On a family of liftings of the Jordan plane

Abstract: We study a family of Hopf algebras arising as liftings of the Jordan plane over the infinite cyclic group. We determine their centres, prime and primitive spectra, and automorphism groups. We show that every prime ideal is completely prime and that every nonzero ideal intersects the centre nontrivially. We construct explicit simple modules corresponding to all primitive ideals and classify the finite-dimensional simple modules. Finally, we prove that these Hopf algebras satisfy the Dixmier--Moeglin equivalence.

• The paper investigates the structure and representation theory of Hopf algebras 𝔘(λ) as liftings of the Jordan plane, establishing the Dixmier–Moeglin equivalence for all members of the family.
• It details the classification of prime and primitive ideals by analyzing central elements and module constructions in both bosonization (λ=0) and nontrivial deformations (λ≠0).
• The analysis further elucidates the automorphism groups and explicit module classifications, offering deep insights into deformation theory and noncommutative algebraic structures.

Structure and Representation Theory of a Family of Liftings of the Jordan Plane

Introduction and Context

The study concerns a distinguished family of Hopf algebras $\mathfrak{U}(\lambda)$, constructed as liftings of the Jordan plane over the group algebra $\mathbb{K} G$ (where $G$ is the infinite cyclic group), following the lifting method of Andruskiewitsch and Schneider. The Jordan plane, $$\mathbb{K}\langle x, y \mid yx-xy = -\frac{1}{2} x^2 \rangle$$, is an Artin–Schelter regular algebra of global dimension two and a prominent example of a Nichols algebra of non-diagonal type with finite Gelfand–Kirillov dimension.

Building upon earlier classification results on pointed Hopf algebras and the explicit structure of liftings for the Jordan and super Jordan planes, the paper investigates both the algebraic and the representation-theoretic aspects of $\mathfrak{U}(\lambda)$. The analysis covers the cases $\lambda=0$ (bosonization) and $\lambda \ne 0$ (nontrivial cocycle deformations) in detail, examining the centers, prime and primitive spectra, automorphism groups, and simple module classifications. The Dixmier–Moeglin equivalence is established for all members of the family.

Construction and Algebraic Properties of $\mathfrak{U}(\lambda)$

The Hopf algebras $\mathfrak{U}(\lambda)$ are defined by generators $g^{\pm1}$, $\mathbb{K} G$0, and $\mathbb{K} G$1, with relations: $\mathbb{K} G$2 with the Hopf structure inherited naturally from the group algebra and the Yetter–Drinfeld module structure. The family parameter $\mathbb{K} G$3 realizes a cocycle deformation, with $\mathbb{K} G$4 corresponding to the bosonization and $\mathbb{K} G$5 to genuinely nontrivial deformations.

A key structural feature is the existence of a central polynomial

$\mathbb{K} G$6

which generates the center in both the bosonization and the liftings ($\mathbb{K} G$7). The algebra admits a Noetherian domain structure and Gelfand–Kirillov dimension three.

Prime Spectrum, Center, and Primitive Ideals

Case $\mathbb{K} G$8: Bosonization

The central element $\mathbb{K} G$9 reduces to $G$0. Prime ideals are described precisely:

• The zero ideal $G$1
• Ideals $G$2 for $G$3
• Ideals of the form $G$4 with $G$5 prime in $G$6

All prime ideals are completely prime; every nonzero ideal intersects the center nontrivially. Primitive ideals coincide with maximal ideals and are exhausted by $G$7 and $G$8 for $G$9, $$\mathbb{K}\langle x, y \mid yx-xy = -\frac{1}{2} x^2 \rangle$$0: every primitive ideal is maximal. Notably, the description of infinite-dimensional simple modules reduces to the study of the simple modules of the first Weyl algebra localized at $$\mathbb{K}\langle x, y \mid yx-xy = -\frac{1}{2} x^2 \rangle$$1.

Case $$\mathbb{K}\langle x, y \mid yx-xy = -\frac{1}{2} x^2 \rangle$$2: Nontrivial Liftings

The central structure remains ($$\mathbb{K}\langle x, y \mid yx-xy = -\frac{1}{2} x^2 \rangle$$3), but the structure of the quotients by central elements changes dramatically. The analysis of $$\mathbb{K}\langle x, y \mid yx-xy = -\frac{1}{2} x^2 \rangle$$4 divides into two regimes:

• Generic ($$\mathbb{K}\langle x, y \mid yx-xy = -\frac{1}{2} x^2 \rangle$$5): The quotient is a simple domain.
• Exceptional ($$\mathbb{K}\langle x, y \mid yx-xy = -\frac{1}{2} x^2 \rangle$$6): The quotient fails to be simple and is not a domain; it decomposes via normal elements $$\mathbb{K}\langle x, y \mid yx-xy = -\frac{1}{2} x^2 \rangle$$7 and $$\mathbb{K}\langle x, y \mid yx-xy = -\frac{1}{2} x^2 \rangle$$8.

