Binomial Artin-Schelter Regular Algebras and Twisted Monoid Algebras

Introduction

In this paper, we study binomial Artin–Schelter (AS) regular algebras through their connections with twisted monoid algebras. Our main interest is in the monoids arising from binomial relations, their groups of fractions and the cocycles that determine the coefficients in these relations.

Let $A$ be a connected graded domain equipped with a faithful grading by a group $G$ such that

The support of this grading forms a submonoid $M$ of $G$, and multiplication in $A$ determines a 2-cocycle on $M$. Consequently, $A$ can be described as a twisted monoid algebra

The monoid $M$ records the relations among the degrees of the generators, while the cocycle $\sigma$ determines the coefficients in their products.

This motivates the study of coefficient-free relations obtained by replacing the nonzero coefficients in the binomial relations by 1. When these relations present $M$, the resulting algebra is the monoid algebra $k[M]$. We investigate when the original algebra can be recovered as a cocycle twist of $k[M]$.

If the monoid 2-cocycle extends to a group 2-cocycle and the group grading is compatible with the connected $\mathbb{N}$-grading, then cocycle twisting preserves Koszul AS-regularity and global dimension [10 Theorem 4.2.1].

An important class of examples comes from monoids of I-type, introduced by Gateva-Ivanova and Van den Bergh [8]. These monoids are closely related to finite involutive nondegenerate set-theoretic solutions of the Yang–Baxter equation [7]. Jespers and Okniński showed that the groups of fractions of monoids of I-type on $n$ generators admit embeddings into $\mathbb{Z}^n \rtimes S_n$ and contain finite-index free abelian subgroups of rank $n$ [11]. In particular, these groups are solvable Bieberbach groups of Hirsch length $n$.

We focus particularly on four-dimensional Koszul AS-regular algebras defined by four generators and six quadratic binomial relations. We started with a list of 168 binomial algebras, which is available on GitHub. For each algebra, we computed its superpotential in the sense of Mori and Smith [13]. We then considered the natural action of the symmetric group $S_4$ on the variables $x_0,x_1,x_2,x_3$. Two superpotentials were considered equivalent if one could be obtained from the other by permuting the variables. This gave 23 representative superpotentials under the action of $S_4$.

For these 23 representatives, we study the associated coefficient-free monoids, their groups of fractions, and the finite permutation groups arising from the corresponding Yang–Baxter solutions. We also compare these groups with known examples in the literature, including those studied by Jespers and Okniński [11] and Buzaglo and Rogalski [5].

We also compute the second group cohomology $H^2(G(M),k^\times)$ for the associated groups of fractions. This group classifies group 2-cocycles up to coboundaries and provides information about cocycle twists of the corresponding monoid algebras.

The paper is organized as follows. In § 2, we recall the necessary background on group gradings, cocycles, binomial AS-regular algebras, monoids of I-type and their groups of fractions. In § 3, we explain the connection between binomial AS-regular algebras and twisted monoid algebras, including the role of cocycles in determining the coefficients of binomial relations. In § 4, we study monoids of I-type and the structure of their groups of fractions. In § 5, we illustrate these constructions through examples in dimensions 2 and 3. In § 6, we investigate the 23 four-dimensional examples and determine their associated permutation groups. In § 7, we compute the second cohomology groups $H^2(G(M),k^\times)$ for the examples considered. In § 8, we use these computations to construct explicit cocycle twists of some algebras. Finally, the appendix § 9 contains the defining relations of the 23 four-dimensional binomial algebras.

Preliminaries

Throughout this paper, let $k$ be an algebraically closed field of characteristic zero. We recall some definitions and results that we will use throughout the paper. We begin with group gradings and cocycles, followed by binomial AS-regular algebras, monoids of I-type and their groups of fractions.

Group gradings and cocycles

Let $G$ be a group. An algebra $A$ is called $G$-graded if

for all $g,h\in G$.

The grading is called faithful if the support

generates $G$ as a group. If $A$ is a domain, then $\operatorname{supp}(A)$ is a submonoid of $G$.

