Nichols algebras with standard braiding

Angiono, Iván

doi:10.2140/ant.2009.3.35

Cited by 41 publications

(54 citation statements)

References 19 publications

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“…Other Nichols algebras of rank 2 were presented in [27,45,53]. Standard type appeared in [22]; this paper contains a self-contained proof of the defining relations of the Nichols algebras of Cartan type at a generic parameter, i.e. the sufficiency of the quantum Serre relations.…”

Section: Attributionmentioning

confidence: 95%

On finite dimensional Nichols algebras of diagonal type

2017

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This is a survey on Nichols algebras of diagonal type with finite dimension, or more generally with arithmetic root system. The knowledge of these algebras is the cornerstone of the classification program of pointed Hopf algebras with finite dimension, or finite Gelfand-Kirillov dimension; and their structure should be indispensable for the understanding of the representation theory, the computation of the various cohomologies, and many other aspects of finite dimensional pointed Hopf algebras. These Nichols algebras were classified in Heckenberger (Adv Math 220:59-124, 2009) as a notable application of the notions of Weyl groupoid and generalized root system (Heckenberger in Invent Math 164: 175-188, 2006; Heckenberger and Yamane in Math Z 259:255-276, 2008). In the first part of this monograph, we give an overview of the theory of Nichols algebras of diagonal type. This includes a discussion of the notion of generalized root system and its appearance in the contexts of Nichols algebras of diagonal type and (modular) Lie superalgebras. In the second and third part, we describe for each Nichols algebra in the list of Heckenberger (2009) Communicated by Efim Zelmanov.

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Section: Attributionmentioning

confidence: 95%

On finite dimensional Nichols algebras of diagonal type

2017

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“…For several items, I also add the presentations known from [15] and explicitly borrowed from [22], including the case number in [22]. From the Nichols-algebra data, I move toward conformal field theory by analyzing the conditions on the screening momenta.…”

Section: Theorem the Central Charge Of The Virasoro Algebra Centralimentioning

confidence: 99%

“…6 The Wakimoto bosonization [42] yields two essentially different three-boson realizations of sℓ(2)-the "symmetric" and the "nonsymmetric" ones, respectively centralizing two fermionic screenings and one bosonic plus one fermionic screening. The names refer to the " j + ↔ j − symmetric" structure of (15) and the "asymmetric" structure of (13). Somewhat broader, the "variously symmetric" realizations are discussed in [40].…”

Section: (57(4)mentioning

confidence: 99%

“…Is any finitedimensional Nichols algebra with diagonal braiding an algebra of screenings in some conformal model? This is a fascinating problem, especially considering the recent remarkable development in the theory of Nichols algebras-originally a "technicality" in Andruskiewitsch and Schneider's program of classification of pointed Hopf algebras, which has grown into a beautiful theory in and of itself (in addition to the papers cited above and the references therein, also see [10,11,12,8,13,14,15,16,17]). Diagonal braiding is assumed in what follows.…”

Section: Introductionmentioning

confidence: 99%

“…All of these were listed by Heckenberger [21] (the general classification, for any rank, was achieved in [7] and was reproduced in a different and independent way in [15,16,17]). These notes are in fact a compilation of the original Heckenberger's list with explicit results on the presentation of some Nichols algebras (obtained in [15] for the standard type and in [22] in several nonstandard cases), and with several CFT constructions added. The extended algebra of a logarithmic model-the octuplet algebra extending the W 3 algebra-is offered in only one case; the other CFT constructions are merely a starting point for finding extended algebras.…”

Section: Introductionmentioning

confidence: 99%

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Virasoro Central Charges for Nichols Algebras

Semikhatov

2014

Mathematical Lectures From Peking University

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ABSTRACT. A Virasoro central charge can be associated with each Nichols algebra with diagonal braiding in a way that is invariant under the Weyl groupoid action. The central charge takes very suggestive values for some items in Heckenberger's list of rank-2 Nichols algebras. In particular, this might be viewed as an indication of the existence of reasonable logarithmic extensions of W 3 ≡ WA 2 , WB 2 , and WG 2 models of conformal field theory. In the W 3 case, the construction of an octuplet extended algebra-a counterpart of the triplet (1, p) algebra-is outlined.

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