---
title: Graded Double Ore Extension
url: https://www.emergentmind.com/topics/graded-double-ore-extension
type: topic
---

# Graded Double Ore Extension

A graded double Ore extension is a connected $\mathbb N$-graded algebra obtained from a graded algebra by adjoining two degree-$1$ variables subject to a matrix-controlled twisting action and a single quadratic relation between the new variables. In the literature it appears in the general form $R_P[x_1,x_2;\sigma,\delta,\tau]$ and in the trimmed form $A_P[y_1,y_2;\sigma]$. For Koszul Artin–Schelter regular input algebras, the construction preserves Koszulity and increases global dimension by $2$; more recent work further gives a minimal free resolution in the presence of nontrivial skew derivations, introduces the $\sigma$-divergence as a homological invariant, computes the Nakayama automorphism explicitly, and realizes the algebra as a derivation-quotient algebra of a twisted superpotential [1810.06778] [2507.15252].

## 1. Defining data and basic variants

Let $R=\bigoplus_{m\ge 0} R_m$ be an $\mathbb N$-graded algebra. A graded right double Ore extension of $R$ in two new variables is an algebra
\[
A \;=\; R_P[x_1,x_2;\sigma,\delta,\tau]
\;=\; R\langle x_1,x_2\rangle/I
\]
with parameters $P=\{p_{12},p_{11}\}\subseteq K$, tail data $\tau=\{\tau_1,\tau_2,\tau_0\}\subseteq R$, a $2\times 2$ matrix of graded $K$-linear maps
\[
\sigma=(\sigma_{ij})_{i,j=1,2}:R\to M_2(R),
\]
and a pair of graded $\sigma$-derivations
\[
\delta=(\delta_1,\delta_2):R\to R^2
\]
satisfying the degree conditions
\[
\sigma_{ij}(R_m)\subseteq R_m,\qquad \delta_j(R_m)\subseteq R_{m+1},
\]
together with the relations
\[
x_jr=\sum_{i=1}^2 \sigma_{ij}(r)x_i+\delta_j(r),\qquad r\in R,\; j=1,2,
\]
and
\[
x_2x_1
=
p_{12}x_1x_2+p_{11}x_1^2+\tau_1x_1+\tau_2x_2+\tau_0.
\]
The grading is specified by $\deg(r)=\deg_R(r)$ and $\deg(x_1)=\deg(x_2)=1$, and all defining relations are required to be homogeneous. In the graded context one further requires freeness as a left $R$-module on the standard monomials, and under mild conditions also as a right $R$-module with the obvious monomial bases [1810.06778].

In the formulation used for Koszul AS-regular algebras, one takes $A=T(V)/(R)$, assumes that $A$ is Koszul Artin–Schelter regular of dimension $d$, fixes $P=\{p_{12}\neq 0,p_{11}\}\subset k$, a graded algebra homomorphism
\[
\sigma:A\to M_2(A),
\]
and a degree-$1$ linear map
\[
\nu=(\nu_1,\nu_2):A\to A\oplus A
\]
satisfying the $\sigma$-derivation rule
\[
\nu(ab)=\sigma(a)\cdot \nu(b)+\nu(a)b,\qquad a,b\in A.
\]
The resulting graded right double Ore extension
\[
B=A_P[y_1,y_2;\sigma,\nu]
\]
is generated by $A$ and degree-$1$ variables $y_1,y_2$ with mixed relations
\[
y_i a=\sigma_{i1}(a)y_1+\sigma_{i2}(a)y_2+\nu_i(a),\qquad i=1,2,
\]
and quadratic relation
\[
y_2y_1=p_{12}y_1y_2+p_{11}y_1^2.
\]
It is required that $B$ be free as a left $A$-module with basis $\{y_1^i y_2^j\mid i,j\ge 0\}$. When $p_{12}\neq 0$, $\sigma$ must be invertible in the matrix sense, and then $B$ is also a left double Ore extension [2507.15252].

A central special case is the trimmed graded double Ore extension. Here one sets $\delta\equiv 0$ and $\tau_0=\tau_1=\tau_2=0$, and writes
\[
B=A_P[y_1,y_2;\sigma].
\]
This is the setting of the Nakayama-automorphism formulas in Zhu–Van Oystaeyen–Zhang and of much of the explicit PBW theory for regular algebras of type $(14641)$ [1401.0330] [2606.08396].

