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Sheaves of Algebras over Boolean Spaces comprehensively covers sheaf theory as applied to universal algebra. Sheaves decompose general algebras into simpler pieces called the stalks. A classical case is commutative von Neumann regular rings, whose stalks are fields. Other classical theorems also extend to shells, a common generalization of rings and lattices. This text presents intuitive ideas from topology such as the notion of metric space and the concept of central idempotent from ring theory. These lead to the abstract notions of complex and factor element, respectively. Factor elements are defined by identities, discovered for shells for the first time, explaining why central elements in rings and lattices have their particular form. Categorical formulations of the many representations by sheaves begin with adjunctions and move to equivalences as the book progresses, generalizing Stone's theorem for Boolean algebras. Half of the theorems provided in the text are new; the rest are presented in a coherent framework, starting with the most general, and proceeding to specific applications. Many open problems and research areas are outlined, including a final chapter summarizing applications of sheaves in diverse fields that were not covered earlier in the book. This monograph is suitable for graduate students and researchers, and it will serve as an excellent reference text for those who wish to learn about sheaves of algebras.

Category theory. Homological algebra --- Algebra --- Topology --- algebra --- topologie

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An outgrowth of a two-week summer session at Jacobs University in Bremen, Germany in August 2009 ("Structures in Lie Theory, Crystals, Derived Functors, Harish-Chandra Modules, Invariants and Quivers"), this volume consists of expository and research articles that highlight the various Lie algebraic methods used in mathematical research today. Key topics discussed include spherical varieties, Littelmann Paths and Kac-Moody Lie algebras, modular representations, primitive ideals, representation theory of Artin algebras and quivers, Kac-Moody superalgebras, categories of Harish-Chandra modules, cohomological methods, and cluster algebras.  List of Contributors:  M. Boos M. Brion J. Fuchs M. Gorelik A. Joseph M. Reineke C. Schweigert V. Serganova A. Seven W. Soergel B. Wilson G. Zuckerman

Category theory. Homological algebra --- Algebra --- Topological groups. Lie groups --- algebra --- topologie (wiskunde)

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In this book we want to explore aspects of coherence in homological algebra, that already appear in the classical situation of abelian groups or abelian categories. Lattices of subobjects are shown to play an important role in the study of homological systems, from simple chain complexes to all the structures that give rise to spectral sequences. A parallel role is played by semigroups of endorelations. These links rest on the fact that many such systems, but not all of them, live in distributive sublattices of the modular lattices of subobjects of the system. The property of distributivity al

Algebra, Homological. --- Homological algebra --- Algebra, Abstract --- Homology theory --- Algebra, Homological --- Lattices, Distributive

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Category theory. Homological algebra --- Ordered algebraic structures --- Algebra --- Algebraic topology --- Geometry --- algebra --- landmeetkunde --- topologie (wiskunde) --- geometrie

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This monograph provides the first up-to-date and self-contained presentation of a recently discovered mathematical structure the Schrödinger-Virasoro algebra. Just as Poincaré invariance or conformal (Virasoro) invariance play a key role in understanding, respectively, elementary particles and two-dimensional equilibrium statistical physics, this algebra of non-relativistic conformal symmetries may be expected to apply itself naturally to the study of some models of non-equilibrium statistical physics, or more specifically in the context of recent developments related to the non-relativistic AdS/CFT correspondence.   The study of the structure of this infinite-dimensional Lie algebra touches upon topics as various as statistical physics, vertex algebras, Poisson geometry, integrable systems and supergeometry as well as representation theory, the cohomology of infinite-dimensional Lie algebras, and the spectral theory of Schrödinger operators. .

Category theory. Homological algebra --- Algebra --- Topological groups. Lie groups --- Mathematical physics --- Statistical physics --- algebra --- topologie (wiskunde) --- theoretische fysica --- wiskunde