Prime spectrum and representations of the super Jordan plane

Abstract: We study the ring-theoretic structure and representation theory of the super Jordan plane $\mathcal{J}$ over fields of characteristic different from $2$. We prove that $\mathcal{J}$ is prime and classify its prime, primitive, and maximal ideals. We determine its classical ring of quotients and classify the finite-dimensional simple modules, while relating infinite-dimensional simple modules to those of the first Weyl algebra. Our approach is based on showing that a localization of $\mathcal{J}$ is a matrix algebra over a localization of the first Weyl algebra.

• The paper establishes a complete description of the prime, primitive, and maximal ideal structures of the super Jordan plane through advanced localization techniques.
• It demonstrates a Morita equivalence by reducing the structure to a localization of the first Weyl algebra via an explicit Ore extension framework.
• In positive characteristic, the work characterizes the Azumaya property, PI degree, and finite simple modules, revealing novel ring-theoretic phenomena.

Prime Spectrum and Representations of the Super Jordan Plane

Introduction and Motivation

The super Jordan plane, denoted $\CJ$, is a Nichols algebra of central interest in the theory of quantum groups and in the classification of pointed Hopf algebras with finite GK-dimension. Unlike the classical Jordan plane, the super Jordan plane is not a domain and presents novel ring-theoretic and representation-theoretic phenomena. The algebra $\CJ$ is generated by $x$ and $y$ subject to $x^2=0$ and $y^2x - xy^2 - xyx = 0$, or equivalently, $x^2=0$ and $y s - s y - x s = 0$ with $s = xy+yx$.

This work achieves a full description of the prime, primitive, and maximal ideal structure of $\CJ$ over fields of characteristic different from $\CJ$0, and classifies its finite- and infinite-dimensional simple modules. The analysis utilizes a localization strategy, revealing that a distinguished localization of $\CJ$1 is Morita equivalent to a localization of the first Weyl algebra, thus demystifying the structure and representations of $\CJ$2 through well-understood properties of Weyl algebras and matrix rings.

Structure and Localization

$\CJ$3 is $\CJ$4-graded with PBW basis $\CJ$5, and has GK-dimension $\CJ$6. The subalgebra generated by $\CJ$7 and $\CJ$8 is isomorphic to the Jordan plane. Key to the investigation is the normal element $\CJ$9. Localization at powers of $x$0, denoted $x$1, allows $x$2 to be described as an Ore extension, and the algebra is shown to be isomorphic to $x$3 for a certain localization $x$4 of the first Weyl algebra.

Explicitly, if $x$5, then in $x$6 the relations $x$7 and $x$8 are established. The subalgebra $x$9, generated by $y$0 and $y$1, is a localization of the Weyl algebra, and the embedding $y$2 is constructed by defining explicit matrix units within $y$3.

This reduction has powerful consequences: it makes the prime spectrum, center, and module theory of $y$4 (and much of $y$5) accessible via standard results in the theory of matrix rings and Weyl algebras.

Ring-Theoretic Properties in Characteristic Zero

Working over an algebraically closed field $y$6 of characteristic zero, the paper establishes:

• The center of $y$7 and its localization $y$8 is $y$9.
• $x^2=0$0 is simple and hence prime.
• The classical ring of quotients $x^2=0$1 satisfies $x^2=0$2, where $x^2=0$3 is the Weyl skewfield.

The prime spectrum of $x^2=0$4 is completely described:

Prime Ideal Type	Structure
Zero ideal	$x^2=0$5
Height 1	$x^2=0$6
Maximal	$x^2=0$7, $x^2=0$8

All nonzero prime ideals are completely prime. The primitive ideals are precisely the zero ideal and the maximal ideals. The maximal spectrum coincides with the primitive spectrum, except for the height-one prime $x^2=0$9.