## 2. PBW structure and relation to graded skew PBW extensions

The defining freeness condition already imposes a PBW-type monomial structure. In the trimmed setting over a quadratic base algebra $\Bbbk_Q[x_1,x_2]$, one fixes the order
\[
x_1\prec x_2\prec y_1\prec y_2,
\]
and a monomial is PBW normal if it has the form
\[
x_1^a x_2^b y_1^c y_2^d,\qquad a,b,c,d\in \mathbb N.
\]
Bergman’s Diamond Lemma together with the trimmed-double-extension hypotheses implies that
\[
\mathcal B=\{x_1^a x_2^b y_1^c y_2^d\mid a,b,c,d\ge 0\}
\]
is a $\Bbbk$-basis, and every word has a unique normal form obtained by repeatedly rewriting the forbidden adjacent pairs $x_2x_1$, $y_2y_1$, and $y_ix_s$ via the quadratic and mixed relations [2606.08396].

Gómez–Suárez gives necessary and sufficient conditions for a graded double Ore extension to be a graded skew PBW extension. If
\[
A=R_P[x_1,x_2;\sigma,\delta,\tau]
\]
is a connected graded double Ore extension of a connected graded algebra $R$, then $A$ is a graded skew PBW extension of $R$ if and only if the following hold:

1. $p_{12}\neq 0$ and $p_{11}=0$;
2. $\sigma_{12}=\sigma_{21}=0$, and $\sigma_{11},\sigma_{22}$ are graded automorphisms of $R$;
3. $\delta_1,\delta_2$ are arbitrary graded $\sigma$-derivations;
4. $\tau_1,\tau_2,\tau_0\in R$ are homogeneous elements of positive degree.

Under these hypotheses the defining relations become exactly those of a skew PBW extension, the ordered monomials $x_1^a x_2^b$ remain linearly independent over $R$, and the PBW property follows [1810.06778].

The explicit trimmed family
\[
B_K(\alpha)=\Bbbk_{-1}[x_1,x_2]_P[y_1,y_2;\sigma],\qquad P=(-1,0),
\]
with
\[
\sigma(x_1)=\begin{pmatrix}x_1&0\\0&x_1\end{pmatrix},
\qquad
\sigma(x_2)=\begin{pmatrix}0&x_2\\ \alpha x_2&0\end{pmatrix},
\]
has defining relations
\begin{align*}
x_2x_1+x_1x_2&=0, & y_2y_1+y_1y_2&=0,\\
y_1x_1-x_1y_1&=0, & y_1x_2-x_2y_2&=0,\\
y_2x_1-x_1y_2&=0, & y_2x_2-\alpha x_2y_1&=0.
\end{align*}
A finite Gröbner–Shirshov calculation yields the PBW basis
\[
\{x_1^a x_2^b y_1^c y_2^d\mid a,b,c,d\ge 0\},
\]
and hence
\[
H_B(t)=\frac{1}{(1-t)^4}.
\]
The same example is neither an iterated Ore extension in the order $(y_1,y_2)$ nor $(y_2,y_1)$ [2604.02456].

## 3. Homological behavior, Koszulity, and minimal resolutions

For Koszul AS-regular input algebras, graded double Ore extensions preserve the main homological regularity properties. Zhu–Van Oystaeyen–Zhang state that if $A$ is Koszul then the trimmed extension $B=A_P[y_1,y_2;\sigma]$ is also Koszul, and if in addition $A$ is Artin–Schelter regular of global dimension $d$, $\sigma$ is invertible, and $p_{12}\neq 0$, then $B$ is AS-regular with
\[
\operatorname{gldim} B=d+2.
\]
Gómez–Suárez formulates the same increase-by-$2$ phenomenon for connected graded skew PBW extensions of Artin–Schelter regular algebras and derives the skew Calabi–Yau consequence for connected skew Calabi–Yau bases [1401.0330] [1810.06778].