A normalized 2-cocycle on $G$ is a function

satisfying $\sigma(g,h)\sigma(gh,\ell)=\sigma(h,\ell)\sigma(g,h\ell)$ for all $g,h,\ell\in G$, and

Two normalized 2-cocycles $\sigma$ and $\tau$ are called cohomologous if there exists a function $f:G\to k^\times$, with $f(1)=1$, such that

for all $g,h\in G. The equivalence classes of normalized 2-cocycles form the second cohomology group $H^2(G,k^\times)$, where $G$ acts trivially on $k^\times$.

A normalized 2-cocycle on a monoid $M$ is defined in the same way, with $g,h,\ell\in M$.

Given a $G$-graded algebra $A$ and a normalized 2-cocycle $\sigma$ on $G$, we define a new multiplication on homogeneous elements by

The cocycle identity ensures that this multiplication is associative. The resulting algebra, denoted by $A^\sigma$, is called a cocycle twist of $A$. It has the same underlying graded vector space as $A$.

Cocycle twists are closely related to Zhang twists [6]. Under suitable assumptions, cocycle twisting preserves Koszul AS-regularity [10].

Binomial Artin–Schelter regular algebras

Recall that a connected $\mathbb{N}$-graded algebra has the form

A connected graded algebra is called Artin–Schelter regular [2] if it has finite global dimension, finite Gelfand–Kirillov dimension, and satisfies the AS-Gorenstein condition. More precisely, if its global dimension is $d$, then

and

for some integer $\ell$, where the Ext groups are taken in the graded category.

An algebra is called Koszul [14] if the trivial module $k$ has a graded free resolution in which the generators at homological degree $i$ occur in internal degree $i$.

In this paper, we consider Koszul Artin–Schelter regular algebras generated in degree one and presented by quadratic binomial relations of the form

We are particularly interested in the coefficient-free relations obtained by setting the nonzero coefficients equal to 1. These relations define a monoid

whose group of fractions we study when it exists.

Monoids of I-type and Yang–Baxter solutions

Monoids of I-type were introduced by Gateva-Ivanova and Van den Bergh [8]. They are closely related to set-theoretic solutions of the Yang–Baxter equation [7].

Let $X$ be a finite set. A set-theoretic solution of the Yang–Baxter equation is a map

satisfying

We follow the terminology of [11 Section 1].

For each $i,j \in X$, write

where

and $\pi_1,\pi_2 : X \times X \longrightarrow X$ are the natural projections. The maps $L_i : X \longrightarrow X$ and $R_j : X \longrightarrow X$ are called the left and right actions, respectively.

The solution is called involutive if $r^2 = \operatorname{id}$, and nondegenerate if every $L_i$ and $R_j$ is a bijection of $X$.

The left actions generate a finite permutation group

The structure monoid associated with $(X,r)$ is

The relationship between these monoids and Yang–Baxter solutions is described in [11 Theorem 1.1].

Groups of fractions and Hirsch length

A group is called virtually abelian if it contains an abelian subgroup of finite index. Similarly, it is called virtually nilpotent if it contains a nilpotent subgroup of finite index.

A group $G$ is called polycyclic if it has a subnormal series

such that each quotient $G_i/G_{i-1}$ is cyclic.

The Hirsch length $h(G)$ is the number of infinite cyclic factors in such a series. It is independent of the chosen series [15 Theorem 5.4.13].

We recall the following standard property of Hirsch length.

For a monoid of I-type on $d$ generators, Theorem 1 gives an exact sequence

where $N \cong\mathbb{Z}^d$ and $F$ is finite. Applying Proposition 1, we obtain

These properties will be useful when studying the groups arising from coefficient-free binomial relations in dimensions 2, 3 and 4.

Binomial Artin–Schelter regular algebras as twisted monoid algebras

We explain how the monoid associated with a binomial algebra records the relations among the degrees of its generators, while a cocycle determines the coefficients appearing in these relations.

Twisted monoid algebras

Let $A$ be a connected graded domain with a faithful grading

by a group $G$, and suppose that $\dim_k A_g \leq1$ for every $g \in G$.

