Quantum groups and quantum shuffles
TL;DR: In this paper, the upper triangular part of the quantized enveloping algebra associated with a symetrizable Cartan matrix is shown to be isomorphic to the subalgebra generated by elements of degree 0 and 1 of the cotensor Hopf algebra associated by a suitable Hopf bimodule on the group algebra.
Abstract: 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
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20 Oct 2010
TL;DR: Benabou, Eilenberg, Kelly and Mac Lane as discussed by the authors proposed the notion of a bilax monoidal functor which plays a central role in this work and showed how ideas in Parts I and II lead to a unified approach to Hopf algebras.
Abstract: 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.
326 citations
Cites methods from "Quantum groups and quantum shuffles..."
...Certain Q-Hopf algebras known as Nichols algebras of diagonal type are central to the construction by Lusztig and by Rosso of quantum enveloping algebras (quantum groups) [174, 226], and play a key role in the classification of pointed Hopf algebras by Andruskiewitsch and Schneider [16, 17, 18, 19]....
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TL;DR: In this paper, the authors 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.
Abstract: 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.
324 citations
TL;DR: In this paper, the authors consider a class of Hopf algebras having as an invariant a generalized Cartan matrix and prove that a Hopf algebra of this kind is finite-dimensional if and only if its generalized matrix is actually a finite Cartan (under some mild hypothesis).
281 citations
TL;DR: Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property They can also be considered as generalizations of ordinary root systems with rich structure and many new examples as discussed by the authors.
233 citations
Cites background from "Quantum groups and quantum shuffles..."
...However note that under some hypotheses all examples are deformations of the upper triangular part of a semisimple Lie algebra [20] [5]....
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...Nichols algebras can be described in many different ways [21], [2], [20]....
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TL;DR: In this paper, a connection between the theory of Nichols algebras and semi-simple Lie algesas is made closer, and 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.
Abstract: 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.
228 citations
Cites methods from "Quantum groups and quantum shuffles..."
...There exist classification results of Rosso [ 10, Theorem 21 ] and Andruskiewitsch and Schneider [1, Theorem 1.1] on Nichols algebras of Cartan type with finite Gel’fand–Kirillov dimension (F3) and finite dimension...
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...relate Nichols algebras to semisimple Lie algebras were proved for example by Rosso [ 10 ] and Andruskiewitsch and Schneider [1]....
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References
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Book•
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04 Nov 1994
TL;DR: In this paper, the authors introduce the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions and present the quantum groups attached to SL2 as well as the basic concepts of the Hopf algebras.
Abstract: Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups attached to SL2 as well as the basic concepts of the theory of Hopf algebras. Coverage also focuses on Hopf algebras that produce solutions of the Yang-Baxter equation and provides an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations.
5,966 citations
Book•
01 Jan 1994
TL;DR: In this paper, the Kac-Moody algebras and quasitriangular Hopf algesas were used to represent the universal R-matrix and the root of unity case.
Abstract: Introduction 1. Poisson-Lie groups and Lie bialgebras 2. Coboundary Poisson-Lie groups and the classical Yang-Baxter equation 3. Solutions of the classical Yang-Baxter equation 4. Quasitriangular Hopf algebras 5. Representations and quasitensor categories 6. Quantization of Lie bialgebras 7. Quantized function algebras 8. Structure of QUE algebras: the universal R-matrix 9. Specializations of QUE algebras 10. Representations of QUE algebras: the generic case 11. Representations of QUE algebras: the root of unity case 12. Infinite-dimensional quantum groups 13. Quantum harmonic analysis 14. Canonical bases 15. Quantum group invariants of knots and 3-manifolds 16. Quasi-Hopf algebras and the Knizhnik-Zamolodchikov equation Appendix. The Kac-Moody algebras.
2,637 citations
TL;DR: The structure theorem of Hopf algebras has been generalized by Borel, Leray, and others as discussed by the authors, and some new proofs of the classical theorems are given, as well as some new results.
