Double Extension Method in AS Regular Algebras

• Double Extension Method is a framework for presenting 4-generator Artin–Schelter regular algebras as double Ore extensions, enabling explicit PBW bases and central computations.
• It unifies the Zhang–Zhang classification with family-specific rewriting, leading to concrete outcomes in center determination and cancellation theory.
• Computational techniques like SageMath and Gröbner–Shirshov bases are integral for automating PBW reductions and validating complex algebraic structures.

The double extension method, in the setting of 4-generator Artin–Schelter regular algebras of type $(14641)$, is the structural framework that presents an algebra as a double Ore extension $R_P[y_1,y_2;\sigma,\delta,\tau]$ of a 2-dimensional Artin–Schelter regular algebra $R$, furnishes a PBW basis, and makes explicit center computations possible (Rubiano, 26 Jan 2026). In this role it links the Zhang–Zhang classification of regular algebras to family-by-family rewriting, commutation, and centrality calculations, and it yields concrete centers, explicit central subalgebras, and applications to the Zariski cancellation problem.

1. Definition and basic data

In the corrected Zhang–Zhang sense used in the center-computation literature, let $R$ be a subalgebra of a $\Bbbk$-algebra $B$. A right double extension of $R$ is an algebra generated by $R$ and two new variables $y_1,y_2$ such that $B$ is generated by $R_P[y_1,y_2;\sigma,\delta,\tau]$0; the new generators satisfy

$R_P[y_1,y_2;\sigma,\delta,\tau]$1

where $R_P[y_1,y_2;\sigma,\delta,\tau]$2 and $R_P[y_1,y_2;\sigma,\delta,\tau]$3; $R_P[y_1,y_2;\sigma,\delta,\tau]$4 is free as a left $R_P[y_1,y_2;\sigma,\delta,\tau]$5-module with basis

$R_P[y_1,y_2;\sigma,\delta,\tau]$6

and the mixed products are controlled by a matrix homomorphism $R_P[y_1,y_2;\sigma,\delta,\tau]$7 and a $R_P[y_1,y_2;\sigma,\delta,\tau]$8-derivation $R_P[y_1,y_2;\sigma,\delta,\tau]$9 through

$R$0

If, in addition, $R$1, $R$2 is free as a right $R$3-module with basis $R$4, and the left/right $R$5-spans match, then it is called simply a double extension (Rubiano, 26 Jan 2026).

The package

$R$6

is the DE-data, with $R$7 the parameter and $R$8 the tail. A particularly important special case is the trimmed double extension, where

$R$9

This trimmed case is central in the analysis of regular algebras of type $R$0, because many classified examples admit presentations of this form.

2. Type $R$1 and the classification framework

The type $R$2 designation refers to connected graded Artin–Schelter regular algebras of global dimension $R$3 with four degree-one generators whose minimal resolution has Betti numbers

$R$4

The numerical pattern $R$5 is therefore the homological signature of the class under study (Rubiano, 26 Jan 2026).

Zhang and Zhang showed that many such algebras arise as double extensions $R$6 where $R$7 is a 2-dimensional Artin–Schelter regular algebra. When $R$8 is 2-dimensional regular, the double extension is itself strongly Noetherian, Auslander regular, Cohen–Macaulay, a domain, and Koszul. Within this framework, the classification used in the center paper consists of 26 families $R$9, and a key structural point is that many algebras of type $\Bbbk$0 can be presented as trimmed double extensions (Rubiano, 26 Jan 2026).

This classification-theoretic role is what makes the method more than a presentation device. It provides a uniform language for moving from the homological classification of regular algebras to explicit structural invariants such as the center.

3. Presentation, System C, and relation to iterated Ore extensions

For double extensions over

$\Bbbk$1

the defining relations are written in the form

$\Bbbk$2

together with four mixed relations

$\Bbbk$3

These coefficients are encoded in the matrix $\Bbbk$4 (Rubiano, 26 Jan 2026).

