Bimonoids For Hyperplane Arrangements

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Bimonoids for Hyperplane Arrangements

Marcelo Aguiar,Swapneel Mahajan
Bimonoids for Hyperplane Arrangements Book PDF

Author : Marcelo Aguiar,Swapneel Mahajan
Publisher : Cambridge University Press
Page : 853 pages
File Size : 46,8 Mb
Release : 2020-03-19
Category : Mathematics
ISBN : 9781108495806

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Bimonoids for Hyperplane Arrangements by Marcelo Aguiar,Swapneel Mahajan Pdf

The goal of this monograph is to develop Hopf theory in a new setting which features centrally a real hyperplane arrangement. The new theory is parallel to the classical theory of connected Hopf algebras, and relates to it when specialized to the braid arrangement. Joyal's theory of combinatorial species, ideas from Tits' theory of buildings, and Rota's work on incidence algebras inspire and find a common expression in this theory. The authors introduce notions of monoid, comonoid, bimonoid, and Lie monoid relative to a fixed hyperplane arrangement. They also construct universal bimonoids by using generalizations of the classical notions of shuffle and quasishuffle, and establish the Borel-Hopf, Poincar -Birkhoff-Witt, and Cartier-Milnor-Moore theorems in this setting. This monograph opens a vast new area of research. It will be of interest to students and researchers working in the areas of hyperplane arrangements, semigroup theory, Hopf algebras, algebraic Lie theory, operads, and category theory.

Coxeter Bialgebras

Marcelo Aguiar,Swapneel Mahajan
Coxeter Bialgebras Book PDF

Author : Marcelo Aguiar,Swapneel Mahajan
Publisher : Cambridge University Press
Page : 897 pages
File Size : 48,9 Mb
Release : 2022-10-31
Category : Mathematics
ISBN : 9781009243735

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Coxeter Bialgebras by Marcelo Aguiar,Swapneel Mahajan Pdf

The goal of this monograph is to develop Hopf theory in the setting of a real reflection arrangement. The central notion is that of a Coxeter bialgebra which generalizes the classical notion of a connected graded Hopf algebra. The authors also introduce the more structured notion of a Coxeter bimonoid and connect the two notions via a family of functors called Fock functors. These generalize similar functors connecting Hopf monoids in the category of Joyal species and connected graded Hopf algebras. This monograph opens a new chapter in Coxeter theory as well as in Hopf theory, connecting the two. It also relates fruitfully to many other areas of mathematics such as discrete geometry, semigroup theory, associative algebras, algebraic Lie theory, operads, and category theory. It is carefully written, with effective use of tables, diagrams, pictures, and summaries. It will be of interest to students and researchers alike.

Quasi-Hopf Algebras

Daniel Bulacu,Stefaan Caenepeel,Florin Panaite,Freddy Van Oystaeyen
Quasi Hopf Algebras Book PDF

Author : Daniel Bulacu,Stefaan Caenepeel,Florin Panaite,Freddy Van Oystaeyen
Publisher : Cambridge University Press
Page : 545 pages
File Size : 45,9 Mb
Release : 2019-02-21
Category : Mathematics
ISBN : 9781108427012

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Quasi-Hopf Algebras by Daniel Bulacu,Stefaan Caenepeel,Florin Panaite,Freddy Van Oystaeyen Pdf

This self-contained book dedicated to Drinfeld's quasi-Hopf algebras takes the reader from the basics to the state of the art.

Basic Category Theory

Tom Leinster
Basic Category Theory Book PDF

Author : Tom Leinster
Publisher : Cambridge University Press
Page : 193 pages
File Size : 44,8 Mb
Release : 2014-07-24
Category : Mathematics
ISBN : 9781107044241

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Basic Category Theory by Tom Leinster Pdf

A short introduction ideal for students learning category theory for the first time.

Hopf Algebras and Tensor Categories

Nicolás Andruskiewitsch,Juan Cuadra,Blas Torrecillas
Hopf Algebras and Tensor Categories Book PDF

Author : Nicolás Andruskiewitsch,Juan Cuadra,Blas Torrecillas
Publisher : American Mathematical Soc.
Page : 347 pages
File Size : 49,6 Mb
Release : 2013-02-21
Category : Mathematics
ISBN : 9780821875643

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Hopf Algebras and Tensor Categories by Nicolás Andruskiewitsch,Juan Cuadra,Blas Torrecillas Pdf

