Dialgebras and Related Operads PDF Download

Author: J.-L. Loday
Publisher: Springer
ISBN: 3540453288
Category : Mathematics
Languages : en
Pages : 138

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Book Description
The main object of study of these four papers is the notion of associative dialgebras which are algebras equipped with two associative operations satisfying some more relations of the associative type. This notion is studied from a) the homological point of view: construction of the (co)homology theory with trivial coefficients and general coefficients, b) the operadic point of view: determination of the dual operad, that is the dendriform dialgebras which are strongly related with the planar binary trees, c) the algebraic point of view: Hopf structure and Milnor-Moore type theorem.

Author: Chengming Bai
Publisher: World Scientific
ISBN: 9814365122
Category : Mathematics
Languages : en
Pages : 318

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Book Description
The book aims to exemplify the recent developments in operad theory, in universal algebra and related topics in algebraic topology and theoretical physics. The conference has established a better connection between mathematicians working on operads (mainly the French team) and mathematicians working in universal algebra (primarily the Chinese team), and to exchange problems, methods and techniques from these two subject areas.

Author: Martin Markl
Publisher: American Mathematical Soc.
ISBN: 0821843621
Category : Mathematics
Languages : en
Pages : 362

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Book Description
Operads are mathematical devices which describe algebraic structures of many varieties and in various categories. From their beginnings in the 1960s, they have developed to encompass such areas as combinatorics, knot theory, moduli spaces, string field theory and deformation quantization.

Author: Jean-Louis Loday
Publisher: Springer Science & Business Media
ISBN: 3642303625
Category : Mathematics
Languages : en
Pages : 649

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Book Description
In many areas of mathematics some “higher operations” are arising. These havebecome so important that several research projects refer to such expressions. Higher operationsform new types of algebras. The key to understanding and comparing them, to creating invariants of their action is operad theory. This is a point of view that is 40 years old in algebraic topology, but the new trend is its appearance in several other areas, such as algebraic geometry, mathematical physics, differential geometry, and combinatorics. The present volume is the first comprehensive and systematic approach to algebraic operads. An operad is an algebraic device that serves to study all kinds of algebras (associative, commutative, Lie, Poisson, A-infinity, etc.) from a conceptual point of view. The book presents this topic with an emphasis on Koszul duality theory. After a modern treatment of Koszul duality for associative algebras, the theory is extended to operads. Applications to homotopy algebra are given, for instance the Homotopy Transfer Theorem. Although the necessary notions of algebra are recalled, readers are expected to be familiar with elementary homological algebra. Each chapter ends with a helpful summary and exercises. A full chapter is devoted to examples, and numerous figures are included. After a low-level chapter on Algebra, accessible to (advanced) undergraduate students, the level increases gradually through the book. However, the authors have done their best to make it suitable for graduate students: three appendices review the basic results needed in order to understand the various chapters. Since higher algebra is becoming essential in several research areas like deformation theory, algebraic geometry, representation theory, differential geometry, algebraic combinatorics, and mathematical physics, the book can also be used as a reference work by researchers.

Author: Kar Ping Shum
Publisher: World Scientific
ISBN: 9811215480
Category : Mathematics
Languages : en
Pages : 498

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Book Description
This volume composed of twenty four research articles which are selected from the keynote speakers and invited lectures presented in the 3rd International Congress in Algebra and Combinatorics (ICAC2017) held on 25-28 August 2017 in Hong Kong and one additional invited article. This congress was specially dedicated to Professor Leonid Bokut on the occasion of his 80th birthday.

