A geometric realization of liftings of Cartan type

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 $B_2$ computed by Beattie, Dascalescu and Raianu, and of type $B_3$ computed by the authors, for Drinfeld-Jimbo type braidings. Moreover, we present all liftings of type $B_θ$ and $D_θ$, for $θ\geq 2$, giving in this way new explicit infinite families of liftings for Drinfeld-Jimbo type braidings.

💡 Research Summary

The paper addresses the long‑standing problem of explicitly describing all liftings of bosonizations of Nichols algebras of diagonal braided vector spaces of Cartan type. In the classical “lifting method” of Andruskiewitsch and Schneider, one starts with a finite‑dimensional pointed Hopf algebra (A) whose coradical is a group algebra (\mathbb{C}\Gamma). The associated graded Hopf algebra (\operatorname{gr}A) is isomorphic to the bosonization (\mathcal B(V)#\mathbb{C}\Gamma), where (\mathcal B(V)) is a finite‑dimensional Nichols algebra generated by a braided vector space (V). The classification of liftings then reduces to solving deformation equations for the power‑root relations (x_{N\alpha}=r_{\alpha}(\mu)) and the quantum Serre relations. In the original works these equations were handled by heavy recursive calculations, which have only been carried out for a few low‑rank cases (type (A_n), (B_2), (B_3)).

The authors propose a completely different strategy that eliminates the need for explicit recursive computations. They restrict to Drinfeld‑Jimbo type braidings, i.e. braiding matrices of the form (q_{ij}=q^{d_i a_{ij}}) where (C=(a_{ij})) is a symmetrizable Cartan matrix and (q) is a primitive root of unity of odd order (N) coprime to 210. For a chosen lattice (M) satisfying (Q\subseteq M\subseteq P) (with (Q) the root lattice and (P) the weight lattice of the corresponding Lie algebra (\mathfrak g)), they construct the quantized universal enveloping algebra (U_q(\mathfrak g)) and its Borel subalgebra (U_q(\mathfrak b^+)).

Two structural facts are central:

1. Central Hopf subalgebra (Z_{\ge M}\subset U_q(\mathfrak b^+)) generated by the elements (K_{Nm}) and (E_{N\alpha}) (for all (m\in M) and positive roots (\alpha)). Under the quantum Frobenius map this subalgebra corresponds to the classical function algebra (\mathcal O(B^+_M)) on the Borel subgroup of the algebraic group (G_M) associated with (\mathfrak g) and the lattice (M).

2. Exact sequence
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