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218
Noncommutative FiniteDimensional Manifolds  I. SPHERICAL MANIFOLDS AND RELATED EXAMPLES
, 2001
"... We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 d ..."
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Cited by 89 (12 self)
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We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 defined by basic Ktheoretic equations. We find a 3parameter family of deformations of the standard 3sphere S 3 and a corresponding 3parameter deformation of the 4dimensional Euclidean space R 4. For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed R 4 u are isomorphic to the algebras introduced by Sklyanin in connection with the YangBaxter equation. Special values of the deformation parameters do not give rise to Sklyanin algebras and we extract a subclass, the θdeformations, which we generalize in any dimension and various contexts, and study in some details. Here, and
Pointed Hopf algebras
 In “New directions in Hopf algebras”, MSRI series Cambridge Univ
, 2002
"... Abstract. This is a survey on pointed Hopf algebras over algebraically closed fields of characteristic 0. We propose to classify pointed Hopf algebras A by first determining the graded Hopf algebra gr A associated to the coradical filtration of A. The A0coinvariants elements form a braided Hopf alg ..."
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Cited by 53 (4 self)
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Abstract. This is a survey on pointed Hopf algebras over algebraically closed fields of characteristic 0. We propose to classify pointed Hopf algebras A by first determining the graded Hopf algebra gr A associated to the coradical filtration of A. The A0coinvariants elements form a braided Hopf algebra R in the category of Yetter–Drinfeld modules over the coradical A0 = Γ, Γ the group of grouplike elements of A, and gr A ≃ R#A0. We call the braiding of the primitive elements of R the infinitesimal braiding of A. If this braiding is of Cartan type [AS2], then it is often possible to determine R, to show that R is generated as an algebra by its primitive elements and finally to compute all deformations or liftings, that is pointed Hopf algebras such that gr A ≃ R#Γ. In the last chapter, as a concrete illustration of the method, we describe explicitly all finitedimensional pointed Hopf algebras A with abelian group of grouplikes G(A) and infinitesimal braiding of type An (up to some exceptional cases). In other words, we compute all the liftings of type An; this result is our main new
Coalgebra extensions and algebra coextensions of Galois type, Commun. Algebra 27
, 1999
"... The notion of a coalgebraGalois extension is defined as a natural generalisation of a HopfGalois extension. It is shown that any coalgebraGalois extension induces a unique entwining map ψ compatible with the right coaction. For the dual notion of an algebraGalois coextension it is also proven th ..."
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Cited by 49 (15 self)
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The notion of a coalgebraGalois extension is defined as a natural generalisation of a HopfGalois extension. It is shown that any coalgebraGalois extension induces a unique entwining map ψ compatible with the right coaction. For the dual notion of an algebraGalois coextension it is also proven that there always exists a unique entwining structure compatible with the right action. 1
Construction of Field Algebras with Quantum Symmetry from Local Observables
, 1996
"... It has been discussed earlier that ( weak quasi) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid) statistics and locality was established. This work addresses to the reco ..."
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Cited by 49 (8 self)
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It has been discussed earlier that ( weak quasi) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid) statistics and locality was established. This work addresses to the reconstruction of quantum symmetries and algebras of field operators. For every algebra A of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti) commutation relations, these fields are demonstrated to obey local braid relation. Contents 1 Introduction 1 2 The Notion of Quantum Symmetry 5 3 Algebraic Methods for Field Construction 9 3.1 Observables and superselection sectors in local quantum field theory . . . . 10 3.2 Localized endomorphisms and fusion structure . . . . . ....
Noncommutative geometry and gravity
"... We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a starproduct. The class of noncommutative spaces studied is very rich. Nonanti ..."
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Cited by 47 (14 self)
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We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a starproduct. The class of noncommutative spaces studied is very rich. Nonanticommutative superspaces are also briefly considered. The differential geometry developed is covariant under deformed diffeomorphisms and it is coordinate independent. The main target of this work is the construction of Einstein’s equations for gravity on noncommutative manifolds.
The standard model on noncommutative spacetime, Eur
 Phys. J. C23
"... We consider the Standard Model on a noncommutative space and expand the action in the noncommutativity parameter θ µν. No new particles are introduced, the structure group is SU(3) × SU(2) × U(1). We derive the leading order action. At zeroth order the action coincides with the ordinary Standard ..."
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Cited by 43 (5 self)
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We consider the Standard Model on a noncommutative space and expand the action in the noncommutativity parameter θ µν. No new particles are introduced, the structure group is SU(3) × SU(2) × U(1). We derive the leading order action. At zeroth order the action coincides with the ordinary Standard Model. At leading order in θ µν we find new vertices which are absent in the Standard Model on commutative spacetime. The most striking features are couplings between quarks, gluons and electroweak bosons and many new vertices in the charged and neutral currents. We find that parity is violated in noncommutative QCD. The Higgs mechanism can be applied. QED is not deformed in the minimal version of the A method for implementing nonAbelian SU(N) YangMills theories on noncommutative spacetime has recently been proposed [1–4]. Previously only U(N) gauge theories were under control, and it was thus only possible to consider extensions of the Standard Model. Recently there has been a lot of activity on model building. The aim of this
Braided Hopf algebras over non abelian finite groups
 Acad. Nac. Ciencias (Córdoba
"... Abstract. In the last years a new theory of Hopf algebras has begun to be developed: that of Hopf algebras in braided categories, or, briefly, braided Hopf algebras. This is a survey of general aspects of the theory with emphasis in H HYD, the Yetter–Drinfeld category over H, where H is the group al ..."
