Super Jordan Plane Algebra

• Super Jordan Plane is a Nichols algebra of a 2-dimensional braided vector space defined by relations x²=0 and a cubic relation, exhibiting polynomial growth.
• It features an Ore-extension and localization at a normal element s that connects its computations to the classical Jordan plane and Weyl algebra.
• In both characteristic 0 and p > 2, its spectrum and representation theory reveal rich structures including prime, artinian, and Azumaya properties.

The super Jordan plane is the algebra over a field $k$ of characteristic different from $2$ defined by

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$

In the literature it is also denoted $\mathcal B$ or $A=B(V(-1,2))$. It is a connected $\mathbb N$-graded Nichols algebra of a $2$-dimensional braided vector space of non-diagonal type, usually described as the $-1$-block. The algebra has Gelfand–Kirillov dimension $2$, is Noetherian, has infinite global dimension, and is not a domain. At the same time, its ring-theoretic, Hopf-theoretic, homological, and representation-theoretic behavior is much richer than that summary suggests: the embedded Jordan plane controls many calculations, the localization at a distinguished normal element identifies a large open part with a matrix algebra over a localization of the first Weyl algebra, and recent work gives a complete description of the prime, primitive, and maximal spectra together with the classical quotient ring and the simple modules (Lu, 30 Jun 2026, Andruskiewitsch et al., 2 Sep 2025, Andruskiewitsch et al., 2015).

1. Definition, presentations, and Nichols origin

A standard presentation uses generators $x,y$ of degree $2$0 and relations

$2$1

If one sets

$2$2

then the defining relations can be written as

$2$3

and from these one deduces

$2$4

In particular, the subalgebra generated by $2$5 and $2$6 is isomorphic to the classical Jordan plane (Lu, 30 Jun 2026).

Several earlier papers use generators $2$7 and the anticommutator

$2$8

so that the same algebra is written as

$2$9

In this notation, a PBW basis is

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$0

while other treatments use the PBW bases

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$1

and

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$2

The Hilbert series recorded in the homological study is

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$3

again reflecting polynomial growth of degree $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$4 (Reca et al., 2017).

The Nichols-algebra origin is fundamental. The super Jordan plane is the Nichols algebra $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$5 attached to a $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$6-dimensional Yetter–Drinfeld module with a non-diagonal Jordan block of eigenvalue $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$7; this is the case $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$8 in the classification of $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$9-dimensional blocks. The adjective “super” is tied to the $\mathcal B$0-grading in which the degree-$\mathcal B$1 generators are odd, while $\mathcal B$2 is even (Andruskiewitsch et al., 2015, Andruskiewitsch et al., 2 Sep 2025).

2. Internal structure and comparison with the classical Jordan plane

The super Jordan plane admits a useful Ore-extension presentation

$\mathcal B$3

with

$\mathcal B$4

and multiplication rule

$\mathcal B$5

This presentation makes straightening arguments effective and yields a normal form in which at most one $\mathcal B$6 survives, followed by powers of $\mathcal B$7, then powers of $\mathcal B$8 (Andruskiewitsch et al., 2 Sep 2025).

The embedded Jordan plane governs much of the structure. Inside $\mathcal B$9, the subalgebra $A=B(V(-1,2))$0 is isomorphic to the classical Jordan plane, and $A=B(V(-1,2))$1 is the Euler derivation on $A=B(V(-1,2))$2. This is why formulas in the super Jordan plane frequently reduce to identities in the ordinary Jordan plane. A characteristic example is the commutation rule

$A=B(V(-1,2))$3

together with the automorphism $A=B(V(-1,2))$4 of $A=B(V(-1,2))$5 given by $A=B(V(-1,2))$6, $A=B(V(-1,2))$7 (Andruskiewitsch et al., 2 Sep 2025).

Despite sharing $A=B(V(-1,2))$8 and Noetherianity with the classical Jordan plane, the super Jordan plane is not a domain. Explicit zero divisors are already visible in

$A=B(V(-1,2))$9

and the relation $\mathbb N$0 forces nilpotence in degree $\mathbb N$1. In the $\mathbb N$2-graded decomposition

$\mathbb N$3

the even zero divisors are precisely those in $\mathbb N$4, while for an odd element $\mathbb N$5, being a zero divisor is equivalent to being nilpotent, equivalently to $\mathbb N$6 (Andruskiewitsch et al., 2 Sep 2025).

Earlier work proved that $\mathbb N$7 is super-prime in the graded sense: if homogeneous $\mathbb N$8 satisfy $\mathbb N$9, then $2$0 or $2$1. It also constructed a super-simple, super-artinian ring of fractions by inverting homogeneous regular elements. This is distinct from ordinary primeness, which was established later by ring-theoretic methods using localization at the normal element $2$2 (Andruskiewitsch et al., 2 Sep 2025, Lu, 30 Jun 2026).