Simple Module Classification in Characteristic Zero

Every finite-dimensional simple $y^2x - xy^2 - xyx = 0$0-module is one-dimensional, corresponding to a maximal ideal $y^2x - xy^2 - xyx = 0$1 for $y^2x - xy^2 - xyx = 0$2. Infinite-dimensional simple modules are characterized by the invertibility of $y^2x - xy^2 - xyx = 0$3; they are faithful and extend to simple $y^2x - xy^2 - xyx = 0$4-modules.

For $y^2x - xy^2 - xyx = 0$5, a family of infinite-dimensional simple modules $y^2x - xy^2 - xyx = 0$6 is constructed, where the structure is governed by the intertwining action of $y^2x - xy^2 - xyx = 0$7 and $y^2x - xy^2 - xyx = 0$8 on an explicit basis $y^2x - xy^2 - xyx = 0$9.

These results clarify the direct analogy and differences with the representation theory of the Weyl algebra and underline the significance of the Morita equivalence established via localization.

Positive Characteristic Case

For $x^2=0$0, the situation varies significantly:

• The centers are $x^2=0$1 and $x^2=0$2.
• $x^2=0$3 is an Azumaya algebra of rank $x^2=0$4 over its center, with all simple modules of dimension $x^2=0$5.
• $x^2=0$6 is a prime PI algebra with PI degree $x^2=0$7, and its classical quotient ring remains a central simple algebra of dimension $x^2=0$8.

The prime spectrum is explicitly described as follows:

Prime Ideal Type	Structure
Height 1 primes	$x^2=0$9
Maximal, $y s - s y - x s = 0$0	$y s - s y - x s = 0$1, $y s - s y - x s = 0$2
Extended from $y s - s y - x s = 0$3 via localization	$y s - s y - x s = 0$4, $y s - s y - x s = 0$5

Primitive ideals coincide with maximal ideals and have two forms:

• One-dimensional simple module quotients at $y s - s y - x s = 0$6.
• $y s - s y - x s = 0$7-dimensional simple module quotients at nonzero $y s - s y - x s = 0$8.

The Azumaya locus is described by those maximal ideals of the center for which $y s - s y - x s = 0$9, corresponding to those $s = xy+yx$0-modules with maximal PI-degree.

Explicit Simple Module Construction in Positive Characteristic

All simple modules are constructed explicitly:

• For any $s = xy+yx$1, $s = xy+yx$2 is one-dimensional and simple.
• For any $s = xy+yx$3 and $s = xy+yx$4, $s = xy+yx$5 is $s = xy+yx$6-dimensional and simple.

Every simple $s = xy+yx$7-module arises via such constructions, and all simple modules are finite-dimensional in positive characteristic.

Implications and Future Directions

This work settles several foundational open questions regarding the ring-theoretic and module-theoretic structure of the super Jordan plane. By linking the structure to that of the Weyl algebra via localization, the analysis circumvents complications introduced by the absence of the domain property in $s = xy+yx$8 and exploits Morita equivalence to full effect.

The explicit classification has consequences for the study of more general Nichols algebras, their associated quantum groups, and braided Hopf algebras. The Azumaya property in positive characteristic may have implications for geometric and deformation-theoretic applications.

There is scope to further clarify equivalences with representation categories of other quantum or super-algebraic structures, as well as to extend the analysis to the related liftings of the super Jordan plane and their role in the broader landscape of pointed Hopf algebra classification.

Conclusion

This paper rigorously determines the prime and primitive spectra, the structure of the classical ring of quotients, and the nature of simple modules for the super Jordan plane algebra $s = xy+yx$9 over arbitrary fields of characteristic not $\CJ$0. The approach, centered on a key localization and leveraging Morita equivalence with localizations of the Weyl algebra, reduces challenging non-domain questions to tractable and well-understood algebraic frameworks. The results provide a definitive understanding of the ordinary ring-theoretic and representation-theoretic properties of $\CJ$1, laying the groundwork for further advances in Nichols algebra theory and related quantum algebraic systems.

Reference: "Prime spectrum and representations of the super Jordan plane" (2606.31731)