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Cited by 40 (11 self)
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Abstract. In the last years a new theory of Hopf algebras has begun to be developed: that of Hopf algebras in braided categories, or, briefly, braided Hopf algebras. This is a survey of general aspects of the theory with emphasis in H HYD, the Yetter–Drinfeld category over H, where H is the group algebra of a non abelian finite group Γ. We discuss a special class of braided graded Hopf algebras from different points of view following Lusztig, Nichols and Schauenburg. We present some finite dimensional examples arising in an unpublished work by Milinski and Schneider. Sinopsis. En los últimos años comenzó a ser desarrollada una nueva teoría de álgebras de Hopf en categorías trenzadas, o brevemente, álgebras de Hopf trenzadas. Presentamos aquí aspectos generales de la teoría con énfasis en H HYD, la categoría de Yetter–Drinfeld sobre H, donde H es el álgebra de grupo de un grupo finito no abeliano Γ. Discutimos una clase especial de álgebras de Hopf trenzadas graduadas desde diferentes puntos de vista, siguiendo a Lusztig, Nichols y Schauenburg. Presentamos algunos ejemplos de dimensión finita que aparecen en un trabajo inédito de Milinski y Schneider. 0. Introduction and notations 0.1. Introduction.
Cyclic cohomology, quantum group symmetries and the local index formula for SUq(2
 J. Inst. Math. Jussieu
"... We analyse the NCspace underlying the quantum group SUq(2) from the spectral point of view which is the basis of noncommutative geometry, and show how the general theory developped in our joint work with H. Moscovici applies to the specific spectral triple defined by Chakraborty and Pal. This provi ..."
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Cited by 36 (2 self)
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We analyse the NCspace underlying the quantum group SUq(2) from the spectral point of view which is the basis of noncommutative geometry, and show how the general theory developped in our joint work with H. Moscovici applies to the specific spectral triple defined by Chakraborty and Pal. This provides the pseudodifferential calculus, the Wodzcikitype residue, and the local cyclic cocycle giving the index formula. The cochain whose coboundary is the difference between the original Chern character and the local one is given by the remainders in the rational approximation of the logarithmic derivative of the Dedekind eta function. This specific example allows to illustrate the general notion of locality in NCG. The formulas computing the residue are ”local”. Locality by stripping all the expressions from irrelevant details makes them computable. The key feature of this spectral triple is its equivariance, i.e. the SUq(2)symmetry. We shall explain how this leads naturally to the general concept of invariant cyclic cohomology in the framework of quantum group symmetries.
Noncommutative geometry of finite groups
 J. Phys. A
, 1996
"... A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left, right and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A dif ..."
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Cited by 32 (3 self)
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A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left, right and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and, moreover, for the formulation of field theories and dynamics on finite sets. Associated with each bicovariant first order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures. For a covariant calculus, there are notions of invariance for linear connections and tensors. All these concepts are explored for finite groups and illustrated with examples. Some results are formulated more generally for arbitrary associative (Hopf) algebras. In particular, the problem of extension of a connection on a bimodule (over an associative algebra) to tensor products is investigated, leading to the class of ‘extensible connections’. It is shown that invariance properties of an extensible connection on a bimodule over a Hopf algebra are carried over to the extension. Furthermore, an invariance property of a connection is also shared by a ‘dual connection ’ which exists on the dual bimodule (as defined in this work).2 1
Differential calculi over quantum groups and twisted cyclic cocycles
 J Geom Phys
"... Abstract. We study some aspects of the theory of noncommutative differential calculi over complex algebras, especially over the Hopf algebras associated to compact quantum groups in the sense of S.L. Woronowicz. Our principal emphasis is on the theory of twisted graded traces and their associated t ..."
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Cited by 29 (0 self)
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Abstract. We study some aspects of the theory of noncommutative differential calculi over complex algebras, especially over the Hopf algebras associated to compact quantum groups in the sense of S.L. Woronowicz. Our principal emphasis is on the theory of twisted graded traces and their associated twisted cyclic cocycles. One of our principal results is a new method of constructing differential calculi, using twisted graded traces. Introduction. A compact group is a compact space with a continuous multiplication satisfying certain extra conditions. In the theory of compact quantum groups developed by S.L. Woronowicz [3, 4, 5, 6, 8], one replaces the compact space by a unital C*algebra A that is in general noncommutative, and replaces the group multiplication by a comultiplication on A satisfying certain cancelation conditions.