3. Localization at $2$3, primeness, and the prime spectrum

The normal element

$2$4

is central to the modern structure theory. Since

$2$5

the multiplicative set

$2$6

defines a localization $2$7. Writing

$2$8

one obtains the identities

$2$9

If $-1$0 is the subalgebra generated by $-1$1 and $-1$2, then

$-1$3

so $-1$4 is a localization of the first Weyl algebra $-1$5. More precisely,

$-1$6

with an explicit matrix realization of $-1$7, $-1$8, and $-1$9 (Lu, 30 Jun 2026).

This localization has immediate consequences. In characteristic $2$0, $2$1 is simple, hence $2$2 is simple and therefore prime. From this, together with the normality of $2$3, it follows that $2$4 is prime although it is not a domain. In characteristic $2$5, $2$6 is Azumaya over its center of rank $2$7, and $2$8 is a prime PI algebra of PI degree $2$9 (Lu, 30 Jun 2026).

The quotient by $x,y$0 is also explicit: $x,y$1 with vector-space decomposition

$x,y$2

Over an algebraically closed field, the prime spectrum of $x,y$3 is

$x,y$4

From this and the localization analysis one gets the full prime spectrum of $x,y$5 (Lu, 30 Jun 2026).

In characteristic $x,y$6, over algebraically closed $x,y$7,

$x,y$8

Every nonzero prime ideal is completely prime,

$x,y$9

and

$2$00

The inclusions

$2$01

describe the Zariski topology, and $2$02 is locally closed and primitive (Lu, 30 Jun 2026).

In characteristic $2$03, over algebraically closed $2$04,

$2$05

Moreover,

$2$06

and the corresponding primitive factors are

$2$07

Thus the open set where $2$08 is invertible is controlled entirely by the commutative spectrum of $2$09 (Lu, 30 Jun 2026).

4. Centers, quotient rings, and Azumaya behavior

The center changes sharply with the characteristic. In characteristic $2$10,

$2$11

In characteristic $2$12,

$2$13

The key identity in the positive-characteristic computation is

$2$14

which is tied to the Weyl-algebra normal-ordering formula

$2$15

in characteristic $2$16 (Lu, 30 Jun 2026).

Because $2$17 is Noetherian, its regular elements satisfy the Ore condition and the classical quotient ring $2$18 exists. In characteristic $2$19, the localization result $2$20 gives

$2$21

where $2$22 is the Weyl skewfield. This is the Artin–Wedderburn form of the classical quotient ring (Lu, 30 Jun 2026).

In characteristic $2$23, if $2$24 and

$2$25

then

$2$26

a central simple algebra of dimension $2$27 over $2$28. The localized algebra $2$29 is Azumaya over its center of rank $2$30, all ideals of $2$31 are extended from the center, and every simple $2$32-module has dimension $2$33 (Lu, 30 Jun 2026).

The Azumaya locus is also explicit. Over algebraically closed $2$34 of characteristic $2$35,

$2$36

and for each such maximal ideal $2$37,

$2$38

Its complement is

$2$39

where $2$40 is not Azumaya (Lu, 30 Jun 2026).

5. Representation theory

The earliest systematic representation-theoretic results concern finite-dimensional modules in characteristic $2$41. Every finite-dimensional simple module is one-dimensional, and the algebra’s indecomposable modules of dimensions $2$42 and $2$43 were classified explicitly, together with two families of indecomposables in arbitrary dimension. In the notation of that work, if $2$44 is the one-dimensional module with

$2$45

then every finite-dimensional simple is some $2$46, and the generalized-eigenspace decomposition for the action of $2$47 yields a block decomposition of the finite-dimensional module category. An indecomposable module can have at most two eigenvalues for $2$48, and if two occur they must be $2$49 and $2$50 (Andruskiewitsch et al., 2017).

Recent work sharpens this picture by combining ordinary ring theory with the localization at $2$51. In characteristic $2$52,

$2$53

and each $2$54 is $2$55-dimensional: $2$56 A simple $2$57-module $2$58 is infinite-dimensional if and only if it is faithful, equivalently if and only if $2$59 acts bijectively on $2$60. In that case, $2$61 extends naturally to a simple $2$62-module and, via

$2$63

corresponds by Morita equivalence to a simple $2$64-module, with $2$65 a localization of the first Weyl algebra (Lu, 30 Jun 2026).

An explicit family of infinite-dimensional simple modules is given, for $2$66, by

$2$67

with

$2$68

$2$69

$2$70

with $2$71. These modules are simple and faithful in characteristic $2$72 (Lu, 30 Jun 2026).

