Deformation Theory Of Algebras And Their Diagrams

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Deformation Theory of Algebras and Their Diagrams

Author : Martin Markl
Publisher : American Mathematical Soc.
ISBN 13 : 0821889796
Total Pages : 129 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Deformation Theory of Algebras and Their Diagrams by : Martin Markl

Download or read book Deformation Theory of Algebras and Their Diagrams written by Martin Markl and published by American Mathematical Soc.. This book was released on 2012 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book brings together both the classical and current aspects of deformation theory. The presentation is mostly self-contained, assuming only basic knowledge of commutative algebra, homological algebra and category theory. In the interest of readability, some technically complicated proofs have been omitted when a suitable reference was available. The relation between the uniform continuity of algebraic maps and topologized tensor products is explained in detail, however, as this subject does not seem to be commonly known and the literature is scarce. The exposition begins by recalling Gerstenhaber's classical theory for associative algebras. The focus then shifts to a homotopy-invariant setup of Maurer-Cartan moduli spaces. As an application, Kontsevich's approach to deformation quantization of Poisson manifolds is reviewed. Then, after a brief introduction to operads, a strongly homotopy Lie algebra governing deformations of (diagrams of) algebras of a given type is described, followed by examples and generalizations.

Deformation Theory of Algebras and Structures and Applications

Author : Michiel Hazewinkel
Publisher : Springer Science & Business Media
ISBN 13 : 9400930577
Total Pages : 1024 pages
Book Rating : 4.4/5 (9 download)

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Book Synopsis Deformation Theory of Algebras and Structures and Applications by : Michiel Hazewinkel

Download or read book Deformation Theory of Algebras and Structures and Applications written by Michiel Hazewinkel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 1024 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is a result of a meeting which took place in June 1986 at 'll Ciocco" in Italy entitled 'Deformation theory of algebras and structures and applications'. It appears somewhat later than is perhaps desirable for a volume resulting from a summer school. In return it contains a good many results which were not yet available at the time of the meeting. In particular it is now abundantly clear that the Deformation theory of algebras is indeed central to the whole philosophy of deformations/perturbations/stability. This is one of the main results of the 254 page paper below (practically a book in itself) by Gerstenhaber and Shack entitled "Algebraic cohomology and defor mation theory". Two of the main philosphical-methodological pillars on which deformation theory rests are the fol lowing • (Pure) To study a highly complicated object, it is fruitful to study the ways in which it can arise as a limit of a family of simpler objects: "the unraveling of complicated structures" . • (Applied) If a mathematical model is to be applied to the real world there will usually be such things as coefficients which are imperfectly known. Thus it is important to know how the behaviour of a model changes as it is perturbed (deformed).

Deformation Theory and Quantum Groups with Applications to Mathematical Physics

Author : Murray Gerstenhaber
Publisher : American Mathematical Soc.
ISBN 13 : 0821851411
Total Pages : 388 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Deformation Theory and Quantum Groups with Applications to Mathematical Physics by : Murray Gerstenhaber

Download or read book Deformation Theory and Quantum Groups with Applications to Mathematical Physics written by Murray Gerstenhaber and published by American Mathematical Soc.. This book was released on 1992 with total page 388 pages. Available in PDF, EPUB and Kindle. Book excerpt: Quantum groups are not groups at all, but special kinds of Hopf algebras of which the most important are closely related to Lie groups and play a central role in the statistical and wave mechanics of Baxter and Yang. Those occurring physically can be studied as essentially algebraic and closely related to the deformation theory of algebras (commutative, Lie, Hopf, and so on). One of the oldest forms of algebraic quantization amounts to the study of deformations of a commutative algebra A (of classical observables) to a noncommutative algebra A*h (of operators) with the infinitesimal deformation given by a Poisson bracket on the original algebra A. This volume grew out of an AMS--IMS--SIAM Joint Summer Research Conference, held in June 1990 at the University of Massachusetts at Amherst. The conference brought together leading researchers in the several areas mentioned and in areas such as ``q special functions'', which have their origins in the last century but whose relevance to modern physics has only recently been understood. Among the advances taking place during the conference was Majid's reconstruction theorem for Drinfel$'$d's quasi-Hopf algebras. Readers will appreciate this snapshot of some of the latest developments in the mathematics of quantum groups and deformation theory.

