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Quantum Cluster Algebra Structures On Quantum Nilpotent Algebras
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Book Synopsis Quantum Cluster Algebra Structures on Quantum Nilpotent Algebras by : K. R. Goodearl
Download or read book Quantum Cluster Algebra Structures on Quantum Nilpotent Algebras written by K. R. Goodearl and published by American Mathematical Soc.. This book was released on 2017-04-25 with total page 134 pages. Available in PDF, EPUB and Kindle. Book excerpt: All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always equal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quantized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have a broad range of applications to these problems, including the construction of quantum cluster algebra structures on quantum unipotent groups and quantum double Bruhat cells (the Berenstein-Zelevinsky conjecture), and treat these problems from a unified perspective. All such applications also establish equality between the constructed quantum cluster algebras and their upper counterparts
Book Synopsis Cluster Algebra Structures on Poisson Nilpotent Algebras by : K. R Goodearl
Download or read book Cluster Algebra Structures on Poisson Nilpotent Algebras written by K. R Goodearl and published by American Mathematical Society. This book was released on 2023-11-27 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.
Book Synopsis Algebraic and Topological Aspects of Representation Theory by : Mee Seong Im
Download or read book Algebraic and Topological Aspects of Representation Theory written by Mee Seong Im and published by American Mathematical Society. This book was released on 2024-01-22 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the virtual AMS Special Session on Geometric and Algebraic Aspects of Quantum Groups and Related Topics, held from November 20–21, 2021. Noncommutative algebras and noncommutative algebraic geometry have been an active field of research for the past several decades, with many important applications in mathematical physics, representation theory, number theory, combinatorics, geometry, low-dimensional topology, and category theory. Papers in this volume contain original research, written by speakers and their collaborators. Many papers also discuss new concepts with detailed examples and current trends with novel and important results, all of which are invaluable contributions to the mathematics community.
Book Synopsis Quantum Grothendieck Rings, Cluster Algebras and Quantum Affine Category O by : Léa Bittmann
Download or read book Quantum Grothendieck Rings, Cluster Algebras and Quantum Affine Category O written by Léa Bittmann and published by . This book was released on 2019 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this thesis is to construct and study some quantum Grothendieck ring structure for the category O of representations of the Borel subalgebra Uq(^b) of a quantum affine algebra Uq(^g). First of all, we focus on the construction of asymptotical standard modules, analogs in the context of the category O of the standard modules in the category of finite-dimensional Uq(^g)-modules. A construction of these modules is given in the case where the underlying simple Lie algebra g is sl2. Next, we define a new quantum torus, which extends the quantum torus containing the quantum Grothendieck ring of the category of finite-dimensional modules. In order todo this, we use notions linked to quantum cluster algebras. In the same spirit, we build a quantum cluster algebra structure on the quantum Grothendieck ring of a monoidal subcategory CZ of the category of finite-dimensional representations. With this quantum torus, we de_ne the quantum Grothendieck ring Kt(O+Z) of a subcategory O+Z of the category O as a quantum cluster algebra. Then, we prove that this quantum Grothendieck ring contains that of the category of finite-dimensional representation. This result is first shown directly in type A, and then in all simply-laced types using the quantum cluster algebra structure of Kt(CZ). Finally, we define (q,t)-characters for some remarkable infinite-dimensional simple representations in the category O+Z. This enables us to write t-deformed analogs of important relations in the classical Grothendieck ring of the category O, which are related to the corresponding quantum integrable systems.
Book Synopsis Advances in Rings and Modules by : Sergio R. López-Permouth
Download or read book Advances in Rings and Modules written by Sergio R. López-Permouth and published by American Mathematical Soc.. This book was released on 2018-09-06 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume, dedicated to Bruno J. Müller, a renowned algebraist, is a collection of papers that provide a snapshot of the diversity of themes and applications that interest algebraists today. The papers highlight the latest progress in ring and module research and present work done on the frontiers of the topics discussed. In addition, selected expository articles are included to give algebraists and other mathematicians, including graduate students, an accessible introduction to areas that may be outside their own expertise.
Book Synopsis Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori by : Xiao Xiong
Download or read book Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori written by Xiao Xiong and published by American Mathematical Soc.. This book was released on 2018-03-19 with total page 130 pages. Available in PDF, EPUB and Kindle. Book excerpt: This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative -torus (with a skew symmetric real -matrix). These spaces share many properties with their classical counterparts. The authors prove, among other basic properties, the lifting theorem for all these spaces and a Poincaré type inequality for Sobolev spaces.