In characteristic $2$73, every simple module is finite-dimensional and has dimension either $2$74 or $2$75. The two explicit families are

$2$76

and

$2$77

The module $2$78 has basis $2$79, where $2$80 is the class of $2$81, together with a decomposition

$2$82

and actions

$2$83

The annihilators of the simple modules are exactly the primitive ideals: $2$84 This identifies the primitive spectrum with the set of annihilators of simples in both characteristics (Lu, 30 Jun 2026).

6. Liftings, bosonizations, and doubles

Within the theory of pointed Hopf algebras, the super Jordan plane is one of the canonical $2$85-dimensional Nichols algebras of non-diagonal type with finite Gelfand–Kirillov dimension. The lifting theorem for $2$86-dimensional blocks states that a pointed Hopf algebra $2$87 has finite GK-dimension and infinitesimal braiding a block if and only if $2$88 is nilpotent-by-finite and $2$89 is a lifting $2$90 of either a Jordan or a super Jordan plane. In the super Jordanian case, with a YD-triple $2$91 satisfying $2$92 and $2$93, the lifting is the quotient of $2$94 by

$2$95

with $2$96 if $2$97. Conversely, every pointed Hopf algebra of finite GK-dimension with infinitesimal braiding $2$98 is of this form (Andruskiewitsch et al., 2015).

Bosonization produces a pointed Hopf algebra

$2$99

generated by $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$00 with

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$01

and

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$02

This Hopf algebra has PBW basis

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$03

hence GK-dimension $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$04. It admits a natural quotient

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$05

where $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$06 is the bosonization of a polynomial braided Hopf algebra and may be viewed as a quantum Borel of $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$07 at $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$08. The finite-dimensional simple $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$09-modules are exactly those coming from $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$10, so they have dimension $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$11 or $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$12 (Andruskiewitsch et al., 2019).

The Drinfeld double of the bosonization gives a further layer of structure. For $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$13, the double of the super Jordan plane bosonization yields Hopf superalgebras, thereby justifying a posteriori the adjective “super”. In characteristic $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$14 there is an exact sequence

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$15

where $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$16 is a supercommutative Hopf algebra and $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$17 has a triangular decomposition involving the super Jordan plane and its dual. In characteristic $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$18, the restricted double fits into

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$19

In the restricted case, the simple $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$20-modules coincide with the simple $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$21-modules: there are exactly $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$22 isomorphism classes, of dimensions $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$23 (Andruskiewitsch et al., 2020).

7. Homological invariants and the Virasoro connection

Over an algebraically closed field of characteristic $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$24, the super Jordan plane $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$25 has a highly explicit homological theory. A minimal $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$26-projective resolution is constructed from the quadratic monomial relation $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$27 and the cubic relation $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$28, organized as a bicomplex whose associated spectral sequence collapses at $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$29. This leads to Hochschild homology groups $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$30 that are infinite-dimensional and periodic of period $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$31 starting at degree $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$32 (Reca et al., 2017).

The Hochschild cohomology ring is equally explicit. One has

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$33

and $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$34 is infinite-dimensional with distinguished generators

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$35

The Gerstenhaber bracket satisfies

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$36

so

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$37

via

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$38

Thus the first Hochschild cohomology is identified with the positive part of the Virasoro algebra together with the Cartan subalgebra $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$39 (Reca et al., 2017).

The higher cohomology groups carry natural $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$40-module structures, and certain ideals and quotients in $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$41 and $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$42 are identified with intermediate-series Virasoro modules $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$43. A central structural feature is periodicity induced by a class $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$44: multiplication by $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$45 yields isomorphisms

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$46

for all $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$47 (Reca et al., 2017).

The Yoneda algebra also displays a mixed quadratic-cubic pattern. Although the super Jordan plane is not quadratic and hence not Koszul, its Ext-algebra with trivial coefficients is generated in degrees $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$48, so $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$49 is $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$50. More precisely,

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$51

with $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$52 and $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$53. For the bosonization $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$54, the cohomology is

$$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$55

generated in degrees $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$56, $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$57, and $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$58; it is finitely generated and graded commutative, but not $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$59 (Reca et al., 2017).

Taken together, these results place the super Jordan plane at the intersection of Nichols algebras, noncommutative ring theory, representation theory, Hopf algebra theory, and homological algebra. The algebra retains the small-growth features of the classical Jordan plane, but the nilpotent relation $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$60, the $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$61-grading, and the Weyl-algebra localization produce a distinct structure: prime but not a domain, simple modules controlled by the behavior of the normal element $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$62, doubles governed by $$\mathcal J = k\langle x,y\rangle/\langle x^2,\; y^2x-xy^2-xyx\rangle .$$63, and Hochschild cohomology organized by the positive Virasoro algebra (Lu, 30 Jun 2026, Reca et al., 2017).