Noncommutative Deformation Theory

Author : Eivind Eriksen
Publisher : CRC Press
ISBN 13 : 1498796028
Total Pages : 242 pages
Book Rating : 4.4/5 (987 download)

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Book Synopsis Noncommutative Deformation Theory by : Eivind Eriksen

Download or read book Noncommutative Deformation Theory written by Eivind Eriksen and published by CRC Press. This book was released on 2017-09-19 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: Noncommutative Deformation Theory is aimed at mathematicians and physicists studying the local structure of moduli spaces in algebraic geometry. This book introduces a general theory of noncommutative deformations, with applications to the study of moduli spaces of representations of associative algebras and to quantum theory in physics. An essential part of this theory is the study of obstructions of liftings of representations using generalised (matric) Massey products. Suitable for researchers in algebraic geometry and mathematical physics interested in the workings of noncommutative algebraic geometry, it may also be useful for advanced graduate students in these fields.

Deformations of Algebraic Schemes

Author : Edoardo Sernesi
Publisher : Springer Science & Business Media
ISBN 13 : 3540306153
Total Pages : 343 pages
Book Rating : 4.5/5 (43 download)

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Book Synopsis Deformations of Algebraic Schemes by : Edoardo Sernesi

Download or read book Deformations of Algebraic Schemes written by Edoardo Sernesi and published by Springer Science & Business Media. This book was released on 2007-04-20 with total page 343 pages. Available in PDF, EPUB and Kindle. Book excerpt: This account of deformation theory in classical algebraic geometry over an algebraically closed field presents for the first time some results previously scattered in the literature, with proofs that are relatively little known, yet relevant to algebraic geometers. Many examples are provided. Most of the algebraic results needed are proved. The style of exposition is kept at a level amenable to graduate students with an average background in algebraic geometry.

Deformation Theory

Author : Robin Hartshorne
Publisher : Springer Science & Business Media
ISBN 13 : 1441915966
Total Pages : 241 pages
Book Rating : 4.4/5 (419 download)

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Book Synopsis Deformation Theory by : Robin Hartshorne

Download or read book Deformation Theory written by Robin Hartshorne and published by Springer Science & Business Media. This book was released on 2009-11-12 with total page 241 pages. Available in PDF, EPUB and Kindle. Book excerpt: The basic problem of deformation theory in algebraic geometry involves watching a small deformation of one member of a family of objects, such as varieties, or subschemes in a fixed space, or vector bundles on a fixed scheme. In this new book, Robin Hartshorne studies first what happens over small infinitesimal deformations, and then gradually builds up to more global situations, using methods pioneered by Kodaira and Spencer in the complex analytic case, and adapted and expanded in algebraic geometry by Grothendieck. The author includes numerous exercises, as well as important examples illustrating various aspects of the theory. This text is based on a graduate course taught by the author at the University of California, Berkeley.

Lie Methods in Deformation Theory

Author : Marco Manetti
Publisher : Springer Nature
ISBN 13 : 9811911851
Total Pages : 576 pages
Book Rating : 4.8/5 (119 download)

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Book Synopsis Lie Methods in Deformation Theory by : Marco Manetti

