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dc.contributor.author
Andruskiewitsch, Nicolas
dc.contributor.other
Cardona, Alexander
dc.contributor.other
Morales, Pedro
dc.contributor.other
Ocampo, Hernan
dc.contributor.other
Paycha, Sylvie
dc.contributor.other
Reyes Lega, Andres Fernando
dc.date.available
2021-07-08T19:17:27Z
dc.date.issued
2018
dc.identifier.citation
Andruskiewitsch, Nicolas; An introduction to Nichols Algebras; Springer International Publishing; 2018; 135-195
dc.identifier.isbn
978-3-319-65426-3
dc.identifier.uri
http://hdl.handle.net/11336/135766
dc.description.abstract
Nichols algebras, Hopf algebras in braided categories with distinguishedproperties, were discovered several times. They appeared for the first time in the thesis of W. Nichols [72], aimed to construct new examples of Hopf algebras. In this same paper, the small quantum group uq (sl3), with q a primitive cubic root of one, was introduced. Independently they arose in the paper [84] by Woronowicz as the invariant part of his non-commutative differential calculus. Later there were two unrelated attempts to characterize abstractly the positive part U^+_q(g) of the quantized enveloping algebra of a simple finite-dimensional Lie algebra g at a generic parameter q. First, Lusztig showed in [64] that U^+_q(g) can be defined through the radical of a suitable invariant bilinear form. Second, Rosso interpreted U^+_q(g) in [74,75] via quantum shuffles. These two viewpoints were conciliated later, as alternative definitions of the same notion of Nichols algebra. Other early appearances of Nichols algebras are in [65, 77]. As observed in [17, 18], Nichols algebras are basic invariants of pointed Hopf algebras, their study being crucial in the classification programof Hopf algebras; see also [10]. More recently, they are the subject of an intriguing proposal in Conformal Field Theory [79]. This is an introduction from scratch to the notion of Nichols algebra. I was invited to give a mini-course of two lessons, 90 min each, at the Geometric, Algebraic and Topological Methods for Quantum Field Theory, Villa de Leyva, Colombia, in July 2015. The theme was Nichols algebras that requires several preliminaries and some experience to be appreciated; a selection of the ideas to be presented was necessary. These notes intend to preserve the spirit of the course, discussing some motivational background material in Sect. 4.1, then dealing with braided vector spaces and braided tensor categories in Sect. 4.2, arriving at last to the definition and main calculation tools of Nichols algebras in Sect. 4.3. I hope that the various examples and exercises scattered through the text would serve the reader to absorb the beautiful concept of Nichols algebra and its many facets. Section 4.4 is a survey of the main examples of, and results on, Nichols algebras that I am aware of; here the pace is faster and the precise formulation of some statementsis referred to the literature. I apologize in advance for any possible omission. This section has intersection with, and is an update of, the surveys [1, 2, 19], to which I refer for further information.
dc.format
application/pdf
dc.language.iso
eng
dc.publisher
Springer International Publishing
dc.rights
info:eu-repo/semantics/restrictedAccess
dc.rights.uri
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.subject
Hopf algebras
dc.subject
Nichols algebras
dc.subject
Quantum Yang-Baxter equation
dc.subject.classification
Matemática Pura
dc.subject.classification
Matemáticas
dc.subject.classification
CIENCIAS NATURALES Y EXACTAS
dc.title
An introduction to Nichols Algebras
dc.type
info:eu-repo/semantics/publishedVersion
dc.type
info:eu-repo/semantics/bookPart
dc.type
info:ar-repo/semantics/parte de libro
dc.date.updated
2021-04-30T19:08:32Z
dc.journal.pagination
135-195
dc.journal.pais
Alemania
dc.description.fil
Fil: Andruskiewitsch, Nicolas. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
dc.relation.alternativeid
info:eu-repo/semantics/altIdentifier/url/https://www.springer.com/la/book/9783319654263
dc.conicet.paginas
341
dc.source.titulo
Quantization, geometry and noncommutative structures in mathematics and physics
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