Let

be the support of the grading. Since $A$ is a domain, $M$ is a submonoid of $G$, and faithfulness implies that $M$ generates $G$.

Proof. For each $m \in M$, choose a nonzero element $u_m \in A_m$, with $u_1=1$. Since $A_m$ is one-dimensional, we have $A_m = k u_m$.

For $m,n \in M$, the product $u_m u_n$ is nonzero and belongs to $A_{mn}$. Thus there is a unique scalar $\sigma(m,n) \in k^{\times}$ such that

Associativity gives

while $u_1=1$ implies

Hence $\sigma$ is a normalized 2-cocycle, and the correspondence $e_m \mapsto u_m$ defines an $M$-graded algebra isomorphism

The cocycle $\sigma$ is defined on $M$. When it extends to a group 2-cocycle on $G$, it also determines a class in $H^2(G,k^{\times})$. $\square$

Binomial relations and their coefficients

Suppose that $A$ is generated by nonzero $G$-homogeneous elements $x_1,\ldots,x_d$ and has quadratic binomial relations

Write $g_j=\deg_G(x_j)$.

For a relation

homogeneity gives

Using the basis from Proposition 2, write

Then

and

Consequently,

Thus the relations among the degrees are recorded by $M$, while the coefficients arise from the cocycle $\sigma$ together with the choice of homogeneous generators.

Removing the coefficients

Suppose

and assume that the coefficient-free relations present the support monoid

Define

By the presentation of $M$,

On the other hand, Proposition 2 gives

Thus $A$ is obtained by twisting the multiplication of $k[M]$, and

where the twists are taken with respect to the $M$-grading.

We next consider when this construction preserves Koszul AS-regularity.

Proof. The $G$-grading defines a right coaction of the group algebra $kG$ on $A$. By the compatibility assumption, this coaction preserves the connected $\mathbb{N}$-grading.

The inverse group cocycle $\sigma^{-1}$ defines a Hopf $2$-cocycle on $kG$. By [10 Theorem 4.2.1], the corresponding cocycle twist preserves Koszul AS-regularity and global dimension.

Therefore $A^{\sigma^{-1}}$ is Koszul AS-regular of the same global dimension as $A$. Since

the result follows. □

Monoids and their groups of fractions

We study the monoids arising from coefficient-free quadratic binomial relations and their associated groups.

Our main examples come from four-dimensional Koszul AS-regular algebras with four generators and six quadratic binomial relations. We consider monoids of the form

where $m_i$ and $n_i$ are words of length two.

The group defined by the same presentation is

This is the universal group of $M$. When $M$ is cancellative and satisfies the Ore condition, it embeds in its group of fractions, which we denote by $G(M)$. In this case,

Binomial monoids of I-type

We first recall a criterion for recognizing binomial monoids of I-type.

The preceding theorem concerns binomial monoids with square-free quadratic relations. More general monoids of I-type may also have relations involving squares. For these monoids, we use the characterization in [11 Theorem 1.1].

Structure of the groups of fractions

The following result records some properties of monoids of I-type and their groups of fractions.

In particular, the group of fractions of a monoid of I-type is virtually nilpotent and polycyclic.

Proof. By Proposition 4, the group of fractions of a monoid of I-type is virtually abelian. The assertion follows by contraposition. □

Algebras of monoids of I-type

The structure of a monoid of I-type also imposes strong restrictions on its monoid algebra.

Furthermore, Gateva-Ivanova and Van den Bergh [8 Theorem 1.4] showed that if $M$ is a monoid of I-type and $\sigma$ is a monoid 2-cocycle, then the twisted monoid algebra $k_{\sigma}[M]$ is Koszul, Noetherian and has finite global dimension.

For our four-dimensional examples, we investigate whether the coefficient-free monoids satisfy the I-type conditions. We also study the structure of their groups of fractions, including solvability, virtual nilpotence and Hirsch length.

In the examples where a group of fractions is identified, we subsequently study

to understand the possible group cocycles associated with the binomial algebras.