Prime ideals fall into the following types:

• $$\mathbb{K}\langle x, y \mid yx-xy = -\frac{1}{2} x^2 \rangle$$9 (trivial)
• Height one primes: $\mathfrak{U}(\lambda)$0, $\mathfrak{U}(\lambda)$1, $\mathfrak{U}(\lambda)$2, $\mathfrak{U}(\lambda)$3
• Maximal ideals: $\mathfrak{U}(\lambda)$4 ($\mathfrak{U}(\lambda)$5), and $\mathfrak{U}(\lambda)$6
• Others arise by appropriate specialization

All prime ideals are completely prime. Every nonzero ideal still intersects the center nontrivially. However, not all primitive ideals are maximal—height-one primes at the non-semistable points in the spectrum become primitive (realized as annihilators of explicit simple modules), but only the central quotients with generic parameter are maximal.

Classification of Simple Modules

Finite-dimensional Simples

In both cases, all finite-dimensional simple modules are one-dimensional, corresponding to annihilators $\mathfrak{U}(\lambda)$7. This rigidity is a direct consequence of the structure of the central element and the relations.

Infinite-dimensional Simples

For generic $\mathfrak{U}(\lambda)$8, simple modules with annihilator $\mathfrak{U}(\lambda)$9 are constructed as cyclic modules over the simple domain $\lambda=0$0, with explicit realization via quotient by specific normal elements (e.g., $\lambda=0$1). The module-theoretic structure is finely articulated; for the exceptional non-generic cases, the analysis relies on the structure of the corresponding non-simple/non-domain quotients.

Height-one primitive ideals $\lambda=0$2, $\lambda=0$3 are realized as annihilators of explicitly constructed simple modules, with module structure deduced from the skew polynomial extension description.

Automorphism Groups

The automorphism group of $\lambda=0$4 is the semidirect product $\lambda=0$5, where $\lambda=0$6, $\lambda=0$7 is order two, and $\lambda=0$8 is as described. In the nontrivial liftings $\lambda=0$9, the automorphism group reduces to $\lambda \ne 0$0 with $\lambda \ne 0$1, reflecting increased structural rigidity due to deformation.

Dixmier--Moeglin Equivalence

A central result is that all Hopf algebras $\lambda \ne 0$2, for any $\lambda \ne 0$3, satisfy the Dixmier–Moeglin equivalence: the classes of locally closed, primitive, and rational prime ideals coincide. This is established via explicit computation of the spectra and application of the Nullstellensatz. For $\lambda \ne 0$4, only the primitivity of maximal ideals is observed, while for $\lambda \ne 0$5, the non-maximal, non-rational primes are explicitly classified.

Implications and Future Prospects

The results provide a comprehensive understanding of the structure and representation theory of a central family of noncommutative Hopf algebras closely related to Nichols algebras of non-diagonal type. The explicit description of all prime and primitive spectra, simple modules, and automorphism groups advances the foundational understanding of pointed Hopf algebras built from nontrivial Nichols algebras.

From a theoretical standpoint, the work demonstrates effective techniques for analyzing noncommutative algebras emerging as cocycle deformations of quantum or Artin–Schelter regular algebras, particularly in identifying centers, prime spectra, and explicit module-theoretic realizations. It reinforces and extends the general expectation that such finite GK-dimensional, affine, Noetherian Hopf algebras satisfy the Dixmier–Moeglin equivalence.

Practically, the explicit module classification afforded by the approach can serve as a template when analyzing module categories over other quantum or braided Hopf algebras. The explicit nature of the constructions suggests potential for combinatorial or computational approaches to module theory in wider settings, including applications to noncommutative algebraic geometry, deformation theory, and quantum group symmetry.

Looking forward, similar analyses may be adapted to other classes of liftings, including those over finite, nilpotent, or more general base groups, and to Nichols algebras of higher complexity or different types. The techniques and results may also inform the study of homological invariants and connections to noncommutative projective geometry.

Conclusion

The paper offers a thorough and explicit analysis of the structure, spectra, automorphism groups, and simple modules of a family of Hopf algebra liftings of the Jordan plane. All prime ideals are completely prime, every nonzero ideal intersects the center nontrivially, and the algebras satisfy the Dixmier–Moeglin equivalence. The results deepen understanding of the interplay between cocycle deformations, central structure, and representation theory in noncommutative Hopf algebras.