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19
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
On qanalogues of bounded symmetric domains and Dolbeault complexes
 Math. Phys. Anal. Geom
, 1998
"... Consider an irreducible Hermitian symmetric space X of noncompact type. Let g and g0 denote the complexifications of the Lie algebras of the automorphism group of X and the stabilizer of a point x ∈ X respectively. Then the center of g0 is 1dimensional (Z(g0) = C · H, H ∈ g0), and g = g−1 ⊕ g0 ⊕ g ..."
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Cited by 27 (13 self)
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Consider an irreducible Hermitian symmetric space X of noncompact type. Let g and g0 denote the complexifications of the Lie algebras of the automorphism group of X and the stabilizer of a point x ∈ X respectively. Then the center of g0 is 1dimensional (Z(g0) = C · H, H ∈ g0), and g = g−1 ⊕ g0 ⊕ g1, where g±1 = {ξ ∈ g  [H,ξ] = ±2ξ} (see, e.g., [8]).
Lectures on graded differential algebras and noncommutative geometry
, 1999
"... These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments. ..."
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Cited by 22 (3 self)
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These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.
Homogeneous algebras, statistics and combinatorics
"... After some generalities on homogeneous algebras, we give a formula connecting the Poincaré series of a homogeneous algebra with the homology of the corresponding Koszul complex generalizing thereby a standard result for quadratic algebras. We then investigate two particular types of cubic algebras: ..."
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Cited by 17 (9 self)
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After some generalities on homogeneous algebras, we give a formula connecting the Poincaré series of a homogeneous algebra with the homology of the corresponding Koszul complex generalizing thereby a standard result for quadratic algebras. We then investigate two particular types of cubic algebras: The first one called the parafermionic (parabosonic) algebra is the algebra generated by the creation operators of the universal fermionic (bosonic) parastatics with D degrees of freedom while the second is the plactic algebra that is the algebra of the plactic monoid with entries in {1, 2,..., D}. In the case D = 2 we describe the relations with the cubic ArtinSchelter algebras. It is pointed out that the natural action of GL(2) on the parafermionic algebra for D = 2 extends as an action of the quantum group GLp,q(2) on the generic cubic ArtinSchelter regular algebra of type S1; p and q being related to the ArtinSchelter parameters. It is claimed that this has a counterpart for any integer D ≥ 2.
Some Aspects of Noncommutative Differential Geometry
"... We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finall ..."
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Cited by 13 (2 self)
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We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finally we formulate a general theory of connections in this framework. 1
Linear Connections on the Quantum Plane
, 1994
"... : A general definition has been proposed recently of a linear connection and a metric in noncommutative geometry. It is shown that to within normalization there is a unique linear connection on the quantum plane and there is no metric. LPTHE Orsay 94/94 October, 1994 * Laboratoire associ'e au CNRS. ..."
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Cited by 10 (6 self)
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: A general definition has been proposed recently of a linear connection and a metric in noncommutative geometry. It is shown that to within normalization there is a unique linear connection on the quantum plane and there is no metric. LPTHE Orsay 94/94 October, 1994 * Laboratoire associ'e au CNRS. 1 Linear connections There have been several models proposed of noncommutative geometries (Connes 1986, DuboisViolette 1988), some of which are based on quantum groups (Woronowicz 1987, Pusz & Woronowicz 1989, Wess & Zumino 1990, Maltsiniotis 1993). A definition of a linear connection which uses only the leftmodule structure of the differential forms has been proposed by Chamseddine et al (1993). An algebra of differential forms has however a natural structure of a bimodule. Recently linear connections have been considered in the particular case of differential calculi based on derivations (DuboisViolette & Michor 1994a,b). and more generally (Mourad 1994) which make essential use of ...
Sudbery A. Quantum Supergroups of GL(nm) type: Differential Forms, Koszul Complex and Berezinians
, 1993
"... Abstract. We introduce and study the Koszul complex for a Hecke Rmatrix. Its cohomologies, called the Berezinian, are used to define quantum superdeterminant for a Hecke Rmatrix. Their behaviour with respect to Hecke sum of Rmatrices is studied. Given a Hecke Rmatrix in ndimensional vector spac ..."
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Cited by 6 (0 self)
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Abstract. We introduce and study the Koszul complex for a Hecke Rmatrix. Its cohomologies, called the Berezinian, are used to define quantum superdeterminant for a Hecke Rmatrix. Their behaviour with respect to Hecke sum of Rmatrices is studied. Given a Hecke Rmatrix in ndimensional vector space, we construct a Hecke Rmatrix in 2ndimensional vector space commuting with a differential. The notion of a quantum differential supergroup is derived. Its algebra of functions is a differential coquasitriangular Hopf algebra, having the usual algebra of differential forms as a quotient. Examples of superdeterminants related to these algebras are calculated. Several remarks about Woronowicz’s theory are made. 0.1. Short description of the paper. 0.1.1. We start with constructing differential Hopf algebras (Section 1). Data for such construction are morphisms in the category of graded differential complexes. 0.1.2. Given a Hecke Rmatrix for a vector space V, we construct in this paper another Hecke Rmatrix R for the space W = V ⊕ V equipped with the differential d = () 0 1 0 0 and the grading σ: W → W, σ = ()
Quantum methods in Algebraic Topology
 Contemporary Mathematics Vol 279, American Math. Society
, 2000
"... In this paper, we present a new version of cochains in Algebraic Topology, starting with “quantum differential forms”. This version provides many examples of modules over the braid group, together with control of the non commutativity of cupproducts on the cochain level. If the quantum parameter q ..."