Low-dimensional examples

We begin with examples in dimensions 2 and 3, illustrating how the coefficient-free relations determine the monoids and their groups of fractions.

In dimension 3, we compare our calculations with the classification of nonabelian group gradings of three-dimensional Artin–Schelter regular algebras obtained by Buzaglo and Rogalski [5 Theorem 3.4 and Table 1].

Dimension 2

Example 1. Consider the polynomial ring

Its monoid is

The associated Yang–Baxter solution is the flip map, with trivial left actions. Thus

Example 2. Consider the coefficient-free algebra

with associated monoid

This monoid is of I-type [11 Example 2.5].

The associated left actions are

so

The group of fractions has presentation

Setting

we have $y=ab$. The relation $x^{2}=y^{2}$ becomes

which is equivalent to

Consequently,

the Klein bottle group. The subgroup

has index 2. Hence

Thus the two groups have the same Hirsch length, although one is abelian and the other is nonabelian.

Dimension 3

We consider a noncommutative example whose group of fractions is a semidirect product.

These examples illustrate that the groups of fractions of monoids of I-type need not be abelian, even though they contain finite-index free abelian subgroups.

We next investigate the corresponding groups for four-dimensional coefficient-free binomial algebras, with particular attention to their structure, Hirsch length and second cohomology groups.

Four-dimensional examples

We now consider the coefficient-free monoids associated with the 23 four-dimensional binomial Artin–Schelter regular algebras listed in Appendix § 9. We study their associated set-theoretic solutions of the Yang–Baxter equation and the permutation groups generated by the left actions.

Let $X = \{0,1,2,3\}$, and write

for the associated solution. The left actions generate a finite permutation group

For monoids of I-type, the structure theorem recalled in Proposition 4 gives an exact sequence

In particular, $G(M)$ is a torsion-free Bieberbach group with Hirsch length 4.

Examples of four-generator monoids of I-type have also been studied in [12 Examples 8.2.14 and 8.4.6]. We consider three algebras from our collection, illustrating different permutation groups.

The commutative algebra

An algebra with cyclic permutation group

For example, the relations give

Therefore

The remaining left actions are obtained similarly. In particular,

Since $L_0$ is a 4-cycle, the permutation group is

Since $M$ is a monoid of $I$-type, its group of fractions fits into the exact sequence

In particular, $G(M)$ is a solvable Bieberbach group with Hirsch length 4.

An algebra with dihedral permutation group

Example 6 (Algebra 21). Consider the coefficient-free relations

Let $M$ be the monoid defined by these relations. The associated solution has left actions

For example, the relations give

Therefore

The remaining left actions are obtained similarly. In particular,

Setting $a=L_1$ and $b=L_0$, we have

Thus the permutation group is

where $D_8$ denotes the dihedral group of order 8.

Since $M$ is a monoid of $I$-type, its group of fractions fits into the exact sequence

In particular, $G(M)$ is a solvable Bieberbach group with Hirsch length 4.

Comparison of the permutation groups

Table 1 summarizes the permutation groups associated with the 23 algebras listed in Appendix A.

Permutation group $F$	Algebras	Count
$\{1\}$	1	1
$C_{2}$	2, 4, 6, 7, 8, 10, 16	7
$C_{3}$	3, 18, 19, 20	4
$C_{4}$	17, 23	2
$C_{2} \times C_{2}$	5, 9, 11, 13, 14, 15, 22	7
$D_{8}$	12, 21	2

The permutation groups in Table 1 are all finite and solvable. For the corresponding monoids of $I$-type, their groups of fractions are torsion-free, virtually abelian Bieberbach groups of Hirsch length 4.

Computing the cocycles

We now study the second cohomology groups of the groups of fractions considered in the previous sections.

Throughout this section, we regard $k^{\times}$ as a trivial $G(M)$-module. We compute $H^{2}(G(M), k^{\times})$ using group homology and the universal coefficient theorem.

