Homological Perturbation Theory and Associativitypedro
ثبت نشده
چکیده
In this paper, we prove various results concerning DGA-algebras in the context of the Homological Perturbation Theory. We distinguish two class of contractions for algebras: full algebra contractions and semi-full algebra contractions. A full algebra contraction is, in particular , a semi-full algebra contraction. Taking a full algebra contraction and an \algebra perturbation" as data of the Basic Perturbation Lemma, the Algebra Perturbation Lemma (or simply, F-APL) of 20] and 29] appears in a natural way. We establish here a perturbation machinery, the Semi-Full Algebra Perturbation Lemma (or, simply, SF-APL) that is a generalization of the previous one in the sense that the application range of SF-APL is wider than that of F-APL. We show four important applications in which this result is essential for the construction of algebra or coalgebra structures in various chain complexes.
منابع مشابه
Homological Perturbation Theory and Mirror Symmetry
We explain how deformation theories of geometric objects such as complex structures, Poisson structures and holomorphic bundle structures lead to differential Gerstenhaber or Poisson algebras. We use homological perturbation theory to obtain A∞ algebra structures and some canonically defined deformations of such structures on the cohomology. We formulate the A∞ algebraic mirror symmetry as the ...
متن کاملDifferential Equations, Spencer Cohomology, and Computing Resolutions
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 i...
متن کاملHomological Perturbation Theory for Algebras over Operads
We extend homological perturbation theory to encompass algebraic structures governed by operads and cooperads. Specifically, for an operad O, we define the notion of an ‘O-algebra contraction’ and we prove that the formulas of the Basic Perturbation Lemma preserve O-algebra contractions. Over a ground ring containing the rational numbers, we give explicit formulas for constructing an O-algebra ...
متن کاملA∞-structure for Lines in a Plane
As an explicit example of an A∞-structure associated to geometry, we construct an A∞-structure for a Fukaya category of finitely many lines (Lagrangians) in R , i.e., we define also non-transversal A∞-products. This construction is motivated by homological mirror symmetry of (two-)tori, where R is the covering space of a two-torus. The strategy is based on an algebraic reformulation of Morse ho...
متن کاملAn Entry From Encyclopaedia of Mathematics Supplement
Kluwer Academic Publishers 2000 Homological Perturbation Theory – A theory that concerns itself with of a collection of techniques for deriving chain complexes which are both smaller and chain homotopy equivalent to a given chain complex (cf. also Complex (in homological algebra)). It is motivated by the desire to find effective algorithms in homological algebra. The cornerstone of the theory i...
متن کامل