On Finite dimensional Nichols algebras of diagonal type

TL;DR

This survey reviews the classification, structure, and properties of finite-dimensional Nichols algebras of diagonal type, emphasizing their role in the theory of pointed Hopf algebras and Lie superalgebras.

Contribution

It provides a comprehensive overview of Nichols algebras of diagonal type, including classification, root systems, relations, and connections to Lie theory, expanding understanding in various characteristics.

Findings

Classification of Nichols algebras of diagonal type completed

Explicit descriptions of root systems and relations provided

Connections to Lie superalgebras in different characteristics established

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 [H-classif RS] as a notable application of the notions of Weyl groupoid and generalized root system [H-Weyl gpd,HY]. 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 [H-classif RS] the following basic information: the generalized root system; its label in terms of Lie theory; the defining relations found in [A-jems,A-presentation]; the PBW-basis; the dimension or the Gelfand-Kirillov dimension; the associated Lie algebra as in [AAR2]. 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.