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Cited by 5 (5 self)
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In this paper, we present a new version of cochains in Algebraic Topology, starting with “quantum differential forms”. This version provides many examples of modules over the braid group, together with control of the non commutativity of cupproducts on the cochain level. If the quantum parameter q is equal to 1, we recover essentially the commutative differential graded algebra of de RhamSullivan forms on a simplicial set [1][12]. For topological applications, we may take either q = 1 if we are dealing with rational coefficients or q = 0 in the general case. In both cases, the quantum formulas are simpler (if q = 0 for instance, the quantum exponential e q (x) is just the function 1/1x). From this viewpoint, we extract a new structure of “neoalgebra ” 1. This structure is detailed in section III of this paper. To a simplicial set X we can associate in a functorial way a neoalgebra ∧ Ω*(X), which cohomology is canonically isomorphic to the usual one with coefficients in k (k might be an arbitrary commutative ring). As a differential graded algebra, Ω*(X) is related to the usual algebra of cochains C*(X) by a (zigzag) sequence of quasiisomorphisms. Using in an essential way some recent results of M.A. Mandell [8] [9], one may then show that ∧ Ω*(X) (up to quasiisomorphisms of neoalgebras) determines the padic homotopy type2 of X (if k = F). The proof relies on the basic fact that p ∧ Ω*(X) may be provided with an Ealgebra structure which is related to the classical one on C*(X) by a sequence of quasiisomorphisms. On a more practical level, we can show how to compute Steenrod operations in mod. p cohomology, as well as homotopy groups of X from the neoalgebraic data on ∧ Ω*(X). Finally in the fourth section of this paper, we see how all the theory can be dualized in the framework of “neocoalgebras”. This paper is mainly expository, although some proofs are sketched. Details will be published elsewhere, as well as applications to homotopy theory (closed model categories, homotopy groups of Moore spaces...). The following URL address:
Differential calculus in braided Abelian categories
, 1997
"... Braided noncommutative differential geometry is studied. We investigate the theory of (bicovariant) differential calculi in braided abelian categories. Previous results on crossed modules and Hopf bimodules in braided categories are used to construct higher order bicovariant differential calculi ov ..."
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Cited by 5 (0 self)
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Braided noncommutative differential geometry is studied. We investigate the theory of (bicovariant) differential calculi in braided abelian categories. Previous results on crossed modules and Hopf bimodules in braided categories are used to construct higher order bicovariant differential calculi over braided Hopf algebras out of first order ones. These graded objects are shown to be braided differential Hopf algebras with universal bialgebra properties. The article extends Woronowicz’s results on (bicovariant) differential calculi to the braided noncommutative case.
COCHAINES QUASICOMMUTATIVES EN TOPOLOGIE ALGEBRIQUE
"... Le but de cet article est de construire sur un ensemble simplicial X des “formes différentielles ” définies sur un anneau commutatif cohérent1 k. Elles permettent de définir une nouvelle structure d’algèbre différentielle graduée2 d*(X) dite quasicommutative, qui est quasiisomorphe à l’algèbre des ..."
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Cited by 3 (0 self)
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Le but de cet article est de construire sur un ensemble simplicial X des “formes différentielles ” définies sur un anneau commutatif cohérent1 k. Elles permettent de définir une nouvelle structure d’algèbre différentielle graduée2 d*(X) dite quasicommutative, qui est quasiisomorphe à l’algèbre des cochaînes classiques sur X (d’où la terminologie). On montre que cette structure détermine (sous certaines conditions de finitude) le type d’homotopie de X si k = Z. En particulier, les opérations de Steenrod sur la cohomologie de X, ainsi que les groupes d’homotopie de X peuvent s’en déduire par des méthodes standard d’algèbre homologique. Notre travail est donc analogue à celui de D. Quillen [24] et D. Sullivan [28] en homotopie rationnelle, où les algèbres différentielles graduées commutatives jouent un rôle essentiel. Il est aussi intimement lié à celui de M.A. Mandell sur le type d’homotopie à l’aide des E∞algèbres [19], que nous utilisons à la fin de l’article. Cette structure quasicommutative sur l’algèbre d*(X) enrichit considérablement la théorie classique des cochaînes (notée traditionnellement C*(X)), comme nous comptons le montrer de manière sommaire dans cette introduction. Elle consiste à se donner de manière naturelle, pour tout couple d’espaces X et Y, un sous kmodule différentiel gradué d*(X) ⊗ d*(Y) de