Deformation Theory Of Algebras And Structures And Applications

Author : Michiel Hazewinkel,Murray Gerstenhaber
Publisher : Springer Science & Business Media
Page : 1024 pages
File Size : 43,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789400930575

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Deformation Theory of Algebras and Structures and Applications by Michiel Hazewinkel,Murray Gerstenhaber Pdf

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).

Author : Murray Gerstenhaber
Publisher : American Mathematical Soc.
Page : 377 pages
File Size : 55,7 Mb
Release : 1992
Category : Mathematics
ISBN : 9780821851418

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Deformation Theory and Quantum Groups with Applications to Mathematical Physics by Murray Gerstenhaber Pdf

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 Drinfeld's quasi-Hopf algebras. Readers will appreciate this snapshot of some of the latest developments in the mathematics of quantum groups and deformation theory.

Author : Martin Markl
Publisher : American Mathematical Soc.
Page : 129 pages
File Size : 47,6 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821889794

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Deformation Theory of Algebras and Their Diagrams by Martin Markl Pdf

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.

Author : Eivind Eriksen,Olav Arnfinn Laudal,Arvid Siqveland
Publisher : CRC Press
Page : 242 pages
File Size : 54,8 Mb
Release : 2017-09-19
Category : Mathematics
ISBN : 9781498796026

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Noncommutative Deformation Theory by Eivind Eriksen,Olav Arnfinn Laudal,Arvid Siqveland Pdf

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.

Author : Hossein Abbaspour,Matilde Marcolli,Thomas Tradler
Publisher : Springer Science & Business Media
Page : 173 pages
File Size : 50,7 Mb
Release : 2010-04-21
Category : Mathematics
ISBN : 9783834896803

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Deformation Spaces by Hossein Abbaspour,Matilde Marcolli,Thomas Tradler Pdf

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.

Author : Marco Manetti
Publisher : Springer Nature
Page : 576 pages
File Size : 42,5 Mb
Release : 2022-08-01
Category : Mathematics
ISBN : 9789811911859

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Lie Methods in Deformation Theory by Marco Manetti Pdf

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.

Author : Edoardo Sernesi
Publisher : Springer Science & Business Media
Page : 343 pages
File Size : 41,8 Mb
Release : 2007-04-20
Category : Mathematics
ISBN : 9783540306153

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Deformations of Algebraic Schemes by Edoardo Sernesi Pdf

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.

Author : Gert-Martin Greuel,Christoph Lossen,Eugenii I. Shustin
Publisher : Springer Science & Business Media
Page : 472 pages
File Size : 45,6 Mb
Release : 2007-02-23
Category : Mathematics
ISBN : 9783540284192

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Introduction to Singularities and Deformations by Gert-Martin Greuel,Christoph Lossen,Eugenii I. Shustin Pdf

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.

Author : Michiel Hazewinkel
Publisher : Springer Science & Business Media
Page : 499 pages
File Size : 49,8 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9789400959941

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Encyclopaedia of Mathematics by Michiel Hazewinkel Pdf

Author : M. Hazewinkel
Publisher : Springer
Page : 967 pages
File Size : 42,6 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781489937957

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Encyclopaedia of Mathematics by M. Hazewinkel Pdf

Author : Dennis Gaitsgory,Nick Rozenblyum
Publisher : American Mathematical Society
Page : 436 pages
File Size : 53,6 Mb
Release : 2020-10-07
Category : Mathematics
ISBN : 9781470452858

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A Study in Derived Algebraic Geometry by Dennis Gaitsgory,Nick Rozenblyum Pdf

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.

Author : Eivind Eriksen,Olav Arnfinn Laudal,Arvid Siqveland
Publisher : CRC Press
Page : 211 pages
File Size : 52,7 Mb
Release : 2017-09-19
Category : Mathematics
ISBN : 9781351652124

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Noncommutative Deformation Theory by Eivind Eriksen,Olav Arnfinn Laudal,Arvid Siqveland Pdf

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.

Author : Daniel Sternheimer,John Rawnsley,Simone Gutt
Publisher : Springer
Page : 392 pages
File Size : 52,8 Mb
Release : 1997-07-31
Category : Mathematics
ISBN : UOM:39015047132207

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Deformation Theory and Symplectic Geometry by Daniel Sternheimer,John Rawnsley,Simone Gutt Pdf

Proceedings of the Ascona Meeting, June 1996

Author : Sergei Silvestrov,Anatoliy Malyarenko,Milica Rančić
Publisher : Springer Nature
Page : 976 pages
File Size : 53,9 Mb
Release : 2020-06-18
Category : Mathematics
ISBN : 9783030418502

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Algebraic Structures and Applications by Sergei Silvestrov,Anatoliy Malyarenko,Milica Rančić Pdf

This book explores the latest advances in algebraic structures and applications, and focuses on mathematical concepts, methods, structures, problems, algorithms and computational methods important in the natural sciences, engineering and modern technologies. In particular, it features mathematical methods and models of non-commutative and non-associative algebras, hom-algebra structures, generalizations of differential calculus, quantum deformations of algebras, Lie algebras and their generalizations, semi-groups and groups, constructive algebra, matrix analysis and its interplay with topology, knot theory, dynamical systems, functional analysis, stochastic processes, perturbation analysis of Markov chains, and applications in network analysis, financial mathematics and engineering mathematics. The book addresses both theory and applications, which are illustrated with a wealth of ideas, proofs and examples to help readers understand the material and develop new mathematical methods and concepts of their own. The high-quality chapters share a wealth of new methods and results, review cutting-edge research and discuss open problems and directions for future research. Taken together, they offer a source of inspiration for a broad range of researchers and research students whose work involves algebraic structures and their applications, probability theory and mathematical statistics, applied mathematics, engineering mathematics and related areas.

Author : Licata Ignazio
Publisher : World Scientific
Page : 740 pages
File Size : 55,7 Mb
Release : 2016-03-30
Category : Science
ISBN : 9781783268337

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Beyond Peaceful Coexistence by Licata Ignazio Pdf

'It may be that a real synthesis of quantum and relativity theories requires not just technical developments but radical conceptual renewal.'J S BellBeyond Peaceful Coexistence: The Emergence of Space, Time and Quantum brings together leading academics in mathematics and physics to address going beyond the 'peaceful coexistence' of space-time descriptions (local and continuous ones) and quantum events (discrete and non-commutative ones). Formidable challenges waiting beyond the Standard Model require a new semantic consistency within the theories in order to build new ways of understanding, working and relating to them. The original A. Shimony meaning of the peaceful coexistence (the collapse postulate and non-locality) appear to be just the tip of the iceberg in relation to more serious fundamental issues across physics as a whole.Chapters in this book present perspectives on emergent, discrete, geometrodynamic and topological approaches, as well as a new interpretative spectrum of quantum theories after Copenhagen, discrete time theories, time-less approaches and 'super-fluid' pictures of space-time.As well as stimulating further research among established theoretical physicists, the book can also be used in courses on the philosophy and mathematics of theoretical physics.