
TL;DR
This paper introduces the distinguished pre-Nichols algebra, an intermediate algebra between the tensor algebra and Nichols algebra, and explores its properties and applications to Hopf algebras.
Contribution
It formally defines the distinguished pre-Nichols algebra and demonstrates its role in constructing new Noetherian pointed Hopf algebras with finite Gelfand-Kirillov dimension.
Findings
Provides a presentation of the algebra with fewer relations
Shows the algebra yields new Hopf algebra examples
Generalizes results on quantum groups at roots of unity
Abstract
We formally define and study the distinguished pre-Nichols algebra of a braided vector space of diagonal type with finite-dimensional Nichols algebra . The algebra is presented by fewer relations than , so it is intermediate between the tensor algebra and . Prominent examples of distinguished pre-Nichols algebras are the positive parts of quantized enveloping (super)algebras and their multiparametric versions. We prove that these algebras give rise to new examples of Noetherian pointed Hopf algebras of finite Gelfand-Kirillov dimension. We investigate the kernel (in the sense of Hopf algebras) of the projection from to , generalizing results of De Concini and Procesi on quantum groups at roots of unity.
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