A geometric realization of liftings of Cartan type

2025/12/18 by D. Bagio, Bagio, D., G. A. García +3
Mathematics · #16T05 #16T20 #17B37 #20G42 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.2512.17000

openalex publication_date 2025/12/18 · openalex created_date 2025/12/23 · openalex updated_date 2026/07/28

Abstract

We introduce a novel approach to compute liftings of bosonizations of Nichols algebras of diagonal braided vector spaces of Cartan type which replaces heavy computations with structural maps related to quantum groups. This provides an answer to a question posed by Andruskiewitsch and Schneider, who classified finite-dimensional complex pointed Hopf algebras over finite abelian groups whose order is coprime with 210. As application and in order to give not-too-technical examples, we recover with our method the liftings of type A_n computed by Andruskiewitsch and Schneider, of type B₂ computed by Beattie, Dascalescu and Raianu, and of type B₃ computed by the authors, for Drinfeld-Jimbo type braidings. Moreover, we present all liftings of type B_θ and D_θ, for θ≥ 2, giving in this way new explicit infinite families of liftings for Drinfeld-Jimbo type braidings.

Citations

• Pointed Hopf algebras revisited, with a view from tensor categories
• Liftings of Nichols algebras of type B₃
• Twisted deformations vs. cocycle deformations for quantum groups
• Multiparameter quantum groups at roots of unity
• Liftings of Nichols algebras of diagonal type II. All liftings are\n cocycle deformations
• Distinguished Pre-Nichols algebras
• Lifting via cocycle deformation
• On Nichols algebras of diagonal type
• Drinfel'd doubles and Shapovalov determinants
• Abelian and non-abelian second cohomologies of quantized enveloping algebras
• Classification of arithmetic root systems
• Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of order p³

Related

• Liftings of Nichols algebras of diagonal type I. Cartan type A
• On Nichols algebras of low dimension
• Lifting of Nichols Algebras of Type B₂, with an Appendix: A\n generalization of the q-binomial theorem
• Liftings of Nichols algebras of type B₃
• Pointed Hopf algebras over non-abelian groups with non-simple standard braidings