Download or read book Lie Methods in Deformation Theory written by Marco Manetti and published by Springer Nature. This book was released on 2022-08-01 with total page 576 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book furnishes a comprehensive treatment of differential graded Lie algebras, L-infinity algebras, and their use in deformation theory. We believe it is the first textbook devoted to this subject, although the first chapters are also covered in other sources with a different perspective. Deformation theory is an important subject in algebra and algebraic geometry, with an origin that dates back to Kodaira, Spencer, Kuranishi, Gerstenhaber, and Grothendieck. In the last 30 years, a new approach, based on ideas from rational homotopy theory, has made it possible not only to solve long-standing open problems, but also to clarify the general theory and to relate apparently different features. This approach works over a field of characteristic 0, and the central role is played by the notions of differential graded Lie algebra, L-infinity algebra, and Maurer–Cartan equations. The book is written keeping in mind graduate students with a basic knowledge of homological algebra and complex algebraic geometry as utilized, for instance, in the book by K. Kodaira, Complex Manifolds and Deformation of Complex Structures. Although the main applications in this book concern deformation theory of complex manifolds, vector bundles, and holomorphic maps, the underlying algebraic theory also applies to a wider class of deformation problems, and it is a prerequisite for anyone interested in derived deformation theory. Researchers in algebra, algebraic geometry, algebraic topology, deformation theory, and noncommutative geometry are the major targets for the book.

Hochschild Cohomology for Algebras

Author : Sarah J. Witherspoon
Publisher : American Mathematical Society
ISBN 13 : 1470462869
Total Pages : 265 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Hochschild Cohomology for Algebras by : Sarah J. Witherspoon

Download or read book Hochschild Cohomology for Algebras written by Sarah J. Witherspoon and published by American Mathematical Society. This book was released on 2020-06-30 with total page 265 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a thorough and self-contained introduction to the theory of Hochschild cohomology for algebras and includes many examples and exercises. The book then explores Hochschild cohomology as a Gerstenhaber algebra in detail, the notions of smoothness and duality, algebraic deformation theory, infinity structures, support varieties, and connections to Hopf algebra cohomology. Useful homological algebra background is provided in an appendix. The book is designed both as an introduction for advanced graduate students and as a resource for mathematicians who use Hochschild cohomology in their work.

Deformation Spaces

Author : Hossein Abbaspour
Publisher : Springer Science & Business Media
ISBN 13 : 3834896802
Total Pages : 174 pages
Book Rating : 4.8/5 (348 download)

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Book Synopsis Deformation Spaces by : Hossein Abbaspour

Download or read book Deformation Spaces written by Hossein Abbaspour and published by Springer Science & Business Media. This book was released on 2010-04-21 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first instances of deformation theory were given by Kodaira and Spencer for complex structures and by Gerstenhaber for associative algebras. Since then, deformation theory has been applied as a useful tool in the study of many other mathematical structures, and even today it plays an important role in many developments of modern mathematics. This volume collects a few self-contained and peer-reviewed papers by experts which present up-to-date research topics in algebraic and motivic topology, quantum field theory, algebraic geometry, noncommutative geometry and the deformation theory of Poisson algebras. They originate from activities at the Max-Planck-Institute for Mathematics and the Hausdorff Center for Mathematics in Bonn.

Introduction to Singularities and Deformations

Author : Gert-Martin Greuel
Publisher : Springer Science & Business Media
ISBN 13 : 3540284192
Total Pages : 482 pages
Book Rating : 4.5/5 (42 download)

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Book Synopsis Introduction to Singularities and Deformations by : Gert-Martin Greuel

Download or read book Introduction to Singularities and Deformations written by Gert-Martin Greuel and published by Springer Science & Business Media. This book was released on 2007-02-23 with total page 482 pages. Available in PDF, EPUB and Kindle. Book excerpt: Singularity theory is a young, rapidly-growing topic with connections to algebraic geometry, complex analysis, commutative algebra, representations theory, Lie groups theory and topology, and many applications in the natural and technical sciences. This book presents the basic singularity theory of analytic spaces, including local deformation theory and the theory of plane curve singularities. It includes complete proofs.