This volume contains the proceedings of the Conference on Hopf Algebras and Tensor Categories, held July 4-8, 2011, at the University of Almeria, Almeria, Spain. The articles in this volume cover a wide variety of topics related to the theory of Hopf algebras and its connections to other areas of mathematics. In particular, this volume contains a survey covering aspects of the classification of fusion categories using Morita equivalence methods, a long comprehensive introduction to Hopf algebras in the category of species, and a summary of the status to date of the classification of Hopf algebras of dimensions up to 100. Among other topics discussed in this volume are a study of normalized class sum and generalized character table for semisimple Hopf algebras, a contribution to the classification program of finite dimensional pointed Hopf algebras, relations to the conjecture of De Concini, Kac, and Procesi on representations of quantum groups at roots of unity, a categorical approach to the Drinfeld double of a braided Hopf algebra via Hopf monads, an overview of Hom-Hopf algebras, and several discussions on the crossed product construction in different settings.

Coherence in Three-Dimensional Category Theory

Nick Gurski
Coherence in Three Dimensional Category Theory Book PDF

Author : Nick Gurski
Publisher : Cambridge University Press
Page : 287 pages
File Size : 47,5 Mb
Release : 2013-03-21
Category : Mathematics
ISBN : 9781107034891

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Coherence in Three-Dimensional Category Theory by Nick Gurski Pdf

Serves as an introduction to higher categories as well as a reference point for many key concepts in the field.

Topics in Hyperplane Arrangements

Marcelo Aguiar,Swapneel Mahajan
Topics in Hyperplane Arrangements Book PDF

Author : Marcelo Aguiar,Swapneel Mahajan
Publisher : American Mathematical Soc.
Page : 611 pages
File Size : 48,7 Mb
Release : 2017-11-22
Category : Algebraic spaces
ISBN : 9781470437114

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Topics in Hyperplane Arrangements by Marcelo Aguiar,Swapneel Mahajan Pdf

This monograph studies the interplay between various algebraic, geometric and combinatorial aspects of real hyperplane arrangements. It provides a careful, organized and unified treatment of several recent developments in the field, and brings forth many new ideas and results. It has two parts, each divided into eight chapters, and five appendices with background material. Part I gives a detailed discussion on faces, flats, chambers, cones, gallery intervals, lunes and other geometric notions associated with arrangements. The Tits monoid plays a central role. Another important object is the category of lunes which generalizes the classical associative operad. Also discussed are the descent and lune identities, distance functions on chambers, and the combinatorics of the braid arrangement and related examples. Part II studies the structure and representation theory of the Tits algebra of an arrangement. It gives a detailed analysis of idempotents and Peirce decompositions, and connects them to the classical theory of Eulerian idempotents. It introduces the space of Lie elements of an arrangement which generalizes the classical Lie operad. This space is the last nonzero power of the radical of the Tits algebra. It is also the socle of the left ideal of chambers and of the right ideal of Zie elements. Zie elements generalize the classical Lie idempotents. They include Dynkin elements associated to generic half-spaces which generalize the classical Dynkin idempotent. Another important object is the lune-incidence algebra which marks the beginning of noncommutative Möbius theory. These ideas are also brought upon the study of the Solomon descent algebra. The monograph is written with clarity and in sufficient detail to make it accessible to graduate students. It can also serve as a useful reference to experts.

Coxeter Groups and Hopf Algebras

Marcelo Aguiar
Coxeter Groups and Hopf Algebras Book PDF

Author : Marcelo Aguiar
Publisher : American Mathematical Soc.
Page : 201 pages
File Size : 46,6 Mb
Release : 2006
Category : Education
ISBN : 9780821853542

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Coxeter Groups and Hopf Algebras by Marcelo Aguiar Pdf

An important idea in the work of G.-C. Rota is that certain combinatorial objects give rise to Hopf algebras that reflect the manner in which these objects compose and decompose. Recent work has seen the emergence of several interesting Hopf algebras of this kind, which connect diverse subjects such as combinatorics, algebra, geometry, and theoretical physics. This monograph presents a novel geometric approach using Coxeter complexes and the projection maps of Tits for constructing and studying many of these objects as well as new ones. The first three chapters introduce the necessary background ideas making this work accessible to advanced graduate students. The later chapters culminate in a unified and conceptual construction of several Hopf algebras based on combinatorial objects which emerge naturally from the geometric viewpoint. This work lays a foundation and provides new insights for further development of the subject.

