    Springer
    Letters in mathematical physics 24 (1992), S. 1-12 
    Details
    ISSN: 1573-0530
    Keywords: 19P55 ; 17B37 ; 81S30 ; 19K56
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We define the notion of a closed star product. A (generalized) star product (deformation of the associative product of functions on a symplectic manifold W) is closed iff integration over W is a trace on the deformed algebra. We show that for these products the cyclic cohomology replaces the Hochschild cohomology in usual star products. We then define the character of a closed star product as the cohomology class (in the cyclic bicomplex) of a well-defined cocycle, and show that, in the case of pseudodifferential operators (standard ordering on the cotangent bundle to a compact Riemannian manifold), the character is defined and given by the Todd class, while in general it fails to satisfy the integrality condition.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00429997
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  • 7
    Electronic Resource
    Electronic Resource
    Cyclic Cohomology and Hopf Algebra Symmetry (2000)
    Springer
    Letters in mathematical physics 52 (2000), S. 1-28 
    Details
    ISSN: 1573-0530
    Keywords: noncommutative geometry ; cyclic cohomology ; Hopf algebras ; quantum groups
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract Cyclic cohomology has been recently adapted to the treatment of Hopf symmetry in noncommutative geometry. The resulting theory of characteristic classes for Hopf algebras and their actions on algebras allows the expansion of the range of applications of cyclic cohomology. It is the goal of this Letter to illustrate these recent developments, with special emphasis on the application to transverse index theory, and point towards future directions. In particular, we highlight the remarkable accord between our framework for cyclic cohomology of Hopf algebras on the one hand and both the algebraic as well as the analytic theory of quantum groups on the other, manifest in the construction of the modular square.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1007698216597
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  • 8
    Electronic Resource
    Electronic Resource
    Geometry from the spectral point of view (1995)
    Springer
    Letters in mathematical physics 34 (1995), S. 203-238 
    Details
    ISSN: 1573-0530
    Keywords: 46L60 ; 46L80 ; 46L87 ; 19K56 ; 58H15 ; 58A12
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract In this Letter, we develop geometry from a spectral point of view, the geometric data being encoded by a triple (A. H. D.) of an algebraA represented in a Hilbert spaceH with selfadjoint operatorD. This point of view is dictated by the general framework of noncommutative geometry and allows us to use geometric ideas in many situations beyond Riemannian geometry.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01872777
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  • 9
    Electronic Resource
    Electronic Resource
    Lessons from Quantum Field Theory: Hopf Algebras and Spacetime Geometries (1999)
    Springer
    Letters in mathematical physics 48 (1999), S. 85-96 
    Details
    ISSN: 1573-0530
    Keywords: quantum field theory ; noncommutative geometry ; renormalization ; Hopf algebras ; foliations ; ODE
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We discuss the prominence of Hopf algebras in recent progress in Quantum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a conceptual understanding of the subtraction procedure. We shall then describe several occurrences of this, or closely related Hopf algebras, in other mathematical domains, such as foliations, Runge-Kutta methods, iterated integrals and multiple zeta values. We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1007523409317
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  • 10
    Electronic Resource
    Electronic Resource
    Cyclic Cohomology and Hopf Algebras (1999)
    Springer
    Letters in mathematical physics 48 (1999), S. 97-108 
    Details
    ISSN: 1573-0530
    Keywords: cyclic cohomology ; Hopf algebras
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract We associate canonically a cyclic module to any Hopf algebra endowed with a modular pair in involution, consisting of a group-like element and a character. This provides the key construction for allowing the extension of cyclic cohomology to Hopf algebras in the nonunimodular case and, further, to developing a theory of characteristic classes for actions of Hopf algebras compatible not only with traces but also with the modular theory of weights. This applies to both ribbon and coribbon algebras as well as to quantum groups and their duals.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1007527510226
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