Book Synopsis Medial/Skeletal Linking Structures for Multi-Region Configurations by : James Damon
Download or read book Medial/Skeletal Linking Structures for Multi-Region Configurations written by James Damon and published by American Mathematical Soc.. This book was released on 2018-01-16 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors consider a generic configuration of regions, consisting of a collection of distinct compact regions in which may be either regions with smooth boundaries disjoint from the others or regions which meet on their piecewise smooth boundaries in a generic way. They introduce a skeletal linking structure for the collection of regions which simultaneously captures the regions' individual shapes and geometric properties as well as the “positional geometry” of the collection. The linking structure extends in a minimal way the individual “skeletal structures” on each of the regions. This allows the authors to significantly extend the mathematical methods introduced for single regions to the configuration of regions.
Book Synopsis Hypercontractivity in Group von Neumann Algebras by : Marius Junge
Download or read book Hypercontractivity in Group von Neumann Algebras written by Marius Junge and published by American Mathematical Soc.. This book was released on 2017-09-25 with total page 102 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, the authors provide a combinatorial/numerical method to establish new hypercontractivity estimates in group von Neumann algebras. They illustrate their method with free groups, triangular groups and finite cyclic groups, for which they obtain optimal time hypercontractive inequalities with respect to the Markov process given by the word length and with an even integer. Interpolation and differentiation also yield general hypercontrativity for via logarithmic Sobolev inequalities. The authors' method admits further applications to other discrete groups without small loops as far as the numerical part—which varies from one group to another—is implemented and tested on a computer. The authors also develop another combinatorial method which does not rely on computational estimates and provides (non-optimal) hypercontractive inequalities for a larger class of groups/lengths, including any finitely generated group equipped with a conditionally negative word length, like infinite Coxeter groups. The authors' second method also yields hypercontractivity bounds for groups admitting a finite dimensional proper cocycle. Hypercontractivity fails for conditionally negative lengths in groups satisfying Kazhdan's property (T).
Book Synopsis AdS/CFT, (Super-)Virasoro, Affine (Super-)Algebras by : Vladimir K. Dobrev
Download or read book AdS/CFT, (Super-)Virasoro, Affine (Super-)Algebras written by Vladimir K. Dobrev and published by Walter de Gruyter GmbH & Co KG. This book was released on 2019-04-01 with total page 246 pages. Available in PDF, EPUB and Kindle. Book excerpt: The De Gruyter Studies in Mathematical Physics are devoted to the publication of monographs and high-level texts in mathematical physics. They cover topics and methods in fields of current interest, with an emphasis on didactical presentation. The series will enable readers to understand, apply and develop further, with sufficient rigor, mathematical methods to given problems in physics. For this reason, works with a few authors are preferred over edited volumes. The works in this series are aimed at advanced students and researchers in mathematical and theoretical physics. They can also serve as secondary reading for lectures and seminars at advanced levels.
Book Synopsis Entire Solutions for Bistable Lattice Differential Equations with Obstacles by : Aaron Hoffman
Download or read book Entire Solutions for Bistable Lattice Differential Equations with Obstacles written by Aaron Hoffman and published by American Mathematical Soc.. This book was released on 2018-01-16 with total page 132 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors consider scalar lattice differential equations posed on square lattices in two space dimensions. Under certain natural conditions they show that wave-like solutions exist when obstacles (characterized by “holes”) are present in the lattice. Their work generalizes to the discrete spatial setting the results obtained in Berestycki, Hamel, and Matuno (2009) for the propagation of waves around obstacles in continuous spatial domains. The analysis hinges upon the development of sub and super-solutions for a class of discrete bistable reaction-diffusion problems and on a generalization of a classical result due to Aronson and Weinberger that concerns the spreading of localized disturbances.
Book Synopsis Property ($T$) for Groups Graded by Root Systems by : Mikhail Ershov
Download or read book Property ($T$) for Groups Graded by Root Systems written by Mikhail Ershov and published by American Mathematical Soc.. This book was released on 2017-09-25 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors introduce and study the class of groups graded by root systems. They prove that if is an irreducible classical root system of rank and is a group graded by , then under certain natural conditions on the grading, the union of the root subgroups is a Kazhdan subset of . As the main application of this theorem the authors prove that for any reduced irreducible classical root system of rank and a finitely generated commutative ring with , the Steinberg group and the elementary Chevalley group have property . They also show that there exists a group with property which maps onto all finite simple groups of Lie type and rank , thereby providing a “unified” proof of expansion in these groups.
