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44
Berikashvili’s functor D and the deformation equation
 PROCEEDINGS OF THE A. RAZMADZE MATHEMATICAL INSTITUTE 119
, 1999
"... Berikashvili’s functor D defined in terms of twisting cochains is related to deformation theory, gauge theory, Chen’s formal power series connections, and the master equation in physics. The idea is advertised that some unification and understanding of the links between these topics is provided by t ..."
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Cited by 8 (7 self)
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Berikashvili’s functor D defined in terms of twisting cochains is related to deformation theory, gauge theory, Chen’s formal power series connections, and the master equation in physics. The idea is advertised that some unification and understanding of the links between these topics is provided by the notion of twisting cochain and the idea of classifying twisting cochains.
Higher homotopies and MaurerCartan algebras: quasiLieRinehart, Gerstenhaber, and BatalinVilkovisky algebras
 PROGRESS IN MATHEMATICS (2004)
, 2004
"... Higher homotopy generalizations of LieRinehart algebras, Gerstenhaber, and BatalinVilkovisky algebras are explored. These are defined in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two ..."
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Cited by 6 (5 self)
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Higher homotopy generalizations of LieRinehart algebras, Gerstenhaber, and BatalinVilkovisky algebras are explored. These are defined in terms of various antisymmetric bilinear operations satisfying weakened versions of the Jacobi identity, as well as in terms of operations involving more than two variables of the Lie triple systems kind. A basic tool is the MaurerCartan algebra—the algebra of alternating forms on a vector space so that Lie brackets correspond to square zero derivations of this algebra—and multialgebra generalizations thereof. The higher homotopies are phrased in terms of these multialgebras. Applications to foliations are discussed: objects which serve as replacements for the Lie algebra of vector fields on the “space of leaves ” and for the algebra of multivector fields are developed, and the spectral sequence of a foliation is shown to arise as a special case of a more general spectral sequence including as well the Hodgede Rham spectral sequence.
Origins and breadth of the theory of higher homotopies
, 2007
"... Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least ..."
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Cited by 6 (2 self)
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Higher homotopies are nowadays playing a prominent role in mathematics as well as in certain branches of theoretical physics. The purpose of the talk is to recall some of the connections between the past and the present developments. Higher homotopies were isolated within algebraic topology at least as far back as the 1940’s. Prompted by the failure of the AlexanderWhitney multiplication of cocycles to be commutative, Steenrod developed certain operations which measure this failure in a coherent manner. Dold and Lashof extended Milnor’s classifying space construction to associative Hspaces, and a careful examination of this extension led Stasheff to the discovery of Anspaces and A∞spaces as notions which control the failure of associativity in a coherent way so that the classifying space construction can still be pushed through. Algebraic versions of higher homotopies have, as we all know, led Kontsevich eventually to the proof of the formality conjecture. Homological perturbation theory (HPT), in a simple form first isolated by Eilenberg and Mac Lane in the early 1950’s, has nowadays become a standard tool to handle algebraic incarnations of higher homotopies. A basic
MINIMAL FREE MULTI MODELS FOR CHAIN ALGEBRAS
, 2004
"... To the memory of G. Chogoshvili Abstract. Let R be a local ring and A a connected differential graded algebra over R which is free as a graded Rmodule. Using homological perturbation theory techniques, we construct a minimal free multi model for A having properties similar to that of an ordinary mi ..."
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Cited by 5 (2 self)
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To the memory of G. Chogoshvili Abstract. Let R be a local ring and A a connected differential graded algebra over R which is free as a graded Rmodule. Using homological perturbation theory techniques, we construct a minimal free multi model for A having properties similar to that of an ordinary minimal model over a field; in particular the model is unique up to isomorphism of multialgebras. The attribute ‘multi ’ refers to the category of multicomplexes. 2000 Mathematics Subject Classification. 18G10, 18G35, 18G55, 55P35, 55P62, 55U15, 57T30. Key words and phrases. Models for differential graded algebras, minimal models for differential graded algebras over local rings, multicomplex, multialgebra, homological perturbations. 2 JOHANNES HUEBSCHMANN
On PMinimal Homological Models of Twisted Tensor Products of Elementary Complexes Localised over a Prime
"... In this paper, working over Z (p) and using algebra perturbation results from [18], pminimal homological models of twisted tensor products (TTPs) of Cartan 's elementary complexes are obtained. Moreover, making use of the notion of indecomposability of a TTP, we deduce that a homological model ..."
