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Monoidal Functors, Species and Hopf Algebras (Crm Monograph Series)

Swapneel Mahajan Marcelo Aguiar

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Description for Monoidal Functors, Species and Hopf Algebras (Crm Monograph Series) Hardcover. Integrating ideas from category theory, algebra and combinatorics, this monograph is organised into three parts: category theory; combinatorics and geometry; and an examination of how ideas in Parts I and II lead to a unified approach to Hopf algebras. Series: CRM Monograph Series. Num Pages: 784 pages, Illustrations. BIC Classification: PBD. Category: (P) Professional & Vocational. Dimension: 229 x 152. .
This research monograph integrates ideas from category theory, algebra and combinatorics. It is organised in three parts. Part I belongs to the realm of category theory. It reviews some of the foundational work of Bénabou, 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 ... Read morestudy 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. Titles in this series are co-published with the Centre de Recherches Mathématiques.|This research monograph integrates ideas from category theory, algebra and combinatorics. It is organised in three parts. Part I belongs to the realm of category theory. It reviews some of the foundational work of Bénabou, 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. Titles in this series are co-published with the Centre de Recherches Mathématiques. Show Less

Product Details

Format

Hardback

Publication date

2010

Publisher

American Mathematical Society

Condition

New

Series

CRM Monograph Series

Number of Pages

784

Place of Publication

Providence, United States

ISBN

9780821847763

SKU

V9780821847763

Shipping Time

Usually ships in 7 to 11 working days

Ref

99-1

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Monoidal Functors, Species and Hopf Algebras (Crm Monograph Series)

Swapneel Mahajan Marcelo Aguiar

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