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314
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 126 (15 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
Compact quantum groups
 Les Houches, Session LXIV, 1995, Quantum Symmetries
, 1998
"... In this paper we review the theory of compact quantum groups. 0 ..."
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Cited by 101 (2 self)
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In this paper we review the theory of compact quantum groups. 0
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 99 (7 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
Symmetries of quantum spaces. Subgroups and quotient . . .
, 2008
"... We prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group. We give the formula for the corresponding multiplicities in the case of the quotient quantum spaces. We describe the subgroups and the quotient spac ..."
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Cited by 98 (0 self)
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We prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group. We give the formula for the corresponding multiplicities in the case of the quotient quantum spaces. We describe the subgroups and the quotient spaces of quantum SU(2) and SO(3) groups.
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 78 (18 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 NICHOLS ALGEBRA OF A SEMISIMPLE YETTERDRINFELD MODULE
, 2008
"... We study the Nichols algebra of a semisimple YetterDrinfeld module and introduce new invariants including the notions of real roots and the Weyl groupoid. The crucial ingredient is a “reflection” defined on arbitrary such Nichols algebras. Our construction generalizes the restriction of Lusztig’s ..."
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Cited by 65 (22 self)
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We study the Nichols algebra of a semisimple YetterDrinfeld module and introduce new invariants including the notions of real roots and the Weyl groupoid. The crucial ingredient is a “reflection” defined on arbitrary such Nichols algebras. Our construction generalizes the restriction of Lusztig’s automorphisms of quantized KacMoody algebras to the nilpotent part. As a direct application we complete the classifications of finitedimensional pointed Hopf algebras over S3, and of finitedimensional Nichols algebras over S4. This theory has led to surprising new results in the classification of finitedimensional pointed Hopf algebras with a nonabelian group of grouplike elements.
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 64 (12 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.
The standard model on noncommutative spacetime
 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 63 (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
Classification of arithmetic root systems
, 2006
"... Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. They can also be considered as generalizations of ordinary root systems with rich structure and many new examples. On the other hand, Nichols algebras are fundamental objects in the constr ..."
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Cited by 63 (8 self)
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Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. They can also be considered as generalizations of ordinary root systems with rich structure and many new examples. On the other hand, Nichols algebras are fundamental objects in the construction of quantized enveloping algebras, in the noncommutative differential geometry of quantum groups, and in the classification of pointed Hopf algebras by the lifting method of Andruskiewitsch and Schneider. In the present paper arithmetic root systems are classified in full generality. As a byproduct many new finite dimensional pointed Hopf algebras are obtained.
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 61 (7 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 . . . . . ....