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Monoidal Categories and the Gerstenhaber Bracket in Hochschild Cohomology

2016-09-06 Mathematics
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Release : 2016-09-06
Genre : Mathematics
Kind : eBook
Book Rating : 955/5 ( reviews)

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Book Synopsis Monoidal Categories and the Gerstenhaber Bracket in Hochschild Cohomology by : Reiner Hermann:

Download or read book Monoidal Categories and the Gerstenhaber Bracket in Hochschild Cohomology written by Reiner Hermann:. This book was released on 2016-09-06. Available in PDF, EPUB and Kindle. Book excerpt: In this monograph, the author extends S. Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology to certain exact and monoidal categories. Therefore the author establishes an explicit description of an isomorphism by A. Neeman and V. Retakh, which links Ext-groups with fundamental groups of categories of extensions and relies on expressing the fundamental group of a (small) category by means of the associated Quillen groupoid. As a main result, the author shows that his construction behaves well with respect to structure preserving functors between exact monoidal categories. The author uses his main result to conclude, that the graded Lie bracket in Hochschild cohomology is an invariant under Morita equivalence. For quasi-triangular bialgebras, he further determines a significant part of the Lie bracket's kernel, and thereby proves a conjecture by L. Menichi. Along the way, the author introduces n-extension closed and entirely extension closed subcategories of abelian categories, and studies some of their properties.

Hochschild Cohomology for Algebras

2019-12-10 Education
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Release : 2019-12-10
Genre : Education
Kind : eBook
Book Rating : 315/5 ( reviews)

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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. This book was released on 2019-12-10. 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.

Lie Theory and Its Applications in Physics

2023-01-29 Mathematics
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Release : 2023-01-29
Genre : Mathematics
Kind : eBook
Book Rating : 511/5 ( reviews)

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Book Synopsis Lie Theory and Its Applications in Physics by : Vladimir Dobrev

Download or read book Lie Theory and Its Applications in Physics written by Vladimir Dobrev. This book was released on 2023-01-29. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents modern trends in the area of symmetries and their applications based on contributions to the Workshop "Lie Theory and Its Applications in Physics" held in Sofia, Bulgaria (on-line) in June 2021. Traditionally, Lie theory is a tool to build mathematical models for physical systems. Recently, the trend is towards geometrization of the mathematical description of physical systems and objects. A geometric approach to a system yields in general some notion of symmetry which is very helpful in understanding its structure. Geometrization and symmetries are meant in their widest sense, i.e., representation theory, algebraic geometry, number theory, infinite-dimensional Lie algebras and groups, superalgebras and supergroups, groups and quantum groups, noncommutative geometry, symmetries of linear and nonlinear partial differential operators, special functions, and others. Furthermore, the necessary tools from functional analysis are included. This is a big interdisciplinary and interrelated field. The topics covered in this Volume are the most modern trends in the field of the Workshop: Representation Theory, Symmetries in String Theories, Symmetries in Gravity Theories, Supergravity, Conformal Field Theory, Integrable Systems, Quantum Computing and Deep Learning, Entanglement, Applications to Quantum Theory, Exceptional quantum algebra for the standard model of particle physics, Gauge Theories and Applications, Structures on Lie Groups and Lie Algebras. This book is suitable for a broad audience of mathematicians, mathematical physicists, and theoretical physicists, including researchers and graduate students interested in Lie Theory.

Extended Abstracts Spring 2015

2016-11-30 Mathematics
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Release : 2016-11-30
Genre : Mathematics
Kind : eBook
Book Rating : 412/5 ( reviews)

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Book Synopsis Extended Abstracts Spring 2015 by : Dolors Herbera

Download or read book Extended Abstracts Spring 2015 written by Dolors Herbera. This book was released on 2016-11-30. Available in PDF, EPUB and Kindle. Book excerpt: This book includes 33 expanded abstracts of selected talks given at the two workshops "Homological Bonds Between Commutative Algebra and Representation Theory" and "Brave New Algebra: Opening Perspectives," and the conference "Opening Perspectives in Algebra, Representations, and Topology," held at the Centre de Recerca Matemàtica (CRM) in Barcelona between January and June 2015. These activities were part of the one-semester intensive research program "Interactions Between Representation Theory, Algebraic Topology and Commutative Algebra (IRTATCA)." Most of the abstracts present preliminary versions of not-yet published results and cover a large number of topics (including commutative and non commutative algebra, algebraic topology, singularity theory, triangulated categories, representation theory) overlapping with homological methods. This comprehensive book is a valuable resource for the community of researchers interested in homological algebra in a broad sense, and those curious to learn the latest developments in the area. It appeals to established researchers as well as PhD and postdoctoral students who want to learn more about the latest advances in these highly active fields of research.

Homology of Normal Chains and Cohomology of Charges

2017-04-25 Mathematics
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Release : 2017-04-25
Genre : Mathematics
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Book Rating : 359/5 ( reviews)

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Book Synopsis Homology of Normal Chains and Cohomology of Charges by : Th. De Pauw

Download or read book Homology of Normal Chains and Cohomology of Charges written by Th. De Pauw. This book was released on 2017-04-25. Available in PDF, EPUB and Kindle. Book excerpt: The authors consider a category of pairs of compact metric spaces and Lipschitz maps where the pairs satisfy a linearly isoperimetric condition related to the solvability of the Plateau problem with partially free boundary. It includes properly all pairs of compact Lipschitz neighborhood retracts of a large class of Banach spaces. On this category the authors define homology and cohomology functors with real coefficients which satisfy the Eilenberg-Steenrod axioms, but reflect the metric properties of the underlying spaces. As an example they show that the zero-dimensional homology of a space in our category is trivial if and only if the space is path connected by arcs of finite length. The homology and cohomology of a pair are, respectively, locally convex and Banach spaces that are in duality. Ignoring the topological structures, the homology and cohomology extend to all pairs of compact metric spaces. For locally acyclic spaces, the authors establish a natural isomorphism between their cohomology and the Čech cohomology with real coefficients.

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