Algebraic Groups

J. S. Milne
Algebraic Groups Book PDF

Author : J. S. Milne
Publisher : Cambridge University Press
Page : 665 pages
File Size : 50,6 Mb
Release : 2017-09-21
Category : Mathematics
ISBN : 9781107167483

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Algebraic Groups by J. S. Milne Pdf

Comprehensive introduction to the theory of algebraic group schemes over fields, based on modern algebraic geometry, with few prerequisites.

Monoidal Functors, Species and Hopf Algebras

Marcelo Aguiar,Swapneel Arvind Mahajan
Monoidal Functors Species and Hopf Algebras Book PDF

Author : Marcelo Aguiar,Swapneel Arvind Mahajan
Publisher : American Mathematical Soc.
Page : 784 pages
File Size : 41,5 Mb
Release : 2010
Category : Mathematics
ISBN : 0821847767

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Monoidal Functors, Species and Hopf Algebras by Marcelo Aguiar,Swapneel Arvind Mahajan Pdf

This research monograph integrates ideas from category theory, algebra and combinatorics. It is organized in three parts. Part I belongs to the realm of category theory. It reviews some of the foundational work of Benabou, Eilenberg, Kelly and Mac Lane on monoidal categories and of Joyal and Street on braided monoidal categories, and proceeds to study higher monoidal categories and higher monoidal functors. Special attention is devoted to the notion of a bilax monoidal functor which plays a central role in this work. Combinatorics and geometry are the theme of Part II. Joyal's species constitute a good framework for the study of algebraic structures associated to combinatorial objects. This part discusses the category of species focusing particularly on the Hopf monoids therein. The notion of a Hopf monoid in species parallels that of a Hopf algebra and reflects the manner in which combinatorial structures compose and decompose. Numerous examples of Hopf monoids are given in the text. These are constructed from combinatorial and geometric data and inspired by ideas of Rota and Tits' theory of Coxeter complexes. Part III is of an algebraic nature and shows how ideas in Parts I and II lead to a unified approach to Hopf algebras. The main step is the construction of Fock functors from species to graded vector spaces. These functors are bilax monoidal and thus translate Hopf monoids in species to graded Hopf algebras. This functorial construction of Hopf algebras encompasses both quantum groups and the Hopf algebras of recent prominence in the combinatorics literature. The monograph opens a vast new area of research. It is written with clarity and sufficient detail to make it accessible to advanced graduate students.

Noncommutative Geometry, Quantum Fields and Motives

Alain Connes,Matilde Marcolli
Noncommutative Geometry Quantum Fields and Motives Book PDF

Author : Alain Connes,Matilde Marcolli
Publisher : American Mathematical Soc.
Page : 785 pages
File Size : 48,5 Mb
Release : 2019-03-13
Category : Electronic
ISBN : 9781470450458

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Noncommutative Geometry, Quantum Fields and Motives by Alain Connes,Matilde Marcolli Pdf

The unifying theme of this book is the interplay among noncommutative geometry, physics, and number theory. The two main objects of investigation are spaces where both the noncommutative and the motivic aspects come to play a role: space-time, where the guiding principle is the problem of developing a quantum theory of gravity, and the space of primes, where one can regard the Riemann Hypothesis as a long-standing problem motivating the development of new geometric tools. The book stresses the relevance of noncommutative geometry in dealing with these two spaces. The first part of the book deals with quantum field theory and the geometric structure of renormalization as a Riemann-Hilbert correspondence. It also presents a model of elementary particle physics based on noncommutative geometry. The main result is a complete derivation of the full Standard Model Lagrangian from a very simple mathematical input. Other topics covered in the first part of the book are a noncommutative geometry model of dimensional regularization and its role in anomaly computations, and a brief introduction to motives and their conjectural relation to quantum field theory. The second part of the book gives an interpretation of the Weil explicit formula as a trace formula and a spectral realization of the zeros of the Riemann zeta function. This is based on the noncommutative geometry of the adèle class space, which is also described as the space of commensurability classes of Q-lattices, and is dual to a noncommutative motive (endomotive) whose cyclic homology provides a general setting for spectral realizations of zeros of L-functions. The quantum statistical mechanics of the space of Q-lattices, in one and two dimensions, exhibits spontaneous symmetry breaking. In the low-temperature regime, the equilibrium states of the corresponding systems are related to points of classical moduli spaces and the symmetries to the class field theory of the field of rational numbers and of imaginary quadratic fields, as well as to the automorphisms of the field of modular functions. The book ends with a set of analogies between the noncommutative geometries underlying the mathematical formulation of the Standard Model minimally coupled to gravity and the moduli spaces of Q-lattices used in the study of the zeta function.