The structure theorem states that these relations define a right double extension precisely when the System C constraints hold,

$\Bbbk$5

The system comes from enforcing compatibility of the mixed relations with multiplicativity of $\Bbbk$6 and with the quadratic relations in $\Bbbk$7 and among $\Bbbk$8. In this formulation, $\Bbbk$9 is the algebraic condition ensuring that the extension is genuinely “double,” while the parameters

$B$0

control whether one has a nontrivial deformation, and whether the algebra is an iterated Ore extension or not (Rubiano, 26 Jan 2026).

The framework is broader than iterated Ore extensions. If $B$1, then the double extension can be written as an iterated Ore extension $B$2; similarly, if $B$3 and $B$4, one can present it in the opposite order. The double extension method is therefore a genuine enlargement of the iterated Ore extension formalism rather than a reformulation of it.

4. Computational realization: PBW, Gröbner–Shirshov, and SageMath

The computational force of the method lies in the PBW control it imposes. In the center computations, the procedure is to start from a known double extension family from Zhang–Zhang, put the algebra into PBW form using the ordering

$B$5

compute rewriting rules and normal forms, derive commutation formulas for powers of generators, and use those formulas to identify central elements and central subalgebras. The paper derives general recursions for

$B$6

and analogous formulas for powers of $B$7 and $B$8, which are then specialized family by family (Rubiano, 26 Jan 2026).

SageMath is used to automate PBW reduction and commutator computations. The implementation uses a free algebra on $B$9, rewriting rules encoding the defining relations, a normal-form routine that repeatedly reduces “bad pairs,” and a commutator routine computing

$R$0

and reducing the result to PBW form. The center must be computed case by case, and the symbolic expansions in the general parameter-dependent case become very large; SageMath is therefore part of the proof strategy, not merely an expository aid (Rubiano, 26 Jan 2026).

A complementary computational approach is provided by the finite Gröbner–Shirshov basis analysis of regular double extension algebras of type $R$1. There the algebras are presented as quotient algebras of a free associative algebra, the degree-lexicographic order on $R$2 is fixed by

$R$3

and finite GS bases are computed for 12 families,

$R$4

By the Composition–Diamond lemma, this yields PBW bases

$R$5

showing that finite GS bases and PBW normal forms are tightly aligned with the double extension viewpoint (Herrera et al., 6 Sep 2025).

5. Centers, central subalgebras, and cancellation

The main structural payoff of the method is the explicit determination of centers and central subalgebras under parameter restrictions. Representative outcomes are summarized below (Rubiano, 26 Jan 2026).

Family	Parameter regime	Center or central subalgebra
$R$6	—	$R$7
$R$8	—	$R$9
$R$0	$R$1	$R$2
$R$3	$R$4 not a root of unity	$R$5
$R$6	$R$7	$R$8
$R$9	$y_1,y_2$0	$y_1,y_2$1
$y_1,y_2$2	$y_1,y_2$3	$y_1,y_2$4
$y_1,y_2$5	$y_1,y_2$6	$y_1,y_2$7

The general pattern is explicit: when parameters are generic, meaning not roots of unity or avoiding special values like $y_1,y_2$8, the center collapses to $y_1,y_2$9; special parameter values produce larger centers generated by explicit homogeneous invariants. This suggests that the double extension presentation isolates the precise commutation regimes in which nontrivial central generators survive.

The family $B$0 is the standard worked example. Under $B$1, the defining relations are

$B$2

$B$3

$B$4

The proof that

$B$5

proceeds by direct commutator computation in PBW normal form. The key central candidates are

$B$6

and bihomogeneous degree arguments together with a PBW basis calculation show that every central element is a polynomial in these two generators (Rubiano, 26 Jan 2026).

The center computations also feed into cancellation theory. The standard notions recalled in the paper are: cancellative, $B$7; strongly cancellative, the same statement with several variables; and universally cancellative, stability under a broader tensoring condition. A key theorem used there is that if $B$8, then $B$9 is universally cancellative. Consequently, every family in the center table with $R_P[y_1,y_2;\sigma,\delta,\tau]$00 provides a new universally cancellative example (Rubiano, 26 Jan 2026).