Dialgebras and Related Operads

Author : J.-L. Loday
Publisher : Springer
ISBN 13 : 3540453288
Total Pages : 138 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Dialgebras and Related Operads by : J.-L. Loday

Download or read book Dialgebras and Related Operads written by J.-L. Loday and published by Springer. This book was released on 2003-07-01 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main object of study of these four papers is the notion of associative dialgebras which are algebras equipped with two associative operations satisfying some more relations of the associative type. This notion is studied from a) the homological point of view: construction of the (co)homology theory with trivial coefficients and general coefficients, b) the operadic point of view: determination of the dual operad, that is the dendriform dialgebras which are strongly related with the planar binary trees, c) the algebraic point of view: Hopf structure and Milnor-Moore type theorem.

Maurer–Cartan Methods in Deformation Theory

Author : Vladimir Dotsenko
Publisher : Cambridge University Press
ISBN 13 : 1108965644
Total Pages : 187 pages
Book Rating : 4.1/5 (89 download)

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Book Synopsis Maurer–Cartan Methods in Deformation Theory by : Vladimir Dotsenko

Download or read book Maurer–Cartan Methods in Deformation Theory written by Vladimir Dotsenko and published by Cambridge University Press. This book was released on 2023-08-31 with total page 187 pages. Available in PDF, EPUB and Kindle. Book excerpt: Covering an exceptional range of topics, this text provides a unique overview of the Maurer-Cartan methods in algebra, geometry, topology, and mathematical physics. It offers a new conceptual treatment of the twisting procedure, guiding the reader through various versions with the help of plentiful motivating examples for graduate students as well as researchers. Topics covered include a novel approach to the twisting procedure for operads leading to Kontsevich graph homology and a description of the twisting procedure for (homotopy) associative algebras or (homotopy) Lie algebras using the biggest deformation gauge group ever considered. The book concludes with concise surveys of recent applications in areas including higher category theory and deformation theory.

A Study in Derived Algebraic Geometry

Author : Dennis Gaitsgory
Publisher : American Mathematical Society
ISBN 13 : 1470452855
Total Pages : 436 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis A Study in Derived Algebraic Geometry by : Dennis Gaitsgory

Download or read book A Study in Derived Algebraic Geometry written by Dennis Gaitsgory and published by American Mathematical Society. This book was released on 2020-10-07 with total page 436 pages. Available in PDF, EPUB and Kindle. Book excerpt: Derived algebraic geometry is a far-reaching generalization of algebraic geometry. It has found numerous applications in other parts of mathematics, most prominently in representation theory. This volume develops deformation theory, Lie theory and the theory of algebroids in the context of derived algebraic geometry. To that end, it introduces the notion of inf-scheme, which is an infinitesimal deformation of a scheme and studies ind-coherent sheaves on such. As an application of the general theory, the six-functor formalism for D-modules in derived geometry is obtained. This volume consists of two parts. The first part introduces the notion of ind-scheme and extends the theory of ind-coherent sheaves to inf-schemes, obtaining the theory of D-modules as an application. The second part establishes the equivalence between formal Lie group(oids) and Lie algebr(oids) in the category of ind-coherent sheaves. This equivalence gives a vast generalization of the equivalence between Lie algebras and formal moduli problems. This theory is applied to study natural filtrations in formal derived geometry generalizing the Hodge filtration.

Deformation Theory and Symplectic Geometry

Author : Daniel Sternheimer
Publisher : Springer
ISBN 13 :
Total Pages : 392 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Deformation Theory and Symplectic Geometry by : Daniel Sternheimer

Download or read book Deformation Theory and Symplectic Geometry written by Daniel Sternheimer and published by Springer. This book was released on 1997-07-31 with total page 392 pages. Available in PDF, EPUB and Kindle. Book excerpt: Proceedings of the Ascona Meeting, June 1996

Elliptic Curves, Hilbert Modular Forms and Galois Deformations

Author : Laurent Berger
Publisher : Springer Science & Business Media
ISBN 13 : 3034806183
Total Pages : 257 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Elliptic Curves, Hilbert Modular Forms and Galois Deformations by : Laurent Berger