Author: Helena Albuquerque
Publisher: Springer Nature
ISBN: 3031327071
Category : Mathematics
Languages : en
Pages : 305

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Book Description
This proceedings volume presents a selection of peer-reviewed contributions from the Second Non-Associative Algebras and Related Topics (NAART II) conference, which was held at the University of Coimbra, Portugal, from July 18–22, 2022. The conference was held in honor of mathematician Alberto Elduque, who has made significant contributions to the study of non-associative structures such as Lie, Jordan, and Leibniz algebras. The papers in this volume are organized into four parts: Lie algebras, superalgebras, and groups; Leibniz algebras; associative and Jordan algebras; and other non-associative structures. They cover a variety of topics, including classification problems, special maps (automorphisms, derivations, etc.), constructions that relate different structures, and representation theory. One of the unique features of NAART is that it is open to all topics related to non-associative algebras, including octonion algebras, composite algebras, Banach algebras, connections with geometry, applications in coding theory, combinatorial problems, and more. This diversity allows researchers from a range of fields to find the conference subjects interesting and discover connections with their own areas, even if they are not traditionally considered non-associative algebraists. Since its inception in 2011, NAART has been committed to fostering cross-disciplinary connections in the study of non-associative structures.

Author: Murray R. Bremner
Publisher: CRC Press
ISBN: 1482248573
Category : Mathematics
Languages : en
Pages : 382

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Book Description
This book presents a systematic treatment of Grobner bases in several contexts. The book builds up to the theory of Grobner bases for operads due to the second author and Khoroshkin as well as various applications of the corresponding diamond lemmas in algebra. Throughout the book, both the mathematical theory and computational methods are emphasized and numerous algorithms, examples, and exercises are provided to clarify and illustrate the concrete meaning of abstract theory.

Author: Pierre Cartier
Publisher: Springer Nature
ISBN: 3030778452
Category : Mathematics
Languages : en
Pages : 277

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Book Description
This book is dedicated to the structure and combinatorics of classical Hopf algebras. Its main focus is on commutative and cocommutative Hopf algebras, such as algebras of representative functions on groups and enveloping algebras of Lie algebras, as explored in the works of Borel, Cartier, Hopf and others in the 1940s and 50s. The modern and systematic treatment uses the approach of natural operations, illuminating the structure of Hopf algebras by means of their endomorphisms and their combinatorics. Emphasizing notions such as pseudo-coproducts, characteristic endomorphisms, descent algebras and Lie idempotents, the text also covers the important case of enveloping algebras of pre-Lie algebras. A wide range of applications are surveyed, highlighting the main ideas and fundamental results. Suitable as a textbook for masters or doctoral level programs, this book will be of interest to algebraists and anyone working in one of the fields of application of Hopf algebras.

Author: Nicolás Andruskiewitsch
Publisher: American Mathematical Soc.
ISBN: 1470456249
Category : Education
Languages : en
Pages : 359

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Book Description
The articles highlight the latest advances and further research directions in a variety of subjects related to tensor categories and Hopf algebras. Primary topics discussed in the text include the classification of Hopf algebras, structures and actions of Hopf algebras, algebraic supergroups, representations of quantum groups, quasi-quantum groups, algebras in tensor categories, and the construction method of fusion categories.

Author: Jean-Louis Loday
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 136

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Book Description
This book introduces the notion of generalized bialgebra, which includes the classical notion of bialgebra (Hopf algebra) and many others, among them the tensor algebra equipped with the deconcatenation as coproduct. The author proves that, under some mild conditions, a connected generalized bialgebra is completely determined by its primitive part. This structure theorem extends the classical Poincare-Birkhoff-Witt theorem and Cartier-Milnor-Moore theorem, valid for cocommutative bialgebras, to a large class of generalized bialgebras. Technically, the author works in the theory of operads which allows him to state his main theorem and permits him to give it a conceptual proof. A generalized bialgebra type is determined by two operads: one for the coalgebra structure $\mathcal{C}$ and one for the algebra structure $\mathcal{A}$. There is also a compatibility relation relating the two. Under some conditions, the primitive part of such a generalized bialgebra is an algebra over some sub-operad of $\mathcal{A}$, denoted $\mathcal{P}$ . The structure theorem gives conditions under which a connected generalized bialgebra is cofree (as a connected $\mathcal{C}$-coalgebra) and can be reconstructed out of its primitive part by means of an enveloping functor from $\mathcal{P}$-algebras to $\mathcal{A}$-algebras. The classical case is $(\mathcal {C, A, P})=(Com, As, Lie)$. This structure theorem unifies several results, generalizing the PBW and the CMM theorems, scattered in the literature. The author treats many explicit examples and suggests a few conjectures.