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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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. 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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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.
Noncommutative εgraded connections
, 2012
"... We introduce the new notion of εgraded associative algebras which takes its root into the notion of commutation factors introduced in the context of Lie algebras [1]. We define and study the associated notion of εderivationbased differential calculus, which generalizes the derivationbased differ ..."
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Cited by 2 (1 self)
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We introduce the new notion of εgraded associative algebras which takes its root into the notion of commutation factors introduced in the context of Lie algebras [1]. We define and study the associated notion of εderivationbased differential calculus, which generalizes the derivationbased differential calculus on associative algebras. A corresponding notion of noncommutative connection is also defined. We illustrate these considerations with various examples of εgraded algebras, in particular some graded matrix algebras and the Moyal algebra. This last example permits also to interpret mathematically a noncommutative gauge field theory.
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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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
Schemes over F1 and Zeta Functions
, 2009
"... We determine the real counting function N(q) (q ∈ [1, ∞)) for the hypothetical “curve” C = Spec Z over F1, whose corresponding zeta function is the complete Riemann zeta function. Then, we develop a theory of functorial F1schemes which reconciles the previous attempts by C. Soulé and A. Deitmar. ..."
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We determine the real counting function N(q) (q ∈ [1, ∞)) for the hypothetical “curve” C = Spec Z over F1, whose corresponding zeta function is the complete Riemann zeta function. Then, we develop a theory of functorial F1schemes which reconciles the previous attempts by C. Soulé and A. Deitmar. Our construction fits with the geometry of monoids of K. Kato, is no longer limited to toric varieties and it covers the case of schemes associated to Chevalley groups. Finally we show, using the monoid of adèle classes over an arbitrary global field, how to apply our functorial theory of Moschemes to interpret conceptually the spectral realization of zeros of Lfunctions.
Gauge theories in noncommutative geometry
, 2011
"... In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and gauge transformations. Two different approaches to noncommut ..."
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In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and gauge transformations. Two different approaches to noncommutative geometry are covered: the one based on derivations and the one based on spectral triples. Examples of noncommutative gauge field theories are given to illustrate the constructions and to display some of the common features. 1 ha l0
Lectures On Graded Differential Algebras And Noncommutative Geometry
, 2000
"... 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. LPTORSAY 99/100 Keywords : graded differential alge ..."
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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. LPTORSAY 99/100 Keywords : graded differential algebras; categories of algebras; bimodules; noncommutative differential calculus; noncommutative symplectic geometry. MSC classification: 16D20, 18B99, 51P05, 53A99, 81Q99. To be published in the Proceedings of the Workshop on Noncommutative Differential Geometry and its Application to Physics, ShonanKokusaimura, Japan, May 31  June 4, 1999. 1 Unit'e Mixte de Recherche du Centre National de la Recherche Scientifique  UMR 8627 1 Contents 1 Introduction 3 2 Graded differential algebras 10 3 Examples related to Lie algebras 15 4 Examples related to associative algebras 20 5 Categories of algebras 26 6 First order differential calculi 31 7 Higher order differential calculi 36 8 Diagonal an...