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 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. Titles in this series are co-published with the Centre de Recherches Mathematiques.|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 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. Titles in this series are co-published with the Centre de Recherches Mathematiques.

Monoidal Functors, Species, and Hopf Algebras

We classify finite-dimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elementsG.A/ is abelian such that all prime divisors of the order of G.A/ are> 7. Since these Hopf algebras turn out to be deformations of a natural class of generalized small quantum groups, our result can be read as an axiomatic description of generalized small quantum groups.

http://annals.math.princeton.edu/wp-content/uploads/annals-v171-n1-p07-p.pdf

On the classification of finite-dimensional pointed Hopf algebras

Finite Quantum Groups and Cartan Matrices

Classification of arithmetic root systems

The theory of Nichols algebras of diagonal type is known to be closely related to that of semi-simple Lie algebras. In this paper the connection between both theories is made closer. For any Nichols algebra of diagonal type invertible transformations are introduced, which remind one of the action of the Weyl group on the root system associated to a semi-simple Lie algebra. They give rise to the definition of a groupoid. As an application an alternative proof of classification results of Rosso, Andruskiewitsch, and Schneider is obtained without using any technical assumptions on the braiding.

The Weyl groupoid of a Nichols algebra of diagonal type

Let U

 q
 + be the “upper triangular part” of the quantized enveloping algebra associated with a symetrizable Cartan matrix We show that U

 q
 + is isomorphic (as a Hopf algebra) to the subalgebra generated by elements of degree 0 and 1 of the cotensor Hopf algebra associated with a suitable Hopf bimodule on the group algebra of Z

 n
  This method gives supersymetric as well as multiparametric versions of U

 q
 + in a uniform way (for a suitable choice of the Hopf bimodule) We give a classification result about the Hopf algebras which can be obtained in this way, under a reasonable growth condition We also show how the general formalism allows to reconstruct higher rank quantized enveloping algebras from U

 q
 sl(2) and a suitable irreducible finite dimensional representation

Quantum groups and quantum shuffles

Given a braided vector space \(\left( {V,\sigma } \right)\) , we show that iterated integrals of operator-valued functions satisfying a certain exchange relation give rise to representations of the quantum shuffle algebra built on \(\left( {V,\sigma } \right)\). Using the quantum shuffle construction of the 'upper triangular part' \(U_q n_{\text{ + }}\) of a quantum shuffle, this provides a simple proof of the result of Bouwknegt, MacCarthy and Pilch saying that integrals of vertex operators acting on certain Fock modules give rise to representations of \(U_q n_{\text{ + }}\).

Marc Rosso

Papers

Quantum groups and quantum shuffles

Integrals of Vertex Operators and uantum Shuffles