Recall that for a group $G$, the universal coefficient theorem [16 Theorem 3.6.5] gives a short exact sequence

In particular, when $H_{1}(G,\mathbb{Z})$ is free abelian, the $\operatorname{Ext}$ term vanishes.

Dimension 2

We begin with the two groups arising in dimension 2.

Example 7. For the polynomial ring in two variables, the group of fractions is

Since

and

the universal coefficient theorem gives

Dimension 3

We next consider the polynomial ring in three variables.

Dimension 4

We now turn to the four-dimensional examples from the previous section. For each of the 23 algebras, we consider the corresponding group of fractions $G(M)$ and compute its first and second integral homology groups, together with $H^{2}(G(M),k^{\times})$.

We begin with a few examples to illustrate the methods used in these computations. The results for all 23 algebras are summarized in Table 2.

Algebra	$F$	$H_1(G(M),\mathbb{Z})$	$H_2(G(M),\mathbb{Z})$	$H^2(G(M),k^\times)$
1	$1$	$\mathbb{Z}^{4}$	$\mathbb{Z}^{6}$	$(k^\times)^{6}$
2	$C_2$	$\mathbb{Z}^{3}$	$\mathbb{Z}^{3}\oplus C_2$	$(k^\times)^{3}\times C_2$
3	$C_3$	$\mathbb{Z}^{2}$	$\mathbb{Z}^{2}$	$(k^\times)^{2}$
4	$C_2$	$\mathbb{Z}^{3}$	$\mathbb{Z}^{3}\oplus C_2$	$(k^\times)^{3}\times C_2$
5	$C_2^{2}$	$\mathbb{Z}^{2}\oplus C_2$	$\mathbb{Z}\oplus C_2\oplus C_4$	$k^\times\times C_2\times C_4$
6	$C_2$	$\mathbb{Z}^{3}\oplus C_2$	$\mathbb{Z}^{3}\oplus C_2$	$(k^\times)^{3}\times C_2$
7	$C_2$	$\mathbb{Z}^{3}$	$\mathbb{Z}^{3}\oplus C_2$	$(k^\times)^{3}\times C_2$
8	$C_2$	$\mathbb{Z}^{3}$	$\mathbb{Z}^{3}\oplus C_2$	$(k^\times)^{3}\times C_2$
9	$C_2^{2}$	$\mathbb{Z}^{2}\oplus C_2$	$\mathbb{Z}\oplus C_2$	$k^\times\times C_2$
10	$C_2$	$\mathbb{Z}^{2}\oplus C_2$	$\mathbb{Z}^{2}\oplus C_2$	$(k^\times)^{2}\times C_2$
11	$C_2^{2}$	$\mathbb{Z}^{2}\oplus C_2$	$\mathbb{Z}\oplus C_2\oplus C_4$	$k^\times\times C_2\times C_4$
12	$D_8$	$\mathbb{Z}\oplus C_2$	$C_2$	$C_2$
13	$C_2^{2}\oplus C_2$	$\mathbb{Z}^{2}\oplus C_2^{2}$	$\mathbb{Z}\oplus C_3^{2}$	$k^\times\times C_3^{2}$
14	$C_2^{2}$	$\mathbb{Z}^{2}\oplus C_2$	$\mathbb{Z}\oplus C_2^{2}$	$k^\times\times C_2^{2}$
15	$C_2^{2}$	$\mathbb{Z}^{2}\oplus C_2$	$\mathbb{Z}\oplus C_2\oplus C_4$	$k^\times\times C_2\times C_4$
16	$C_2$	$\mathbb{Z}^{2}\oplus C_2$	$\mathbb{Z}^{2}\oplus C_2$	$(k^\times)^{2}\times C_2$
17	$C_4$	$\mathbb{Z}\oplus C_2^{2}$	$\mathbb{Z}\oplus C_2$	$k^\times\times C_2$
18	$C_3$	$\mathbb{Z}^{2}\oplus C_3$	$\mathbb{Z}^{2}\oplus C_3$	$(k^\times)^{2}\times C_3$
19	$C_3$	$\mathbb{Z}^{2}$	$\mathbb{Z}^{2}$	$(k^\times)^{2}$
20	$C_3$	$\mathbb{Z}^{2}$	$\mathbb{Z}^{2}$	$(k^\times)^{2}$
21	$D_8$	$\mathbb{Z}\oplus C_4$	$C_4$	$C_4$
22	$C_2^{2}$	$\mathbb{Z}\oplus C_4$	$C_4$	$C_4$
23	$C_4$	$\mathbb{Z}\oplus C_4$	$\mathbb{Z}$	$k^\times$