For the nontrivial derivation case, let
\[
\cdots\to W_d\otimes A\to W_{d-1}\otimes A\to \cdots \to V\otimes A\to A\to k_A\to 0
\]
be the Koszul resolution of the trivial $A$-module. To lift this to
\[
B=A_P[y_1,y_2;\sigma,\nu],
\]
one introduces a quadruple of linear maps
\[
\delta_{i,r}:W_i\to (W_i\otimes V)^2,\qquad
\delta_{i,l}:W_i\to (V\otimes W_i)^2,
\]
\[
\upsilon_{i,r}:W_i\to W_{i+1}\otimes V,\qquad
\upsilon_{i,l}:W_i\to V\otimes W_{i+1},
\]
satisfying compatibilities with $\sigma$, the differentials, and the Ore homotopies. From these data one defines maps
\[
f_i:W_i\otimes B(-2)\to (W_i\otimes B(-1))^2,
\]
\[
g_i:(W_i\otimes B(-1))^2\to W_i\otimes B,
\]
\[
h_i:W_i\otimes B(-2)\to W_{i+1}\otimes B,
\]
forms two iterated mapping cones, and obtains a complex
\[
F_\bullet:\;
0\to W_d\otimes B(-2)\to
W_{d-1}\otimes B(-2)\oplus (W_d\otimes B(-1))^2
\to\cdots\to
B\oplus (W_1\otimes B(-1))^2
\to B\to k_B\to 0.
\]
Theorem 2.14 asserts that $F_\bullet$ is exact and minimal; hence $B$ is Koszul of global dimension $d+2$ [2507.15252].

The pilot family $B_K(\alpha)$ gives a fully explicit type-$(14641)$ resolution. Its trivial right module has a linear resolution
\[
0\to B(-4)\xrightarrow{d_4} B(-3)^4\xrightarrow{d_3} B(-2)^6\xrightarrow{d_2} B(-1)^4\xrightarrow{d_1} B\to k_B\to 0,
\]
with differentials written in matrix form and all entries in positive degree. The graded Betti numbers are
\[
\beta^B_{i,i}(k_B)=1,4,6,4,1,\qquad \beta^B_{i,j}(k_B)=0\;(j\neq i),
\]
equivalently
\[
P^B_{k_B}(s,t)=(1+st)^4.
\]
The same paper also computes $2$-step linear resolutions for the cyclic modules
\[
M_x=B/(x_1,x_2)B,\qquad M_y=B/(y_1,y_2)B,
\]
with Betti numbers
\[
\beta^B_{0,0}=1,\qquad \beta^B_{1,1}=2,\qquad \beta^B_{2,2}=1
\]
for each module [2604.02456].

## 4. $\sigma$-divergence and explicit Nakayama automorphisms

When $A$ is AS-regular and $\dim W_d=1$, choose a basis $\omega\in W_d$. Then the top-degree components of the auxiliary maps satisfy
\[
\delta_{d,r}(\omega)=(\omega\otimes \delta_{r;1},\;\omega\otimes \delta_{r;2})^T,
\qquad
\delta_{d,l}(\omega)=(\delta_{l;1}\otimes \omega,\;\delta_{l;2}\otimes \omega)^T,
\]
which define vectors $\delta_r,\delta_l\in V^2$. Although the quadruple of auxiliary maps is non-unique, the combination
\[
\operatorname{div}_\sigma \nu
:=
\delta_r+\mu_A^{\oplus 2}\bigl(\sigma^{-T}\cdot \delta_l\bigr)\in V^2
\]
is independent of choices. This invariant is called the $\sigma$-divergence of $\nu$. Homologically, it measures the failure of the obvious part of the Ore homotopies to be a chain-map at the top degree and plays the same rôle as the usual divergence in the $1$-variable Ore extension case [2507.15252].

Since a graded double Ore extension of a Koszul AS-regular algebra is again AS-regular, it has a Nakayama automorphism. In the nontrivial derivation case, Theorem 3.9 gives a closed formula. On the embedded copy of $A$,
\[
\mu_B|_A=(\det \sigma)^{-1}\circ \mu_A.
\]
On the column vector $y=(y_1,y_2)^T$,
\[
\mu_B(y)
=
-\,(J^T)^{-1}J\;\hdet \sigma\;y
-\,(J^T)^{-1}J\;\bigl(\operatorname{div}_\sigma \nu\bigr),
\]
where
\[
J=\begin{pmatrix}-p_{11}&-p_{12}\\ 1&0\end{pmatrix},
\]
and $\hdet \sigma\in k^\times$ is the homological determinant of $\sigma$ defined by its action on $W_d$ [2507.15252].

