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42
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 66 (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 24 (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
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.
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.
The twisted EilenbergZilber theorem
 In ‘Simposio di Topologia (Messina, 1964)’, Edizioni Oderisi, Gubbio
, 1965
"... The purpose of this paper is to give a simpler proof of a theorem of E.H. Brown [Bro59], that if F → E → B is a fibre space, then there is a differential on the graded group X = C(B) ⊗Λ C(F) such that X with this differential is chain equivalent to to C(E) (where C(E) denotes the normalised singular ..."
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Cited by 7 (0 self)
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The purpose of this paper is to give a simpler proof of a theorem of E.H. Brown [Bro59], that if F → E → B is a fibre space, then there is a differential on the graded group X = C(B) ⊗Λ C(F) such that X with this differential is chain equivalent to to C(E) (where C(E) denotes the normalised singular chains of E over a ring Λ). We work in the context of (semisimplicial) twisted cartesian products (thus we assume as do the proofs of the theorem given in [Gug60, Shi62, Szc61] the results of [BGM59] on the relation between fibre spaces and twisted cartesian products). In fact we prove a general result on filtered chain complexes; this result applies to give proofs not only of Brown’s theorem but also of a theorem of G. Hirsch, [Hir53]. Our proof is suggested by the formulae (1) of [Shi62, Ch. II, §1]. Let (X,d), (Y,d) be chain complexes over a ring Λ. Let (Y,d) ∇ f − → (X,d) − → (Y,d) be chain maps and let Φ: X → X be a chain homotopy such that Let X,Y have filtrations (1.1) f ∇ = 1; (1.2) ∇f = 1 + dΦ + Φd; (1.3) fΦ = 0; (1.4) Φ ∇ = 0; (1.5) Φ 2 = 0; (1.6) ΦdΦ = −Φ. and let ∇,f,Φ all preserve these filtrations. 0 = F −1 X ⊆ F 0 X ⊆ · · · ⊆ F p X ⊆ F p+1 X ⊆ · · · (1) 0 = F −1 Y ⊆ F 0 Y ⊆ · · · ⊆ F p Y ⊆ F p+1 Y ⊆ · · · (2) Example 1 Let B,F be (semisimplicial) complexes, let (X,d) = C(B × F), the normalised chains of B × F, let (Y,d) = C(B) ⊗Λ C(F), and let ∇,f,Φ be the natural maps of the EilenbergZilber theorem as constructed explicitly in [EML53]. The relations (1.1)(1.4) are proved in [EML53] while (1.5), (1.6) are easily proved (cf. [Shi62, p.114]). The filtrations on X,Y are induced by the filtration of B by its skeletons. The fact that ∇,f,Φ preserve filtrations is a consequence of naturality of these maps (cf. [Moo56, Ch. 5, p.13]). We now wish to compare C(B ×F) with C(B ×τ F) where B ×τ F coincides with B ×F as a complex except that ∂0 in B ×τ F is given by ∂0(b,x) = (∂0b,τ(b,x)), b ∈ Bp,x ∈ Fp. Then the filtered groups of C(B × F) and C(B ×τ F) coincide but the latter has a differential d τ. If τ satisfies the normalisation condition τ(s0b ′,x) = ∂0x,