Frontiers in Number Theory, Physics, and Geometry II

Pierre E. Cartier,Bernard Julia,Pierre Moussa,Pierre Vanhove
Frontiers in Number Theory Physics and Geometry II Book PDF

Author : Pierre E. Cartier,Bernard Julia,Pierre Moussa,Pierre Vanhove
Publisher : Springer
Page : 0 pages
File Size : 50,9 Mb
Release : 2006-12-06
Category : Mathematics
ISBN : 3540303073

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Frontiers in Number Theory, Physics, and Geometry II by Pierre E. Cartier,Bernard Julia,Pierre Moussa,Pierre Vanhove Pdf

Ten years after a 1989 meeting of number theorists and physicists at the Centre de Physique des Houches, a second event focused on the broader interface of number theory, geometry, and physics. This book is the first of two volumes resulting from that meeting. Broken into three parts, it covers Conformal Field Theories, Discrete Groups, and Renormalization, offering extended versions of the lecture courses and shorter texts on special topics.

Fundamentals of the Theory of Structured Dependence between Stochastic Processes

Tomasz R. Bielecki,Jacek Jakubowski,Mariusz Niew?glowski
Fundamentals of the Theory of Structured Dependence between Stochastic Processes Book PDF

Author : Tomasz R. Bielecki,Jacek Jakubowski,Mariusz Niew?glowski
Publisher : Cambridge University Press
Page : 279 pages
File Size : 53,9 Mb
Release : 2020-08-27
Category : Business & Economics
ISBN : 9781107154254

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Fundamentals of the Theory of Structured Dependence between Stochastic Processes by Tomasz R. Bielecki,Jacek Jakubowski,Mariusz Niew?glowski Pdf

Comprehensive presentation of the technical aspects and applications of the theory of structured dependence between random processes.

Geometric Configurations of Singularities of Planar Polynomial Differential Systems

Joan C. Artés,Jaume Llibre,Dana Schlomiuk,Nicolae Vulpe
Geometric Configurations of Singularities of Planar Polynomial Differential Systems Book PDF

Author : Joan C. Artés,Jaume Llibre,Dana Schlomiuk,Nicolae Vulpe
Publisher : Springer Nature
Page : 699 pages
File Size : 50,5 Mb
Release : 2021-07-19
Category : Mathematics
ISBN : 9783030505707

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Geometric Configurations of Singularities of Planar Polynomial Differential Systems by Joan C. Artés,Jaume Llibre,Dana Schlomiuk,Nicolae Vulpe Pdf

This book addresses the global study of finite and infinite singularities of planar polynomial differential systems, with special emphasis on quadratic systems. While results covering the degenerate cases of singularities of quadratic systems have been published elsewhere, the proofs for the remaining harder cases were lengthier. This book covers all cases, with half of the content focusing on the last non-degenerate ones. The book contains the complete bifurcation diagram, in the 12-parameter space, of global geometrical configurations of singularities of quadratic systems. The authors’ results provide - for the first time - global information on all singularities of quadratic systems in invariant form and their bifurcations. In addition, a link to a very helpful software package is included. With the help of this software, the study of the algebraic bifurcations becomes much more efficient and less time-consuming. Given its scope, the book will appeal to specialists on polynomial differential systems, pure and applied mathematicians who need to study bifurcation diagrams of families of such systems, Ph.D. students, and postdoctoral fellows.

Compound Renewal Processes

A. A. Borovkov
Compound Renewal Processes Book PDF

Author : A. A. Borovkov
Publisher : Cambridge University Press
Page : 128 pages
File Size : 50,9 Mb
Release : 2022-06-30
Category : Mathematics
ISBN : 9781009115605

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Compound Renewal Processes by A. A. Borovkov Pdf

Compound renewal processes (CRPs) are among the most ubiquitous models arising in applications of probability. At the same time, they are a natural generalization of random walks, the most well-studied classical objects in probability theory. This monograph, written for researchers and graduate students, presents the general asymptotic theory and generalizes many well-known results concerning random walks. The book contains the key limit theorems for CRPs, functional limit theorems, integro-local limit theorems, large and moderately large deviation principles for CRPs in the state space and in the space of trajectories, including large deviation principles in boundary crossing problems for CRPs, with an explicit form of the rate functionals, and an extension of the invariance principle for CRPs to the domain of moderately large and small deviations. Applications establish the key limit laws for Markov additive processes, including limit theorems in the domains of normal and large deviations.