In the trimmed case, Zhu–Van Oystaeyen–Zhang obtain the corresponding formula without the derivation term. If $v=\nu_A$ denotes the Nakayama automorphism of $A$, then
\[
\nu_B|_A=(\det \sigma)^{-1}\circ v,
\]
and
\[
\begin{pmatrix}\nu_B(y_1)\\ \nu_B(y_2)\end{pmatrix}
=
(\hdet \sigma)\,P^{-1}
\begin{pmatrix}y_1\\ y_2\end{pmatrix}.
\]
Thus the restriction to $A$ is governed by $\det \sigma$, while the action on the adjoined variables is governed by $\hdet \sigma$ and the parameter matrix $P$ [1401.0330].

## 5. Twisted superpotentials and Calabi–Yau criteria

For Koszul AS-regular algebras, the derivation-quotient viewpoint is intrinsic. The detailed formulation in the nontrivial derivation case states that
\[
B\cong D(\hat W,d)
\]
for a single $(d+2)$-form
\[
\hat W\in \hat V^{\otimes(d+2)},
\qquad
\hat V=V\oplus k\{y_1,y_2\},
\]
constructed from $\omega\in V^{\otimes d}$, the matrix $J$, $\sigma$, and the four families
\[
\delta_{i,r},\ \delta_{i,l},\ \upsilon_{i,r},\ \upsilon_{i,l}.
\]
Moreover $\hat W$ is $\mu_B$-twisted in the sense that
\[
\tau_{d+2}^{d+1}(\mu_B\otimes \operatorname{id}^{\otimes(d+1)})(\hat W)
=
(-1)^{d+1}\hat W,
\]
and
\[
B=T(\hat V)/(\partial^d\hat W).
\]
Its leading term is
\[
\bigl(y_2y_1-p_{12}y_1y_2-p_{11}y_1^2\bigr)\otimes \omega,
\]
while the lower-order corrections involve the $\delta$’s and $\upsilon$’s and encode the commutation rules between the $y$’s and $A$ [2507.15252].

In the trimmed case, the Nakayama formula yields a concrete Calabi–Yau criterion. A connected graded AS-regular algebra is Calabi–Yau exactly when its Nakayama automorphism is inner, and in positive degrees the only inner automorphisms of a connected graded algebra are trivial. Therefore
\[
B=A_P[y_1,y_2;\sigma]
\quad\text{is Calabi--Yau}\quad
\Longleftrightarrow
\begin{cases}
\det \sigma=v,\\
\hdet \sigma=P.
\end{cases}
\]
This criterion is given explicitly in Theorem 3.12 of Zhu–Van Oystaeyen–Zhang [1401.0330].

A worked example with nontrivial skew derivation appears for
\[
A=k\langle x_1,x_2\rangle /(x_2x_1-px_1x_2),
\qquad p^2=-1,
\]
with
\[
\sigma(x_1)=
\begin{pmatrix}
0&px_2\\
-px_2&0
\end{pmatrix},
\qquad
\sigma(x_2)=
\begin{pmatrix}
0&px_1\\
px_1&0
\end{pmatrix},
\]
\[
\nu(x_1)=\binom{0}{-p\,x_1\otimes x_2},
\qquad
\nu(x_2)=\binom{p\,x_1\otimes x_2}{0}.
\]
In this case
\[
\det \sigma(x_1)=-px_1,\qquad \det \sigma(x_2)=px_2,
\]
\[
\hdet \sigma=\operatorname{diag}(p,-p),
\qquad
\operatorname{div}_\sigma \nu=0,
\]
and Theorem 3.9 yields $\mu_B=\operatorname{id}$, so $B$ is Calabi–Yau [2507.15252].