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Cited by 4 (3 self)
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In this paper, working over Z (p) and using algebra perturbation results from [18], pminimal homological models of twisted tensor products (TTPs) of Cartan 's elementary complexes are obtained. Moreover, making use of the notion of indecomposability of a TTP, we deduce that a homological model of a indecomposable pminimal TTP of length # (# 2) of exterior and divided power algebras is a tensor product of kindecomposable (k #) pminimal TTPs of exterior and divided power algebras.
Ideal perturbation lemma
 Comm. Algebra
"... Abstract. We explain the essence of perturbation problems. The key to understanding is the structure of chain homotopy equivalence – the standard one must be replaced by a finer notion which we call a strong chain homotopy equivalence. We formulate an Ideal Perturbation Lemma and show how both new a ..."
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Cited by 4 (1 self)
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Abstract. We explain the essence of perturbation problems. The key to understanding is the structure of chain homotopy equivalence – the standard one must be replaced by a finer notion which we call a strong chain homotopy equivalence. We formulate an Ideal Perturbation Lemma and show how both new and classical (including the Basic Perturbation Lemma) results follow from this ideal statement. 1. Introduction and
Homological Perturbation Theory And Computability Of Hochschild And Cyclic Homologies Of Cdgas
, 1997
"... . We establish an algorithm computing the homology of commutative dierential graded algebras (briey, CDGAs). The main tool in this approach is given by the Homological Perturbation Theory particularized for the algebra category (see [21]). Taking into account these results, we develop and rene some ..."
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Cited by 3 (1 self)
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. We establish an algorithm computing the homology of commutative dierential graded algebras (briey, CDGAs). The main tool in this approach is given by the Homological Perturbation Theory particularized for the algebra category (see [21]). Taking into account these results, we develop and rene some methods already known about the computation of the Hochschild and cyclic homologies of CDGAs. In the last section of the paper, we analyze the plocal homology of the iterated bar construction of a CDGA (p prime). 1. Introduction. The description of eÆcient algorithms of homological computation might be considered as a very important question in Homological Algebra, in order to use those processes mainly in the resolution of problems on algebraic topology; but this subject also inuence directly on the development of non so closedareas as Cohomological Physics (in this sense, we nd useful references in [12], [24], [25]) and Secondary Calculus ([14], [27], [28]). Working in the context ...
DIFFERENTIAL EQUATIONS, SPENCER COHOMOLOGY, AND COMPUTING RESOLUTIONS
"... Abstract. We propose a new point of view of the Spencer cohomology appearing in the formal theory of differential equations based on a dual approach via comodules. It allows us to relate the Spencer cohomology with standard constructions in homological algebra and, in particular, to express it as a ..."
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Abstract. We propose a new point of view of the Spencer cohomology appearing in the formal theory of differential equations based on a dual approach via comodules. It allows us to relate the Spencer cohomology with standard constructions in homological algebra and, in particular, to express it as a Cotor. We discuss concrete methods for its construction based on homological perturbation theory. Appears in Georgian Math. J., vol. 9, No. 4, 2002, 723772. 1.
On the cohomology of the holomorph of a finite cyclic group
 J. of Algebra
"... Abstract. We determine the mod 2 cohomology algebra of the holomorph of any finite cyclic group whose order is a power of 2. 2000 Mathematics Subject Classification. 20J05 20J06. Key words and phrases. Holomorph of a group, mod 2 cohomology of the holomorph of a cyclic group, homological perturbatio ..."
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Cited by 3 (3 self)
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Abstract. We determine the mod 2 cohomology algebra of the holomorph of any finite cyclic group whose order is a power of 2. 2000 Mathematics Subject Classification. 20J05 20J06. Key words and phrases. Holomorph of a group, mod 2 cohomology of the holomorph of a cyclic group, homological perturbations and group cohomology. 2 JOHANNES HUEBSCHMANN 1. Outline The holomorph of a group is the semidirect product of the group with its automorphism group, with respect to the obvious action. The automorphism group of a nontrivial finite cyclic group of order r is well known to be cyclic if and only if the number r is of the kind r = 4, r = pρ, r = 2pρ where p is an odd prime; in these cases, the holomorph is thus a split metacyclic group. The mod p cohomology algebra of an arbitrary metacyclic group has been determined in [2]. In this note we will determine the mod 2 cohomology of the holomorph of a cyclic group whose order is a power of 2. Since for odd p the pprimary part of the automorphism group of any cyclic group is cyclic, we thus get a complete