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36
Introduction to SH Lie algebras for physicists,” Int
 J. Theor. Phys
, 1993
"... Much of point particle physics can be described in terms of Lie algebras and ..."
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Cited by 127 (14 self)
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Much of point particle physics can be described in terms of Lie algebras and
Perturbation theory in differential homological algebra
 Illinois J. Math
, 1989
"... Perturbation theory is a particularly useful way to obtain relatively small differential complexes representing a given chain homotopy type. An important part of the theory is “the basic perturbation lemma ” [RB], [G1], [LS] which is stated in terms of modules M and N of the same homotopy type. It ..."
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Cited by 65 (10 self)
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Perturbation theory is a particularly useful way to obtain relatively small differential complexes representing a given chain homotopy type. An important part of the theory is “the basic perturbation lemma ” [RB], [G1], [LS] which is stated in terms of modules M and N of the same homotopy type. It has been known for some time that it would be useful to have a perturbation
A fixed point approach to homological perturbation theory
, 1991
"... We show that the problem handled by classical homological perturbation theory can be reformulated as a fixed point problem leading to new insights into the nature of its solutions. We show, under mild conditions that the solution is essentially unique. x1. ..."
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Cited by 23 (9 self)
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We show that the problem handled by classical homological perturbation theory can be reformulated as a fixed point problem leading to new insights into the nature of its solutions. We show, under mild conditions that the solution is essentially unique. x1.
Formal solution of the master equation via HPT and deformation theory
, 1999
"... Abstract. We construct a solution of the master equation by means of standard tools from homological perturbation theory under just the hypothesis that the ground field be of characteristic zero, thereby avoiding the formality assumption of the relevant dg Lie algebra. To this end, we endow the homo ..."
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Cited by 21 (11 self)
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Abstract. We construct a solution of the master equation by means of standard tools from homological perturbation theory under just the hypothesis that the ground field be of characteristic zero, thereby avoiding the formality assumption of the relevant dg Lie algebra. To this end, we endow the homology H(g) of any differential graded Lie algebra g over a field of characteristic zero with an shLie structure such that g and H(g) are shequivalent. We discuss our solution of the master equation in the context of deformation theory. Given the extra structure appropriate to the extended moduli space of complex structures on a CalabiYau manifold, the known solutions result as a special case.
Transferring Algebra Structures Up to Homology Equivalence
 Math. Scand
, 1998
"... Given a strong deformation retract M of an algebra A, there are several apparently distinct ways ([9],[19], [13], [24],[15], [18], [17]) of constructing a coderivation on the tensor coalgebra of M in such a way that the resulting complex is quasi isomorphic to the classical (differential tor) [7] ba ..."
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Cited by 15 (3 self)
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Given a strong deformation retract M of an algebra A, there are several apparently distinct ways ([9],[19], [13], [24],[15], [18], [17]) of constructing a coderivation on the tensor coalgebra of M in such a way that the resulting complex is quasi isomorphic to the classical (differential tor) [7] bar construction of A. We show that these methods are equivalent and are determined combinatorially by an inductive formula first given in a very special setting in [16]. Applications to de Rham theory and Massey products are given. 1 Preliminaries and Notation Throughout this paper, R will denote a commutative ring with unit. The term (co)module is used to mean a differential graded (co)module over R and maps between modules are graded maps. When we write\Omega we mean\Omega R . The usual (Koszul) sign conventions are assumed. The degree of a homogeneous element m of some module is denoted by jmj. Algebras are assumed to be connected and coalgebras simply connected. (Co)algebras are assumed to have (co)units.(Co)algebras are, unless otherwise stated, assumed to be (co)augmented. The differential in an (co)algebra is a graded (co)derivation. The Rmodule of maps from M to N (for Rmodules M and N) is denoted by hom(M;N) (if the context requires it, we will use a subscript to denote the ground ring). The differential in this module is given by D(f) = df \Gamma (\Gamma1) jf j fd. Note that D is a derivation with respect to the composition operation whenever it is defined. In particular, End(M) = hom(M;M) is an algebra. If A is an algebra and C is a coalgebra, the module hom(C; A) is an algebra with 1 respect to the operation defined by the following diagram C f [ g  A C\Omega C \Delta ? f\Omega g  A\Omega A 6 m (1) This product is called the cup or convolution...