Book Synopsis On Sudakov's Type Decomposition of Transference Plans with Norm Costs by : Stefano Bianchini
Download or read book On Sudakov's Type Decomposition of Transference Plans with Norm Costs written by Stefano Bianchini and published by American Mathematical Soc.. This book was released on 2018-02-23 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost with , probability measures in and absolutely continuous w.r.t. . The key idea in this approach is to decompose (via disintegration of measures) the Kantorovich optimal transportation problem into a family of transportation problems in , where are disjoint regions such that the construction of an optimal map is simpler than in the original problem, and then to obtain by piecing together the maps . When the norm is strictly convex, the sets are a family of -dimensional segments determined by the Kantorovich potential called optimal rays, while the existence of the map is straightforward provided one can show that the disintegration of (and thus of ) on such segments is absolutely continuous w.r.t. the -dimensional Hausdorff measure. When the norm is not strictly convex, the main problems in this kind of approach are two: first, to identify a suitable family of regions on which the transport problem decomposes into simpler ones, and then to prove the existence of optimal maps. In this paper the authors show how these difficulties can be overcome, and that the original idea of Sudakov can be successfully implemented. The results yield a complete characterization of the Kantorovich optimal transportation problem, whose straightforward corollary is the solution of the Monge problem in each set and then in . The strategy is sufficiently powerful to be applied to other optimal transportation problems.
Book Synopsis Spatially Independent Martingales, Intersections, and Applications by : Pablo Shmerkin
Download or read book Spatially Independent Martingales, Intersections, and Applications written by Pablo Shmerkin and published by American Mathematical Soc.. This book was released on 2018-02-22 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors define a class of random measures, spatially independent martingales, which we view as a natural generalization of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. The authors pair the random measures with deterministic families of parametrized measures , and show that under some natural checkable conditions, a.s. the mass of the intersections is Hölder continuous as a function of . This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones. Among other things, for large classes of random fractals they establish (a) very strong versions of the Marstrand-Mattila projection and slicing results, as well as dimension conservation, (b) slicing results with respect to algebraic curves and self-similar sets, (c) smoothness of convolutions of measures, including self-convolutions, and nonempty interior for sumsets, and (d) rapid Fourier decay. Among other applications, the authors obtain an answer to a question of I. Łaba in connection to the restriction problem for fractal measures.
Book Synopsis La Formule des Traces Locale Tordue by : Colette Moeglin
Download or read book La Formule des Traces Locale Tordue written by Colette Moeglin and published by American Mathematical Soc.. This book was released on 2018-02-23 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: A note to readers: This book is in French. The text has two chapters. The first one, written by Waldspurger, proves a twisted version of the local trace formula of Arthur over a local field. This formula is an equality between two expressions, one involving weighted orbital integrals, the other one involving weighted characters. The authors follow Arthur's proof, but the treatement of the spectral side is more complicated in the twisted situation. They need to use the combinatorics of the “Morning Seminar”. The authors' local trace formula has the same consequences as in Arthur's paper on elliptic characters. The second chapter, written by Moeglin, gives a symmetric form of the local trace formula as in Arthur's paper on Fourier Transform of Orbital integral and describes any twisted orbital integral, in the p-adic case, as integral of characters.
Book Synopsis Tensor Products and Regularity Properties of Cuntz Semigroups by : Ramon Antoine
Download or read book Tensor Products and Regularity Properties of Cuntz Semigroups written by Ramon Antoine and published by American Mathematical Soc.. This book was released on 2018-02-23 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Cuntz semigroup of a -algebra is an important invariant in the structure and classification theory of -algebras. It captures more information than -theory but is often more delicate to handle. The authors systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a -algebra , its (concrete) Cuntz semigroup is an object in the category of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, the authors call the latter -semigroups. The authors establish the existence of tensor products in the category and study the basic properties of this construction. They show that is a symmetric, monoidal category and relate with for certain classes of -algebras. As a main tool for their approach the authors introduce the category of pre-completed Cuntz semigroups. They show that is a full, reflective subcategory of . One can then easily deduce properties of from respective properties of , for example the existence of tensor products and inductive limits. The advantage is that constructions in are much easier since the objects are purely algebraic.
Book Synopsis Nonsmooth Differential Geometry-An Approach Tailored for Spaces with Ricci Curvature Bounded from Below by : Nicola Gigli
Download or read book Nonsmooth Differential Geometry-An Approach Tailored for Spaces with Ricci Curvature Bounded from Below written by Nicola Gigli and published by American Mathematical Soc.. This book was released on 2018-02-23 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author discusses in which sense general metric measure spaces possess a first order differential structure. Building on this, spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting the author to define Hessian, covariant/exterior derivatives and Ricci curvature.
Book Synopsis The Stability of Cylindrical Pendant Drops by : John McCuan
Download or read book The Stability of Cylindrical Pendant Drops written by John McCuan and published by American Mathematical Soc.. This book was released on 2018-01-16 with total page 122 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author considers the stability of certain liquid drops in a gravity field satisfying a mixed boundary condition. He also considers as special cases portions of cylinders that model either the zero gravity case or soap films with the same kind of boundary behavior.
