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Hopf 2-cocycles of type $A_2$

José Ignacio Sánchez

TL;DR

The paper advances the classification of finite-dimensional pointed Hopf algebras of diagonal Cartan type A2 by explicitly computing Hopf 2-cocycles that realize liftings as cocycle deformations. It develops a general setting to obtain a section γ and a braided Hopf 2-cocycle σ, and applies it to Cartan type A2 with q a primitive root of unity, separating generic liftings (Serre relations preserved) from atypical liftings (Serre relations deformed at N=3). It shows that many liftings cannot be obtained as exponentials of Hochschild 2-cocycles, and provides complete, explicit formulas for σ in both the generic and atypical cases, together with GAP code to reproduce the computations. The results complete the Hopf cocycle description for Cartan type A2 and lay a foundation for extending to higher rank Cartan types like Aθ, enhancing the cocycle-based lifting method for pointed Hopf algebras. The work thus deepens understanding of deformation patterns, purity phenomena, and computational tools in the Hopf-algebraic lifting program.

Abstract

We compute the Hopf 2-cocycles involved in the classification of pointed Hopf algebras of diagonal type $A_2$. When the quantum Serre relations are deformed, we characterize those cocycles that can be recovered from Hochschild cohomology, via exponentiation. We identify some hypotheses that allow us to present general formulas that apply in our setting.

Hopf 2-cocycles of type $A_2$

TL;DR

The paper advances the classification of finite-dimensional pointed Hopf algebras of diagonal Cartan type A2 by explicitly computing Hopf 2-cocycles that realize liftings as cocycle deformations. It develops a general setting to obtain a section γ and a braided Hopf 2-cocycle σ, and applies it to Cartan type A2 with q a primitive root of unity, separating generic liftings (Serre relations preserved) from atypical liftings (Serre relations deformed at N=3). It shows that many liftings cannot be obtained as exponentials of Hochschild 2-cocycles, and provides complete, explicit formulas for σ in both the generic and atypical cases, together with GAP code to reproduce the computations. The results complete the Hopf cocycle description for Cartan type A2 and lay a foundation for extending to higher rank Cartan types like Aθ, enhancing the cocycle-based lifting method for pointed Hopf algebras. The work thus deepens understanding of deformation patterns, purity phenomena, and computational tools in the Hopf-algebraic lifting program.

Abstract

We compute the Hopf 2-cocycles involved in the classification of pointed Hopf algebras of diagonal type . When the quantum Serre relations are deformed, we characterize those cocycles that can be recovered from Hochschild cohomology, via exponentiation. We identify some hypotheses that allow us to present general formulas that apply in our setting.

Paper Structure

This paper contains 24 sections, 23 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Some advances
  3. Results
  4. Preliminaries
  5. Cleft objects and Hopf 2-cocycles
  6. Cocycle deformations
  7. Exponential cocycles
  8. Nichols algebras and braided cocycles
  9. Braided vector spaces of diagonal type
  10. Lifting of type $A_2$
  11. A good setting
  12. Hopf algebras of Cartan type $A_2$: generic liftings
  13. The coproduct
  14. The section
  15. Explicit computations in $\mathcal{E}_{\boldsymbol\lambda}$
  16. ...and 9 more sections

Key Result

Lemma 2.3

AAG Let $\mathfrak{q}$, $\mathfrak{B}_{\mathfrak{q}}$ and $\mathcal{E}_{\boldsymbol\lambda}$ be as above. Hence, $\lambda_{112}$ or $\lambda_{122}$ may be nonzero if and only if $\mathfrak{q}=\left(\right)$, $q\in\mathbb{G}_3'$. Thus:

Theorems & Definitions (54)

  • Example 1.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • Example 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 44 more