6. Broader mathematical uses of double extension

The phrase double extension is used in several adjacent literatures, but with different formal meanings. In Poisson algebra, a Poisson double extension is defined as a Poisson-theoretic analogue of a double Ore extension, and suitable double Ore extensions are shown to be deformation quantizations of Poisson double extensions. In that setting the polynomial ring $R_P[y_1,y_2;\sigma,\delta,\tau]$01 carries a Poisson bracket whose form mirrors the DE-data, and the construction clarifies the relation between double Ore extensions, semiclassical limits, modular derivations, and iterated Poisson polynomial extensions (Lou et al., 2016).

In the theory of restricted Lie (super)algebras with a non-degenerate invariant symmetric bilinear form, the double extension method enlarges

$R_P[y_1,y_2;\sigma,\delta,\tau]$02

by means of a central extension and a derivation. The bracket is modified by a central $R_P[y_1,y_2;\sigma,\delta,\tau]$03-cocycle, $R_P[y_1,y_2;\sigma,\delta,\tau]$04 is central, and $R_P[y_1,y_2;\sigma,\delta,\tau]$05 acts by a derivation $R_P[y_1,y_2;\sigma,\delta,\tau]$06. Under the appropriate $R_P[y_1,y_2;\sigma,\delta,\tau]$07-property, the extension is restricted, and conversely any restricted NIS-(super)algebra with non-trivial center arises as such a $R_P[y_1,y_2;\sigma,\delta,\tau]$08-extension subject to an extra condition on the central element (Benayadi et al., 2018).

A closely parallel construction appears for multiplicative restricted Hom-Lie algebras. There the enlarged algebra is

$R_P[y_1,y_2;\sigma,\delta,\tau]$09

with $R_P[y_1,y_2;\sigma,\delta,\tau]$10 central, $R_P[y_1,y_2;\sigma,\delta,\tau]$11 acting by a restricted derivation $R_P[y_1,y_2;\sigma,\delta,\tau]$12, and the bracket on $R_P[y_1,y_2;\sigma,\delta,\tau]$13 corrected by the cocycle $R_P[y_1,y_2;\sigma,\delta,\tau]$14. The method preserves restrictedness under a $R_P[y_1,y_2;\sigma,\delta,\tau]$15-property hypothesis, and every irreducible restricted quadratic Hom-Lie algebra with nonzero center is proved to be the double extension of another restricted quadratic Hom-Lie algebra (Mao et al., 2023).

For commutative $R_P[y_1,y_2;\sigma,\delta,\tau]$16-ary superalgebras with a skew-symmetric invariant form, the method is recast in the derived bracket formalism. The central theorem states that any irreducible but not simple algebra is isomorphic to a generalized double extension, and hence such algebras can be built inductively by orthogonal sums and generalized double extensions (Vishnyakova, 2016). In flat pseudo-Riemannian $R_P[y_1,y_2;\sigma,\delta,\tau]$17-Lie algebras, an analogous extension

$R_P[y_1,y_2;\sigma,\delta,\tau]$18

is reconstructed from a smaller algebra $R_P[y_1,y_2;\sigma,\delta,\tau]$19 using cocycles $R_P[y_1,y_2;\sigma,\delta,\tau]$20, a derivation $R_P[y_1,y_2;\sigma,\delta,\tau]$21, and parameters $R_P[y_1,y_2;\sigma,\delta,\tau]$22, yielding generation theorems in the Lorentzian and $R_P[y_1,y_2;\sigma,\delta,\tau]$23-signature settings (Torres-Gomez et al., 2024).

Across these settings, the common feature is not a single universal definition, but a recurrent structural pattern: an algebra is enlarged in two coupled directions, and the enlarged presentation becomes the mechanism for classification, normal-form control, or the computation of structural invariants.