Abstract: induced by the product M x M e M. The structure theorem of Hopf concerning such algebras has been generalized by Borel, Leray, and others. This paper gives a comprehensive treatment of Hopf algebras and some surrounding topics. New proofs of the classical theorems are given, as well as some new results. The paper is divided into eight sections with the following titles: 1. Algebras and modules. 2. Coalgebras and comodules. 3. Algebras, coalgebras, and duality. 4. Elementary properties of Hopf algebras. 5. Universal algebras of Lie algebras. 6. Lie algebras and restricted Lie algebras. 7. Some classical theorems. 8. Morphisms of connected coalgebras into connected algebras. The first four sections are introductory in nature. Section 5 shows that, over a field of characteristic zero, the category of graded Lie algebras is isomorphic with the category of primitively generated Hopf algebras. In ? 6, a similar result is obtained in the case of characteristic p # 0, but with graded Lie algebras replaced by graded restricted Lie algebras. Section 7 studies conditions when a Hopf algebra with commutative multiplication splits either as a tensor product of algebras with a single generator or a tensor product of
1,570 citations
TL;DR: In this paper, a general theory of non-commutative differential geometry on quantum groups is developed, where bicovariant bimodules as objects analogous to tensor bundles over Lie groups are studied.
Abstract: The paper deals with non-commutative differential geometry. The general theory of differential calculus on quantum groups is developed. Bicovariant bimodules as objects analogous to tensor bundles over Lie groups are studied. Tensor algebra and external algebra constructions are described. It is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra, so the external derivative of higher order differential forms is well defined and obeys the usual properties. The proper form of the Cartan Maurer formula is found. The vector space dual to the space of left-invariant differential forms is endowed with a bilinear operation playing the role of the Lie bracket (commutator). Generalized antisymmetry relation and Jacobi identity are proved.
1,248 citations
"Quantum groups and quantum shuffles..." refers background or methods in this paper
... W . Nichols shows that, taking tensor products over H, Hopf bimodules form a tensor category E. The structure of Hopf bimodules is clarified if one considers the subspaces of left or right coinvariants: M L af m2 M ; d LO mUa 1 mgand M R af m2 M ; d RO mUa m 1g. In fact, a classical result of Sweedler says that M is isomorphic, as left module and comodule, to the trivial one H M L , or, as a right ......
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...It is very remarkable that these cotensor constructions were already done by W . Nichols ([N]) more than 20 years ago! What was missing at that time was interest in braids, and to realize that for each n, the braid group on n strands naturally acts on the homogenous component of degree n of the cotensor Hopf algebra....
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...This Hopf algebra was introduced by W . Nichols under the name bialgebra (or Hopf algebra) of type one....
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Book•
01 Jan 1985TL;DR: Gelfand-Kirillov dimension of algebras Gelfand and Kirillov dimensions of related algesbras Localization Modules Graded and filtered algebraes and modules Almost commutative algebaes Weyl algebras Enveloping algebroes of solvable Lie algebeas Polynomial identitiy algeblas Growth of groups New developments Bibliography Index as discussed by the authors.
Abstract: Introduction Growth of algebras Gelfand-Kirillov dimension of algebras Gelfand-Kirillov dimension of related algebras Localization Modules Graded and filtered algebras and modules Almost commutative algebras Weyl algebras Enveloping algebras of solvable Lie algebras Polynomial identitiy algebras Growth of groups New developments Bibliography Index.
793 citations
"Quantum groups and quantum shuffles..." refers methods in this paper
...One can formulate these twisted bialgebras structures in a unifom way using a universal construction in the Braid category (cf [ Ka ]) made linear . Recall that objects in the braid category B are the ‘‘direct sums’’ of positive integers n, and that the morphisms MorOn; mU are 0, if n6a m, and the group algebra of Bn otherwise....
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