Resolutions via homological perturbation
 J. Symbolic Comp
, 1991
"... The purpose of this paper is to review an algorithm for computing “small ” resolutions in homological algebra, to provide examples of its use as promised in [L1], [LS], and to illustrate the use of computer algebra in an area not usually associated with that subject. Comparison of the complexes prod ..."
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Cited by 14 (5 self)
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The purpose of this paper is to review an algorithm for computing “small ” resolutions in homological algebra, to provide examples of its use as promised in [L1], [LS], and to illustrate the use of computer algebra in an area not usually associated with that subject. Comparison of the complexes produced by the method discussed here with those produced by other methods shows
Homological perturbations, equivariant cohomology, and Koszul duality
"... Dedicated to the memory of V.K.A.M. Gugenheim Abstract. Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher homotopies ..."
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Cited by 10 (6 self)
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Dedicated to the memory of V.K.A.M. Gugenheim Abstract. Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher homotopies typically arise when a huge object, e. g. a chain complex defining various invariants of a certain geometric situation, is cut to a small model, and the higher homotopies can then be dealt with concisely in the language of shstructures (strong homotopy structures). This amounts to precise ways of handling the requisite additional structure encapsulating the various coherence conditions. Given e. g. two augmented differential graded algebras A1 and A2, an shmap from A1 to A2 is a twisting cochain from the reduced bar construction BA1 of A1 to A2 and, in this manner, the class of morphisms of augmented differential graded algebras is extended to that of shmorphisms. In the present paper, we explore small models for equivariant (co)homology via differential homological algebra techniques including homological perturbation theory which, in turn, is a standard tool to handle
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.
A cubical model of a fibration
 J. Pure Appl. Algebra
"... Abstract. In the paper the notion of a truncating twisting function from a simplicial set to a cubical set and the corresponding notion of twisted Cartesian product of these sets are introduced. The latter becomes a cubical set. This construction together with the theory of twisted tensor products f ..."
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Cited by 8 (7 self)
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Abstract. In the paper the notion of a truncating twisting function from a simplicial set to a cubical set and the corresponding notion of twisted Cartesian product of these sets are introduced. The latter becomes a cubical set. This construction together with the theory of twisted tensor products for homotopy Galgebras allows to obtain multiplicative models for fibrations. 1.
Next generation computer algebra systems AXIOM and the scratchpad concept: applications to research in algebra
 In Analysis, algebra, and computers in mathematical research (Lule˚a
, 1992
"... One way in which mathematicians deal with infinite amounts of data is symbolic representation. A simple example is the quadratic equation x = −b±√b2−4ac 2a, a formula which uses symbolic representation to describe the solutions to an infinite class of equations. Most computer algebra systems can dea ..."
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Cited by 7 (4 self)
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One way in which mathematicians deal with infinite amounts of data is symbolic representation. A simple example is the quadratic equation x = −b±√b2−4ac 2a, a formula which uses symbolic representation to describe the solutions to an infinite class of equations. Most computer algebra systems can deal with polynomials with symbolic coefficients, but what if symbolic exponents are called for (e.g., 1+t i)? What if symbolic limits on summations are also called for (e.g., 1+t+...+t i = � j tj)? The “Scratchpad Concept ” is a theoretical ideal which allows the implementation of objects at this level of abstraction and beyond in a mathematically consistent way. The AXIOM computer algebra system is an implementation of a major part of the Scratchpad Concept. AXIOM (formerly called Scratchpad) is a language with extensible parameterized types and generic operators which is based on the notions of domains and categories [Lambe1], [JenksSutor]. By examining some aspects of the AXIOM system, the Scratchpad Concept will be illustrated. It will be shown how some complex problems in homological algebra were solved through the use of this system. New paradigms are evolving in computer science. There is a thrust towards typed