## 6. Explicit families, normal ordering, and combinatorial structure

The Zhang–Zhang classification contains $26$ families
\[
\mathfrak F\in \{A,B,\dots,Z\}
\]
of trimmed double extension regular algebras of type $(14641)$, each Artin–Schelter regular of global dimension $4$ with resolution length pattern $(1$–$4$–$6$–$4$–$1)$. Up to relabeling, the internal data are summarized by pairs
\[
(q_{\mathfrak F},r_{\mathfrak F}),\qquad
(p_{\mathfrak F},s_{\mathfrak F}),
\]
through the relations
\[
x_2x_1=q_{\mathfrak F}x_1x_2+r_{\mathfrak F}x_1^2,
\qquad
y_2y_1=p_{\mathfrak F}y_1y_2+s_{\mathfrak F}y_1^2.
\]
The internal regimes include quantum-plane type when $r_{\mathfrak F}=0$ or $s_{\mathfrak F}=0$, and Jordan type for exactly two families:
\[
A:\ (q_A,r_A)=(1,0),\ (p_A,s_A)=(1,1),
\]
\[
H:\ (q_H,r_H)=(1,1),\ (p_H,s_H)=(-1,0).
\]
Thus in families $A$ and $H$ one of the two internal systems is a Jordan plane [2606.08396].

The normal-ordering theory is explicit. For all $m,n\in \mathbb N$,
\[
x_2^n x_1^m
=
\sum_{k=0}^n \Lambda_{q_{\mathfrak F},r_{\mathfrak F}}(n,m;k)\;x_1^{m+k}x_2^{n-k},
\]
\[
y_2^n y_1^m
=
\sum_{k=0}^n \Lambda_{p_{\mathfrak F},s_{\mathfrak F}}(n,m;k)\;y_1^{m+k}y_2^{n-k},
\]
where
\[
\Lambda_{q,r}(n,m;k)
=
r^k q^{\,m(n-k)}
\begin{bmatrix}n\\ k\end{bmatrix}_q
[m]_q^{\overline{k}},
\qquad
[m]_q^{\overline{k}}=\prod_{j=0}^{k-1}[m+j]_q.
\]
For the mixed relations one has
\[
y_jx_i=\sum_{t,\ell=1}^2 a_{ji,t\ell}\,x_t y_\ell,
\]
and the corresponding recursive coefficient systems $W_{\mathfrak F,i}^{(a,b)}$, $\Theta_{\mathfrak F}$, $m_{\mathfrak F}$, $R_{\mathfrak F,N}^{(a,b,c,d)}$, and $E_{\mathfrak F,N}^L$ control the normal form of a single crossing, mixed blocks, products of PBW monomials, powers of a normal block, and noncommutative multinomials [2606.08396].

The coefficient behavior separates into distinct combinatorial regimes. When $r_{\mathfrak F}=0$ or $s_{\mathfrak F}=0$, the internal $\Lambda$-coefficients reduce to scalar quantum-plane factors. If $q_{\mathfrak F}=-1,r_{\mathfrak F}=0$ or $p_{\mathfrak F}=-1,s_{\mathfrak F}=0$, one obtains skew-commutative signs. In the Jordan subcases one has
\[
\Lambda_{1,1}(n,m;k)=\binom{n}{k}\,m^{\overline{k}},
\]
identified in the paper as Lah–Whitney numbers, with triangular recurrence
\[
L_A(n,m;k)
=
(m+k-1)L_A(n-1,m;k-1)+L_A(n-1,m;k).
\]
The mixed $W$- and $\Theta$-arrays admit a weighted lattice-path interpretation in which local crossings contribute weights from the crossing kernels and internal reorderings contribute quantum or Whitney factors [2606.08396].

For computational purposes, the symbolic PBW machinery for all $26$ families is implemented in the ancillary SageMath file `double_ore_pbw.sage`, which applies the six rewriting rules under the fixed order $x_1\prec x_2\prec y_1\prec y_2$ and computes normal forms, products, powers, commutators, and center-tests in specified bidegrees [2606.08396]. The explicit family $B_K(\alpha)$ shows that even when the total Betti numbers are forced by AS-regularity, the matrices in the minimal resolution detect the parameter $\alpha$ through the mixing blocks of $d_2$, $d_3$, and $d_4$, so the defining double-extension data remain visible in the highest syzygies [2604.02456].

Source: https://www.emergentmind.com/topics/graded-double-ore-extension