Since $H_1(G(M),\mathbb{Z})$ is free abelian, the Ext term in the universal coefficient theorem vanishes. Therefore,

By the $K$ünneth formula [9 Theorem 3B.6],

Since $H_1(G(M),\mathbb{Z})$ is free abelian, the $\operatorname{Ext}$ term in the universal coefficient theorem vanishes. Therefore,

where

In particular, if $\operatorname{char}(k)\ne2$, then

With respect to the basis $e_0 \wedge e_1,\quad e_0 \wedge e_2,\quad e_1 \wedge e_2,$ the induced action is

Both $\psi-I$ and $\wedge^{2}\psi-I$ have Smith normal form $\operatorname{diag}(1,1,0)$.

Therefore,

The sequence splits, giving

Since $H_1(G(M),\mathbb{Z})$ is free abelian, the $\operatorname{Ext}$ term in the universal coefficient theorem vanishes. Hence

We carry out similar computations for all 23 four-dimensional binomial algebras listed in Appendix A. The resulting homology and cohomology groups are summarized in Table 2.

Explicit cocycle twists of some algebras

In Section 7, we computed $H^2(G(M), k^\times)$ for the 23 binomial algebras. We now use these computations to construct families of algebras with nonzero coefficients in their defining relations.

As explained in Section 3, a group 2-cocycle determines the coefficients of the binomial relations in a twisted monoid algebra. We illustrate this construction for Algebras 1, 2 and 3.

The three nonzero parameters correspond to the factors $(k^\times)^3$, while $\varepsilon$ comes from the order-two factor of the cohomology group. In particular, the coefficients of the first two relations must agree, and the coefficient of the fourth relation must satisfy $\varepsilon^2=1$.

Setting $p=q=r=\varepsilon=1$ recovers Algebra 2.

The constructions above give families of four-dimensional Koszul Artin–Schelter regular algebras through cocycle twists. Similar computations can be carried out for the remaining binomial algebras to obtain further families. In future work, we would like to compute the Kodaira–Spencer maps and discrete invariants of these families, following the methods in [4, 3]. We would also like to investigate whether these families belong to the components of the moduli space studied in our previous work, or whether they give rise to new families. Another possible direction is to extend these computations to dimension five, using the classification of Yang–Baxter solutions in [1 Table 2.1].

Defining relations of the binomial algebras

The following tables list the defining relations of the four-dimensional binomial Artin–Schelter regular algebras considered in this paper.

