Topic
Hopf algebra
About: Hopf algebra is a research topic. Over the lifetime, 7254 publications have been published within this topic receiving 148806 citations.
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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
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01 Jan 1993TL;DR: In this paper, the authors define integrals and semisimplicity of subalgebras, and define a set of properties of finite-dimensional Hopf algebra and smash products.
Abstract: Definitions and examples Integrals and semisimplicity Freeness over subalgebras Action of finite-dimensional Hopf algebras and smash products Coradicals and filtrations Inner actions Crossed products Galois extensions Duality New constructions from quantum groups Some quantum groups.
2,659 citations
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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
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01 Apr 2000TL;DR: In this paper, the authors define Hopf algebras as "quasitriangular Hopf-algebraes" and introduce matrix quantum groups and bicrossproduct hopf alges.
Abstract: Introduction 1. Definition of Hopf algebras 2. Quasitriangular Hopf algebras 3. Quantum enveloping algebras 4. Matrix quantum groups 5. Quantum random walks and combinatorics 6. Bicrossproduct Hopf algebras 7. Quantum double and double cross products 8. Lie bialgebras and Poisson brackets 9. Representation theory 10. Braided groups and q-deformation References Symbols Indexes.
2,219 citations
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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