Abstract
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) the following basic information: the generalized root system; its label in terms of Lie theory; the defining relations found in Angiono (J Eur Math Soc 17:2643–2671, 2015; J Reine Angew Math 683:189–251, 2013); the PBW-basis; the dimension or the Gelfand–Kirillov dimension; the associated Lie algebra as in Andruskiewitsch et al. (Bull Belg Math Soc Simon Stevin 24(1):15–34, 2017). Indeed the second part deals with Nichols algebras related to Lie algebras and superalgebras in arbitrary characteristic, while the third contains the information on Nichols algebras related to Lie algebras and superalgebras only in small characteristic, and the few examples yet unidentified in terms of Lie theory.
Highlights
Nichols algebras play a fundamental role in the classification of pointed Hopf algebras, see [20] and Sect. 1.7 below
We summarize the relation between generalized root systems and Nichols algebras: Remark 2.31 (a) The classification of the arithmetic Nichols algebras of diagonal type was achieved in [46], as said
In the celebrated article [46], the classification of the Nichols algebras of diagonal type with arithmetic root system was presented in the form of several tables
Summary
3. The adequate setting for braided vector spaces is that of braided tensor categories [56]; there is a natural notion of Hopf algebra in such categories. Pre-Nichols algebras of the braided vector space (V, c) [26,69]; these are graded connected. Post-Nichols algebras of the braided vector space (V, c) [7]; these are graded connected. The classification of the braided vector spaces (V, c) of diagonal type with finite-dimensional B(V ) was obtained in [46]. In Parts II and III we give the list of all finite-dimensional Nichols algebra of diagonal type (with connected Dynkin diagram) classified in [46] and for each of them, its fundamental information.
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