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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 48 (5 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.
Universal qDifferential Calculus and qAnalog of Homological Algebra
 Acta Math. Univ. Comenian
, 1996
"... . We recall the definition of qdifferential algebras and discuss some representative examples. In particular we construct the qanalog of the Hochschild coboundary. We then construct the universal qdifferential envelope of a unital associative algebra and study its properties. The paper also conta ..."
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Cited by 34 (19 self)
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. We recall the definition of qdifferential algebras and discuss some representative examples. In particular we construct the qanalog of the Hochschild coboundary. We then construct the universal qdifferential envelope of a unital associative algebra and study its properties. The paper also contains general results on d N = 0. 1. Introduction and Algebraic Preliminaries At the origin of this paper there is the longstanding physicallymotivated interest of one of the authors (R.K.) on Z 3 graded structures and differential calculi [RK] although here the point of view is somehow different. There is also the observation that the simplicial (co)homology admits Z N versions leading to cyclotomic homology [Sark] and that, more generally, this suggests that one can introduce "qanalog of homological algebra" for each primitive root q of the unity [Kapr]. Moreover the occurrence of various notions of "qanalog" in connection with quantum groups suggests to include in the formulation t...
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 19 (1 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
The FrölicherNijenhuis Bracket In Non Commutative Differential Geometry
, 1993
"... this paper. We carry over to a quite general noncommutative setting some of the basic tools of differential geometry. From the very beginning we use the setting of convenient vector spaces developed by Frolicher and Kriegl. The reasons for this are the following: If the noncommutative theory shoul ..."
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Cited by 13 (7 self)
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this paper. We carry over to a quite general noncommutative setting some of the basic tools of differential geometry. From the very beginning we use the setting of convenient vector spaces developed by Frolicher and Kriegl. The reasons for this are the following: If the noncommutative theory should contain some version of differential geometry, a manifold M should be represented by the algebra C
Generalized differential spaces with d N = 0 and the qdifferential calculus
 N = 0 and graded qdifferential algebras, Contemporary Mathematics 219, American Mathematical Society
, 1996
"... We present some results concerning the generalized homologies associated with nilpotent endomorphisms d such that d N = 0 for some integer N ≥ 2. We then introduce the notion of graded qdifferential algebra and describe some examples. In particular we construct the qanalog of the simplicial differ ..."
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Cited by 10 (2 self)
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We present some results concerning the generalized homologies associated with nilpotent endomorphisms d such that d N = 0 for some integer N ≥ 2. We then introduce the notion of graded qdifferential algebra and describe some examples. In particular we construct the qanalog of the simplicial differential on forms, the qanalog of the Hochschild differential and the qanalog of the universal differential envelope of an associative unital algebra. 1
More on the FrölicherNijenhuis Bracket In Non Commutative Differential Geometry
, 1996
"... . In commutative differential geometry the FrolicherNijenhuis bracket computes all kinds of curvatures and obstructions to integrability. In [1] the FrolicherNijenhuis bracket was developed for universal differential forms of noncommutative algebras, and several applications were given. In this pa ..."
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Cited by 9 (3 self)
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. In commutative differential geometry the FrolicherNijenhuis bracket computes all kinds of curvatures and obstructions to integrability. In [1] the FrolicherNijenhuis bracket was developed for universal differential forms of noncommutative algebras, and several applications were given. In this paper this bracket and the FrolicherNijenhuis calculus will be developed for several kinds of differential graded algebras based on derivations, which were introduced by [6]. Table of contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Convenient vector spaces . . . . . . . . . . . . . . . . . . . . . . 3 3. Preliminaries: graded differential algebras, derivations, and operations of Lie algebras . . . . . . . . . . . . . . . . . . . . 6 4. Derivations on universal differential forms . . . . . . . . . . . . . . . 8 5. The FrolicherNijenhuis calculus on Chevalley type cochains . . . . . . . 11 6. Description of all derivations in the Chevalley differential...
Connections on Central Bimodules
, 1995
"... We define and study the theory of derivationbased connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also ..."
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Cited by 8 (1 self)
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We define and study the theory of derivationbased connections on a recently introduced class of bimodules over an algebra which reduces to the category of modules whenever the algebra is commutative. This theory contains, in particular, a noncommutative generalization of linear connections. We also discuss the different noncommutative versions of differential forms based on derivations. Then we investigate reality conditions and a noncommutative generalization of pseudoriemannian structures.
The ChernWeil homomorphism in non commutative differential geometry, in preparation
"... In this short review article we sketch some developments which should ultimately lead to the analogy of the ChernWeil homomorphism for principle bundles in the realm of non commutative differential geometry. Principal bundles there should have Hopf algebras as structure ‘cogroups’. Since the usual ..."
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Cited by 4 (1 self)
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In this short review article we sketch some developments which should ultimately lead to the analogy of the ChernWeil homomorphism for principle bundles in the realm of non commutative differential geometry. Principal bundles there should have Hopf algebras as structure ‘cogroups’. Since the usual machinery of Lie algebras,
The FrölicherNijenhuis bracket for derivation based non commutative differential forms
 E.S.I. Preprint 133, 1994 and L.P.T.H.E.ORSAY 94/41
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Some aspects of noncommutative dierential geometry
 in Contemporary Mathematics 203
, 1997
"... We discuss in some generality aspects of noncommutative dierential geometry associated with reality conditions and with dierential calculi. We then describe the dierential calculus based on derivations as generalization of vector elds, and we show its relations with quantum mechanics. Finally we f ..."
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Cited by 1 (1 self)
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We discuss in some generality aspects of noncommutative dierential geometry associated with reality conditions and with dierential calculi. We then describe the dierential calculus based on derivations as generalization of vector elds, and we show its relations with quantum mechanics. Finally we formulate a general theory of connections in this framework. 1