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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 26 (18 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...
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.
LOCAL AND GLOBAL PROPERTIES OF THE WORLD
, 1997
"... physical theory. Abstract. The essence of the method of physics is inseparably connected with the problem of interplay between local and global properties of the universe. In the present paper we discuss this interplay as it is present in three major departments of contemporary physics: general rela ..."
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Cited by 2 (0 self)
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physical theory. Abstract. The essence of the method of physics is inseparably connected with the problem of interplay between local and global properties of the universe. In the present paper we discuss this interplay as it is present in three major departments of contemporary physics: general relativity, quantum mechanics and some attempts at quantizing gravity (especially geometrodynamics and its recent successors in the form of various pregeometry conceptions). It turns out that all big interpretative issues involved in this problem point towards the necessity of changing from the standard spacetime geometry to some radically new, most probably nonlocal, generalization. We argue that the recent noncommutative geometry offers attractive possibilities, and give us a 1 conceptual insight into its algebraic foundations. Noncommutative spaces are, in general, nonlocal, and their applications to physics, known at present, seem very promising. One would expect that beneath the Planck threshold there reigns a “noncommutative pregeometry”, and only when crossing this threshold the usual spacetime geometry emerges. 1
Noncommutative generalization of SU(n)principal ber bundles: a review
, 709
"... Abstract. This is an extended version of a communication made at the international conference Noncommutative Geometry and Physics held at Orsay in april 2007. In this proceeding, we make a review of some noncommutative constructions connected to the ordinary ber bundle theory. The noncommutative alg ..."
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Abstract. This is an extended version of a communication made at the international conference Noncommutative Geometry and Physics held at Orsay in april 2007. In this proceeding, we make a review of some noncommutative constructions connected to the ordinary ber bundle theory. The noncommutative algebra is the endomorphism algebra of a SU(n)vector bundle, and its di erential calculus is based on its Lie algebra of derivations. It is shown that this noncommutative geometry contains some of the most important constructions introduced and used in the theory of connections on vector bundles, in particular, what is needed to introduce gauge models in physics, and it also contains naturally the essential aspects of the Higgs elds and its associated mechanics of mass generation. It permits one also to extend some previous constructions, as for instance symmetric reduction of (here noncommutative) connections. From a mathematical point of view, these geometricoalgebraic considerations highlight some new point on view, in particular we introduce a new construction of the Chern characteristic classes. LPTOrsay/0757
d N = 0
, 2008
"... We study the generalized homology associated with a nilpotent endomorphism d satisfying d N = 0. For simplicial modules we construct such nilpotent endomorphisms and we prove a general result relating the corresponding generalized homologies to the ordinary homology. We also discuss the generalizati ..."
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We study the generalized homology associated with a nilpotent endomorphism d satisfying d N = 0. For simplicial modules we construct such nilpotent endomorphisms and we prove a general result relating the corresponding generalized homologies to the ordinary homology. We also discuss the generalization of the notion of graded differential algebra in this context.
d N = 0
, 2008
"... We study the generalized homology associated with a nilpotent endomorphism d satisfying d N = 0. For simplicial modules we construct such nilpotent endomorphisms and we prove a general result relating the corresponding generalized homologies to the ordinary homology. We also discuss the generalizati ..."
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We study the generalized homology associated with a nilpotent endomorphism d satisfying d N = 0. For simplicial modules we construct such nilpotent endomorphisms and we prove a general result relating the corresponding generalized homologies to the ordinary homology. We also discuss the generalization of the notion of graded differential algebra in this context. L.P.T.H.E.ORSAY 97/53 qalg/9710021
and
, 2008
"... 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 contain ..."
Abstract
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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 Z3graded 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 ZN versions leading to cyclotomic
LOCAL AND GLOBAL PROPERTIES OF THE WORLD
, 1997
"... It is often not sufficiently appreciated how kind nature has been in supplying us with ‘subsystems ’ of the universe which possess characteristic properties (literally in the sense ‘proper to the ..."
Abstract
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It is often not sufficiently appreciated how kind nature has been in supplying us with ‘subsystems ’ of the universe which possess characteristic properties (literally in the sense ‘proper to the