Download or read book Elliptic Curves, Hilbert Modular Forms and Galois Deformations written by Laurent Berger and published by Springer Science & Business Media. This book was released on 2013-06-13 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: The notes in this volume correspond to advanced courses held at the Centre de Recerca Matemàtica as part of the research program in Arithmetic Geometry in the 2009-2010 academic year. The notes by Laurent Berger provide an introduction to p-adic Galois representations and Fontaine rings, which are especially useful for describing many local deformation rings at p that arise naturally in Galois deformation theory. The notes by Gebhard Böckle offer a comprehensive course on Galois deformation theory, starting from the foundational results of Mazur and discussing in detail the theory of pseudo-representations and their deformations, local deformations at places l ≠ p and local deformations at p which are flat. In the last section,the results of Böckle and Kisin on presentations of global deformation rings over local ones are discussed. The notes by Mladen Dimitrov present the basics of the arithmetic theory of Hilbert modular forms and varieties, with an emphasis on the study of the images of the attached Galois representations, on modularity lifting theorems over totally real number fields, and on the cohomology of Hilbert modular varieties with integral coefficients. The notes by Lassina Dembélé and John Voight describe methods for performing explicit computations in spaces of Hilbert modular forms. These methods depend on the Jacquet-Langlands correspondence and on computations in spaces of quaternionic modular forms, both for the case of definite and indefinite quaternion algebras. Several examples are given, and applications to modularity of Galois representations are discussed. The notes by Tim Dokchitser describe the proof, obtained by the author in a joint project with Vladimir Dokchitser, of the parity conjecture for elliptic curves over number fields under the assumption of finiteness of the Tate-Shafarevich group. The statement of the Birch and Swinnerton-Dyer conjecture is included, as well as a detailed study of local and global root numbers of elliptic curves and their classification.

Lie Models in Topology

Author : Urtzi Buijs
Publisher : Springer Nature
ISBN 13 : 3030544303
Total Pages : 283 pages
Book Rating : 4.0/5 (35 download)

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Book Synopsis Lie Models in Topology by : Urtzi Buijs

Download or read book Lie Models in Topology written by Urtzi Buijs and published by Springer Nature. This book was released on 2020-12-15 with total page 283 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the birth of rational homotopy theory, the possibility of extending the Quillen approach – in terms of Lie algebras – to a more general category of spaces, including the non-simply connected case, has been a challenge for the algebraic topologist community. Despite the clear Eckmann-Hilton duality between Quillen and Sullivan treatments, the simplicity in the realization of algebraic structures in the latter contrasts with the complexity required by the Lie algebra version. In this book, the authors develop new tools to address these problems. Working with complete Lie algebras, they construct, in a combinatorial way, a cosimplicial Lie model for the standard simplices. This is a key object, which allows the definition of a new model and realization functors that turn out to be homotopically equivalent to the classical Quillen functors in the simply connected case. With this, the authors open new avenues for solving old problems and posing new questions. This monograph is the winner of the 2020 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics.

From the Basic Homotopy Lemma to the Classification of C*-algebras

Author : Huaxin Lin
Publisher : American Mathematical Soc.
ISBN 13 : 1470434903
Total Pages : 240 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis From the Basic Homotopy Lemma to the Classification of C*-algebras by : Huaxin Lin

Download or read book From the Basic Homotopy Lemma to the Classification of C*-algebras written by Huaxin Lin and published by American Mathematical Soc.. This book was released on 2017-08-11 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book examines some recent developments in the theory of -algebras, which are algebras of operators on Hilbert spaces. An elementary introduction to the technical part of the theory is given via a basic homotopy lemma concerning a pair of almost commuting unitaries. The book presents an outline of the background as well as some recent results of the classification of simple amenable -algebras, otherwise known as the Elliott program. This includes some stable uniqueness theorems and a revisiting of Bott maps via stable homotopy. Furthermore, -theory related rotation maps are introduced. The book is based on lecture notes from the CBMS lecture sequence at the University of Wyoming in the summer of 2015.