Algebra 1	Algebra 2	Algebra 3
$x_{0}x_{1}-x_{1}x_{0}=0$<br>$x_{0}x_{2}-x_{2}x_{0}=0$<br>$x_{0}x_{3}-x_{3}x_{0}=0$<br>$x_{1}x_{2}-x_{2}x_{1}=0$<br>$x_{1}x_{3}-x_{3}x_{1}=0$<br>$x_{2}x_{3}-x_{3}x_{2}=0$	$x_{0}x_{1}-x_{1}x_{0}=0$<br>$x_{0}x_{2}-x_{2}x_{0}=0$<br>$x_{0}x_{3}-x_{3}x_{0}=0$<br>$x_{1}x_{2}-x_{2}x_{1}=0$<br>$x_{1}x_{3}-x_{3}x_{2}=0$<br>$x_{2}x_{3}-x_{3}x_{1}=0$	$x_{0}x_{1}-x_{1}x_{0}=0$<br>$x_{0}x_{2}-x_{2}x_{0}=0$<br>$x_{0}x_{3}-x_{3}x_{2}=0$<br>$x_{1}x_{2}-x_{2}x_{1}=0$<br>$x_{1}x_{3}-x_{3}x_{0}=0$<br>$x_{2}x_{3}-x_{3}x_{1}=0$
Algebra 4	Algebra 5	Algebra 6
$x_{0}x_{1}-x_{1}x_{0}=0$<br>$x_{0}x_{2}-x_{2}x_{1}=0$<br>$x_{0}x_{3}-x_{3}x_{1}=0$<br>$x_{1}x_{2}-x_{2}x_{0}=0$<br>$x_{1}x_{3}-x_{3}x_{0}=0$<br>$x_{2}x_{3}-x_{3}x_{2}=0$	$x_{0}x_{1}-x_{1}x_{0}=0$<br>$x_{0}x_{2}-x_{3}x_{1}=0$<br>$x_{0}x_{3}-x_{2}x_{1}=0$<br>$x_{1}x_{2}-x_{3}x_{0}=0$<br>$x_{1}x_{3}-x_{2}x_{0}=0$<br>$x_{2}x_{3}-x_{3}x_{2}=0$	$x_{0}^{2}-x_{1}^{2}=0$<br>$x_{0}x_{2}-x_{2}x_{0}=0$<br>$x_{0}x_{3}-x_{3}x_{0}=0$<br>$x_{1}x_{2}-x_{2}x_{1}=0$<br>$x_{1}x_{3}-x_{3}x_{1}=0$<br>$x_{2}x_{3}-x_{3}x_{2}=0$
Algebra 7	Algebra 8	Algebra 9
$x_{0}^{2}-x_{1}^{2}=0$<br>$x_{0}x_{2}-x_{2}x_{0}=0$<br>$x_{0}x_{3}-x_{3}x_{1}=0$<br>$x_{1}x_{2}-x_{2}x_{1}=0$<br>$x_{1}x_{3}-x_{3}x_{0}=0$<br>$x_{2}x_{3}-x_{3}x_{2}=0$	$x_{0}^{2}-x_{1}^{2}=0$<br>$x_{0}x_{2}-x_{2}x_{1}=0$<br>$x_{0}x_{3}-x_{3}x_{1}=0$<br>$x_{1}x_{2}-x_{2}x_{0}=0$<br>$x_{1}x_{3}-x_{3}x_{0}=0$<br>$x_{2}x_{3}-x_{3}x_{2}=0$	$x_{0}^{2}-x_{1}^{2}=0$<br>$x_{0}x_{2}-x_{2}x_{0}=0$<br>$x_{0}x_{3}-x_{3}x_{0}=0$<br>$x_{1}x_{2}-x_{3}x_{1}=0$<br>$x_{1}x_{3}-x_{2}x_{1}=0$<br>$x_{2}x_{3}-x_{3}x_{2}=0$
Algebra 10	Algebra 11	Algebra 12
$x_{0}^{2}-x_{1}^{2}=0$<br>$x_{0}x_{2}-x_{3}x_{0}=0$<br>$x_{0}x_{3}-x_{2}x_{0}=0$<br>$x_{1}x_{2}-x_{3}x_{1}=0$<br>$x_{1}x_{3}-x_{2}x_{1}=0$<br>$x_{2}x_{3}-x_{3}x_{2}=0$	$x_{0}^{2}-x_{1}^{2}=0$<br>$x_{0}x_{2}-x_{3}x_{1}=0$<br>$x_{0}x_{3}-x_{2}x_{1}=0$<br>$x_{1}x_{2}-x_{3}x_{0}=0$<br>$x_{1}x_{3}-x_{2}x_{0}=0$<br>$x_{2}x_{3}-x_{3}x_{2}=0$	$x_{0}^{2}-x_{2}x_{3}=0$<br>$x_{0}x_{2}-x_{3}x_{0}=0$<br>$x_{0}x_{3}-x_{1}x_{2}=0$<br>$x_{1}^{2}-x_{3}x_{2}=0$<br>$x_{1}x_{3}-x_{2}x_{1}=0$<br>$x_{2}x_{0}-x_{3}x_{1}=0$
Algebra 13	Algebra 14	Algebra 15
$x_{0}^{2}-x_{1}^{2}=0$<br>$x_{0}x_{2}-x_{2}x_{0}=0$<br>$x_{0}x_{3}-x_{3}x_{0}=0$<br>$x_{1}x_{2}-x_{2}x_{1}=0$<br>$x_{1}x_{3}-x_{3}x_{1}=0$<br>$x_{2}^{2}-x_{3}^{2}=0$	$x_{0}^{2}-x_{1}^{2}=0$<br>$x_{0}x_{2}-x_{2}x_{0}=0$<br>$x_{0}x_{3}-x_{3}x_{1}=0$<br>$x_{1}x_{2}-x_{2}x_{1}=0$<br>$x_{1}x_{3}-x_{3}x_{0}=0$<br>$x_{2}^{2}-x_{3}^{2}=0$	$x_{0}^{2}-x_{1}^{2}=0$<br>$x_{0}x_{2}-x_{2}x_{1}=0$<br>$x_{0}x_{3}-x_{3}x_{1}=0$<br>$x_{1}x_{2}-x_{2}x_{0}=0$<br>$x_{1}x_{3}-x_{3}x_{0}=0$<br>$x_{2}^{2}-x_{3}^{2}=0$

Algebra 16	Algebra 17	Algebra 18
$x_0^2-x_1^2=0$	$x_0^2-x_3^2=0$	$x_0^2-x_1x_2=0$
$x_0x_2-x_3x_1=0$	$x_0x_2-x_1x_3=0$	$x_0x_1-x_2^2=0$
$x_0x_3-x_2x_1=0$	$x_0x_3-x_2x_1=0$	$x_0x_3-x_3x_0=0$
$x_1x_2-x_3x_0=0$	$x_1^2-x_2^2=0$	$x_1^2-x_2x_0=0$
$x_1x_3-x_2x_0=0$	$x_1x_2-x_3x_0=0$	$x_1x_3-x_3x_1=0$
$x_2^2-x_3^2=0$	$x_2x_0-x_3x_1=0$	$x_2x_3-x_3x_2=0$
Algebra 19	Algebra 20	Algebra 21
$x_0^2-x_1x_2=0$	$x_0^2-x_1x_2=0$	$x_0^2-x_3x_1=0$
$x_0x_1-x_2^2=0$	$x_0x_1-x_2^2=0$	$x_0x_1-x_1x_2=0$
$x_0x_3-x_3x_2=0$	$x_0x_3-x_3x_1=0$	$x_0x_2-x_2x_0=0$
$x_1^2-x_2x_0=0$	$x_1^2-x_2x_0=0$	$x_1^2-x_3^2=0$
$x_1x_3-x_3x_0=0$	$x_1x_3-x_3x_2=0$	$x_1x_3-x_2^2=0$
$x_2x_3-x_3x_1=0$	$x_2x_3-x_3x_0=0$	$x_2x_3-x_3x_0=0$
Algebra 22	Algebra 23	Algebra 23
$x_0^2-x_3x_1=0$	$x_0^2-x_3x_1=0$	$x_0^2-x_3x_1=0$
$x_0x_1-x_2x_3=0$	$x_0x_1-x_2x_3=0$	$x_0x_1-x_2x_3=0$
$x_0x_2-x_1^2=0$	$x_0x_2-x_3^2=0$	$x_0x_2-x_3^2=0$
$x_1x_2-x_3x_0=0$	$x_1^2-x_2x_0=0$	$x_1^2-x_2x_0=0$
$x_1x_3-x_2^2=0$	$x_1x_2-x_3x_0=0$	$x_1x_2-x_3x_0=0$
$x_2x_0-x_3^2=0$	$x_2^2-x_3x_1=0$	$x_2^2-x_3x_1=0$

Department of Mathematics, Carleton University, Ottawa, Ontario, Canada

Email address: [email protected]